State Observer For Nonlinear Systems Research Articles

A class of adaptive predefined‐time extended state observers for high‐order strict‐feedback systems

AbstractFast state estimation for nonlinear systems has been a key issue in control theory and the recently developed predefined‐time stability acts as an effective tool for the problem. However, the studies of predefined‐time state observer for nonlinear systems are rare. In this article, we consider the predefined‐time extended state observer (PTESO) design for a class of high‐order strict‐feedback systems subject to Hölder continuous nonlinearities and lumped disturbance. By developing an adaptive mechanism and using time‐varying functions, a class of adaptive PTESOs is designed to estimate the unavailable states and the lumped disturbance, in which the convergence time can be tightly and explicitly preset by only one parameter, irrelevant to the initial conditions. The adaptive mechanism is used to improve the transient performance under wide‐range lumped disturbance. Moreover, thanks to the design of the time‐varying gains, the PTESO could realize fast observation with trivial peaking observation errors. Finally, simulation results are provided to illustrate the effectiveness of the proposed observer.

Data‐driven state observation for nonlinear systems based on online learning

AbstractFor controlling nonlinear processes represented by state‐space models, a state observer is needed to estimate the states from the trajectories of measured variables. While model‐based observer synthesis is traditionally challenging due to the difficulty of solving pertinent partial differential equations, this article proposes an efficient model‐free, data‐driven approach for state observation, which is suitable for data‐driven nonlinear control without accurate nonlinear models. Specifically, by using a Chen–Fliess series representation of the observer dynamics, state observation is endowed with an online least squares regression formulation that can be solved by gradient flow with performance guarantees. When the target state trajectories for regression are unavailable, by exploiting the Kazantzis–Kravaris/Luenberger observer structure, state observation is reduced to a dimensionality reduction problem amenable to an online implementation of kernel principal component analysis. The proposed approach is demonstrated by a limit cycle dynamics and a chaotic system.

On convergence of extended state observers for nonlinear systems with non-differentiable uncertainties

The analysis of convergence of extended state observers (ESOs) requires the total disturbance to be differentiable. However, this requirement is not satisfied in many control engineering practices. In this paper, we attempt to analyze the convergence of ESOs for nonlinear systems with non-differentiable uncertainties. A decomposition method is first presented to divide the non-differentiable total disturbance into a differentiable signal and a bounded but non-differentiable signal. Based on this decomposition, we give out the convergence of both nonlinear and linear ESOs (NLESO/LESO), low- and high-power ESOs (LPESO/HPESO), and fixed-time ESO (FxESO). We also derive the explicit formulas for the estimation errors of these ESOs. Simulations and experiments demonstrate the correctness of the analysis results.

Model based reduced-order observers for a class of mechatronic systems with mitigation of disturbances effects using the Attractive Ellipsoid Method

State-estimation of mechatronic systems is essential because, in addition to position and velocity, there are electric and magnetic signals that must be measured, and can lead to sophisticated and expensive instrumentation systems. This paper addresses the problem of state-estimation for underactuated mechanical systems whose active joints are driven by permanent magnet dc-motors. Luenberger-type structures are designed, where the measured state variables and the dynamic model are used to construct the unavailable ones, in presence of uncertainties and disturbances as inputs. To avoid the peaking phenomenon is one of the main challenges on state observation for nonlinear systems. Underactuated mechatronic systems are not immune to this problem, which is exacerbated when external disturbances are taken into account. The major purpose here is to use a robust reduced observer to estimate the unmeasured state variables. The Attractive Ellipsoid Method provides reliable estimation of unmeasured state signals because it reduces the influence of external disturbances on the estimate error signals. This ensures that the estimated signals converge to an invariant set of minimal size around the real ones. The first design is based on the quasi-linear representation that meets the observability condition. The second considers the entire nonlinear model, which necessitates the availability of the entire position vector. The major findings are demonstrated experimentally on a non-traditional mechatronic system: the linear double pendulum system whose last joint is constrained by two linear springs. This shows a significant improvement of the estimation response, concerning robust techniques based on the linear approximation, towards its application in designing an adaptive observer to provide state and parameters estimation simultaneously, which is essential in robust and adaptive control systems.

Applying sliding mode technique to design extended state observer for nonlinear systems

AbstractThis paper applies sliding mode and stable pole‐zero cancelation techniques to design a finite time extended state observer for a class of single‐input single‐output nonlinear systems in the presence of unknown uncertainties and disturbances. The proposed estimation approaches, including first‐order and second‐order sliding mode extended state observers, can provide more design flexibility and achieve good performance compared with the conventional extended state observers. Moreover, it has a good form for theoretical stability analysis in which the estimation errors can be converged to a neighborhood of zero while remaining uniformly bounded in finite time. The characteristics of the developed observer is presented in the frequency domain, followed by the corresponding analysis in the time domain. Finally, simulation results demonstrate the validity of the proposed method.

Orbital stabilization of nonlinear systems via the immersion and invariance technique

SummaryImmersion and invariance is a technique for the design of stabilizing and adaptive controllers and state observers for nonlinear systems. In all these applications the problem considered is the stabilization of equilibrium points. Motivated by some modern applications, we show that the technique can also be used to solve the problem of orbital stabilization, where the final objective is to generate periodic solutions that are attractive. The feasibility of our result is illustrated by means of some classical mechanical engineering and power electronics examples.

On State Observers for Nonlinear Systems: A New Design and a Unifying Framework

In this paper we propose a new observer design technique for nonlinear systems. It combines the well-known Kazantzis-Kravaris-Luenberger observer and the recently introduced parameter estimation-based observer, which become special cases of it---extending the realm of applicability of both methods. A second contribution of the paper is the proof that these designs can be recast as particular cases of immersion and invariance observers---providing in this way a unified framework for their analysis and design. Simulation results of a physical system that illustrates the superior performance of the proposed observer compared to other existing observers are presented.

Exponential State Observers for Nonlinear Systems with Incremental Quadratic Constraints and Output Nonlinearities

This paper considers the exponential observer design for a class of nonlinear systems with output nonlinearities. The nonlinear terms in the systems are assumed to satisfy incremental quadratic constraints which include many commonly encountered nonlinearities in existing literature as some special cases. We construct a circle-criterion-based observer by injecting both the linear and the nonlinear output error terms into the observer system. Sufficient conditions ensuring the exponential stability of the proposed observer are established and formulated in terms of linear matrix inequalities. Finally, the advantages and effectiveness of the proposed design approach are illustrated through two examples.

Closed-loop control of tumor growth by means of anti-angiogenic administration.

A tumor growth model accounting for angiogenic stimulation and inhibition is here considered, and a closed-loop control law is presented with the aim of tumor volume reduction by means of anti-angiogenic administration. To this end the output-feedback linearization theory is exploited, with the feedback designed on the basis of a state observer for nonlinear systems. Measurements are supposed to be acquired at discrete sampling times, and a novel theoretical development in the area of time-delay systems is applied in order to derive a continuous-time observer in spite of the presence of sampled measurements. The overall control scheme allows to set independently the control and the observer parameters thanks to the structural properties of the tumor growth model. Simulations are carried out in order to mimic a real experimental framework on mice. These results seem extremely promising: they provide very good performances according to the measurements sampling interval suggested by the experimental literature, and show a noticeable level of robustness against the observer initial estimate, as well as against the uncertainties affecting the model parameters.

A chain observer for nonlinear long constant delay systems: A matrix inequality approach

This paper proposes a new state observer for nonlinear systems with non negligible and different time delays in the state, output and input and Lipschitz nonlinearity. The proposed observer is composed of a chain of observers with similar algorithm. Each of them estimates the states in different time instants, while the last observer estimates the state in the present time. Conditions for stability of the observer are translated into linear matrix inequalities (LMIs) which facilitates the design of observer gains. Simulation results demonstrate effectiveness of this observer.

Enhanced predictive functional control based on equivalent input disturbance and generalized extended state observer for nonlinear systems with fast dynamics

An approach for improving tracking performance of the predictive functional control (PFC) based on equivalent input disturbance (EID) and generalized extended state observer (GESO) is presented in this paper. Control structure of the proposed method includes two parts, a PFC controller and a state observer, and both of them can be designed separately. The PFC controller is employed to optimize the tracking control of a nonlinear system and its tracking performance heavily depends on the accuracy of the predictive model. Whereas, various internal perturbations and external disturbances always make the predictive model significantly deviate from the nominal model. To address this problem, a state observer is introduced into the proposed approach. All of the above-mentioned uncertainties are regarded as a lumped disturbance, and the lumped disturbance is estimated by EID–GESO as well as the system states in an integrated manner. The estimated disturbance can be eliminated in a negative feedback loop and then, a relatively accurate predictive model for the PFC controller is offered. Closed-loop stability of the composite control is also provided. Compared with previously related work, the notable feature of the proposed design is that the observer-based PFC is extended to nonintegral chain systems subject to mismatched uncertainties and better performance is obtained. Finally, the proposed approach is applied to a missile longitudinal autopilot design, and a comparison with some prominent methods in the presence of significant uncertainties demonstrates the robustness and effectiveness of the proposed design.

Limitations for Nonlinear Observation Over Erasure Channel

In this technical note, we study the problem of state observation of nonlinear systems over an erasure channel. The notion of mean square exponential stability is used to analyze the stability property of observer error dynamics. The main results of this technical note prove, fundamental limitation arises for mean square exponential stabilization of the observer error dynamics, expressed in terms of probability of erasure, and positive Lyapunov exponents of the system. Positive Lyapunov exponents are a measure of average expansion of nearby trajectories on an attractor set for nonlinear systems. Hence, the dependence of limitation results on the Lyapunov exponents highlights the important role played by nonequilibrium dynamics on the attractor set in observation over an erasure channel. The limitation on observation is also related to measure-theoretic entropy of the system, which is another measure of dynamical complexity. The limitation result for the observation of linear systems is obtained as a special case, where Lyapunov exponents are shown to emerge as the natural generalization of eigenvalues from linear systems to nonlinear systems.

A Neuro‐Augmented Observer for Robust Fault Detection in Nonlinear Systems

A new fault detection method using neural‐networks‐augmented state observer for nonlinear systems is presented in this paper. The novelty of the approach is that instead of approximating the entire nonlinear system with neural network, we only approximate the unmodeled part that is left over after linearization, in which a radial basis function (RBF) neural network is adopted. Compared with conventional linear observer, the proposed observer structure provides more accurate estimation of the system state. The state estimation error is proved to asymptotically approach zero by the Lyapunov method. An aircraft system demonstrates the efficiency of the proposed fault detection scheme, simulation results of which show that the proposed RBF neural network‐based observer scheme is effective and has a potential application in fault detection and identification (FDI) for nonlinear systems.

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.

A new approach to nonlinear filtering via a mixed state observer and polynomial scheme

AbstractThis paper investigates the state estimation problem for a stochastic nonlinear differential systems, driven by standard Wiener processes. A novel algorithm is proposed, based on a mixed observer-filter approach, which can be resumed in three steps: the first step makes use of a state observer for nonlinear systems, applied to the system equations disregarding the noises. The observed state constitutes the trajectory around which a Carleman approximation of the stochastic differential system is achieved. This second step allows to define an approximate representation of the system by means of a bilinear system (i.e. linear drift and multiplicative noise), for which a filtering algorithm is available from the literature, third step.

Extended State Observer for Nonlinear Systems with Uncertainty

AbstractThe extended state observer first proposed by Jingqing Han in [J. Q. Han, The extended state observer for a class of uncertain systems, Control and Decision, 10(1)(1995), 85-88] is the main interim loop of active disturbance rejection control that is taking off as a technology after numerous successful applications in engineering. Unfortunately, there is no a 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 disturbed system of a asymptotical stable system, with which the effect of total disturbance error is eliminated by high-gain.

Sliding mode state observation for non-linear systems

In this paper a class of non-linear systems with uncertainty and a Lipschitz part is considered. The non-linear part satisfies the Lipschitz condition, whilst the uncertain part is a bounded function. A sliding mode observer is designed to yield feedforward compensation input to stabilize the error estimation system with and without a Lipschitz non-linearity. Sufficient conditions for stability of the Thau observer are also proposed. These conditions ensure the stability of the non-linear observer by selecting a suitable observer gain matrix. In general, when the system contains unmatched uncertainty, ultimate boundedness is obtained. The ultimate boundedness radius depends upon the defined ‘distance of the unmatched uncertainty’. Some sufficient conditions may ensure the stability of a non-linear observer for systems with uncertainties.

Cascaded Design of the State Observer for Nonlinear Systems in the Presence of External Perturbations

Applicability of the block approach to the problem of observation of nonlinear systems in the presence of uncontrollable external perturbations was demonstrated. A block-observable form of the general nonlinear systems was obtained with regard for perturbations. It underlies two types of cascaded procedures for designing the state observers in the class of systems with separable motions: (i) with discontinuous controls and (ii) with high-gain feedbacks. The first procedure relies on the method of equivalent control to restore in a finite time the nonmeasurable components of the state vector and also perturbations. The second procedure estimates the nonmeasurable components of the state vector in the prelimit situation with a given accuracy and retention of the decomposition of the design procedure.

A new approach to state observation of nonlinear systems with delayed output

The article presents a new approach for the construction of a state observer for nonlinear systems when the output measurements are available for computations after a nonnegligible time delay. The proposed observer consists of a chain of observation algorithms reconstructing the system state at different delayed time instants (chain observer). Conditions are given for ensuring global exponential convergence to zero of the observation error for any given delay in the measurements. The implementation of the observer is simple and computer simulations demonstrate its effectiveness.

State Observers for Nonlinear Systems with Large Output Delays

A state observation algorithm for nonlinear systems with a non negligible output delay is presented in this paper. The problem of output delay occurs not only when measurements are transmitted through a low-rate communication channel: in many application it is the measurement process itself that takes some time to be worked out. A state observation algorithm that reconstructs the present state on the basis of past measurements is here proposed, and sufficient conditions are given that ensure global exponential convergence of the observed state. The proposed algorithm, when applied to linear systems, ensures exponential convergence for any size of the delay. The implementation of the observer is easy and computer simulations show its effectiveness. For the sake of simplicity, only the case of systems with one input and one output is treated in the paper.