Observer Design For Nonlinear Systems Research Articles

Neural Network Adaptive Observer design for Nonlinear Systems with Partially and Completely Unknown Dynamics Subject to Variable Sampled and Delay Output Measurement

This paper proposes a novel Neural Network Adaptive Observer (NNAO) for Nonlinear Systems with Partially and Completely Unknown Dynamics (NSPCUD), subject to variable sampled and delayed output. The method involves designing a neural network observer for partially unknown nonlinear systems with sampled and delayed outputs, using a radial basis function (RBF) neural network to approximate the system’s unknown part. A new weight update algorithm is proposed, along with a closed-loop output predictor for coping with variable samples, and a closed-loop integral compensation to handle variable delay. This approach is then extended to cover completely unknown systems as well. Numerical simulations and comparisons between the proposed method and previous methods on autonomous ground vehicle models were conducted to verify the effectiveness of the proposed NNAO.

Generalized Multiple Delay-dependent H∞, Functional Observer Design for Nonlinear System

Generalized Multiple Delay-dependent H∞, Functional Observer Design for Nonlinear System

A High-Gain Observer Design for Nonlinear System with Delayed Measurements: Application to a Quadrotor UAV

The paper deals with the design of a high-gain observer for nonlinear system with delayed output measurements and bounded disturbances. The new structure of the proposed observer exhibits good performance in terms of allowing a higher MAVTD (Maximum Allowable Value of the Time-Delay) compared to the existing works, where the MAVTD is limited due to the high value of the high-gain parameter. Based on the LMI technique and using Lyapunov-Krasovskii function, an explicit relation between MAVTD and the LMI tuning parameters is deduced from the stability analysis of the proposed observer. The efficiency of the proposed structure is illustrated through an application to a Quadrotor UAV and a comparison with standard and cascade high-gain observers is performed.

LMI-Based H∞ Observer Design for Nonlinear Lipschitz System

The problem of circle criterion-based H∞ observer design for nonlinear systems is addressed in this article. An LMI-based observer is developed for the nonlinear Lipschitz systems with linear outputs under the presence of noise/disturbances. The proposed LMI condition is obtained by using reformulated Lipschitz property, standard Young's relation and a newly defined positive definite matrix multiplier. The inclusion of the matrix multiplier adds additional numbers of decision variables in the LMI, which provides extra degrees of freedom from a feasibility point of view. Further, the performance of the proposed observer is validated using numerical examples.

Adaptive observer for a nonlinear system with partially unknown state matrix and delayed measurements

Problem of an adaptive state observer design for nonlinear system with unknown time-varying parameters and under condition of delayed measurements is considered. State observation problem was raised by many researchers (see for example Sanx et al. (2019)). In this paper the results proposed in Bobtsov et al. (2021b), Bobtsov et al. (2021a), Bobtsov et al. (2022a), Bobtsov et al. (2022b) are developed. The problem is solved under assumption that the state matrix can be represented as sum of known and unknown parts. The output vector is measured with a known constant delay. An adaptive observer which reconstructs unmeasured state and unknown time-varying parameter is proposed.

Zonotopic observer designs for uncertain Takagi–Sugeno fuzzy systems

This paper addresses the zonotopic observer design for nonlinear systems affected by uncertainties, i.e., state disturbances and measurement noises using Takagi–Sugeno (TS) fuzzy technique. The system uncertainties are considered as unknown but bounded, which are handled via a set-membership framework. For state estimation purposes, we develop an algorithm to recursively compute the zonotope containing the mismatching nonlinear term caused by unmeasured nonlinearities. Then, two methods are proposed to design the zonotopic observer gains. The first method is based on the minimization of the F−radius of zonotopes, for which the membership-function-dependent observer gain must be completely computed online. For the second method, an ℋ∞ approach is used together with a nonquadratic Lyapunov function to determine the observer gain. Then, the zonotopic observer design is reformulated a convex optimization problem under linear matrix inequalities (LMIs), which can be effectively solved with numerical solvers. An autonomous vehicle application is provided to demonstrate and analyze the effectiveness of both proposed methods.

On high-gain observer design for nonlinear systems with delayed output measurements

In this paper, we propose a high-gain observer design for nonlinear systems with time-varying delayed output measurements. Based on a recent high-gain like observer design method, called HG/LMI observer, a larger bound of the time-delay is allowed compared to that obtained by using the standard high-gain methodology. Such a HG/LMI observer adopts a lower value of the tuning parameter, which results in the reduction of the value of the observer gain, and an increase in the maximum bound of the delay required to ensure exponential convergence. Indeed, an explicit relation between the maximum bound of the delay and the observer tuning parameter is derived by using a Lyapunov–Krasovskii functional jointly with the Halanay inequality. Such a relation shows clearly the superiority of HG/LMI observer design methodology. An application to nonlinear systems with sampled measurements is provided. Furthermore, the proposed methodology is extended to systems with nonlinear outputs. This extension provides more general synthesis conditions and encompasses the linear case as a particular situation. Finally, two numerical examples are proposed to illustrate the performance of the proposed observer design procedure, and comparison to standard approaches is also provided.

Interval observer design for nonlinear systems using simplified contraction theory

In this work, vector framework based contraction theory to design interval observers for a class of nonlinear systems having disturbances, inputs and outputs is exploited. The main feature is that it does not require the formulation of error dynamics to show the convergence properties and need not require the construction of a Lyapunov candidate function without any idea of the structure of the function. Specifically, it performs the convergence analysis through a comparison system which has specified properties. Furthermore, this theory is exploited to design dynamic output feedback control through the obtained state bounds from the constructed interval observer and the system outputs, to make the interval observer to be globally asymptotically stable. Examples with simulation outcomes are provided to validate the theoretical results.

A neural network adaptive interval observer design for nonlinear systems

A design method of neural network adaptive interval observer for the continuous-time unknown nonlinear system is proposed. The bounds of unknown nonlinear functions are described by the state and the input, so that the bounds of the nonlinear function vector are unmeasurable. Therefore, this method can be used to observe systems with higher degree of nonlinearity and any priori knowledge about systems. In this work, the hyper basis function neural network (HBFNN) is presented to estimate the boundary (including lower and upper bounds) of the unknown nonlinear function vector. Furthermore, the non-negativity and stability of the error dynamic system are given, and the convergence and non-negativity are also guaranteed. Compared with the traditional interval observer, the efficiency of the proposed approach is verified through a numerical simulation example.

Observer Design for Nonlinear Systems with Equivariance

Equivariance is a common and natural property of many nonlinear control systems, especially those associated with models of mechatronic and navigation systems. Such systems admit a symmetry, associated with the equivariance, that provides structure enabling the design of robust and high-performance observers. A key insight is to pose the observer state to lie in the symmetry group rather than on the system state space. This allows one to define a global intrinsic equivariant error but poses a challenge in defining internal dynamics for the observer. By choosing an equivariant lift of the system dynamics for the observer internal model, we show that the error dynamics have a particularly nice form. Applying the methodology of extended Kalman filtering to the equivariant error state yields a filter we term the equivariant filter. The geometry of the state-space manifold appears naturally as a curvature modification to the classical Riccati equation for extended Kalman filtering. The equivariant filter exploits the symmetry and respects the geometry of an equivariant system model, and thus yields high-performance, robust filters for a wide range of mechatronic and navigation systems.

An Adaptive Observer Design for Nonlinear Systems Affected by Unknown Disturbance with Simultaneous Actuator and Sensor Faults. Application to a CSTR

Continuous Stirred Tank Reactor (CSTR) is an important system in the chemical and biological industries. It's characterized by a complex nonlinear behavior and is usually affected by faults and disturbances. Therefore, the states and faults estimation of a CSTR is always a challenging task for automated process researchers and engineers. This paper proposes an adaptive observer. This paper proposes an adaptive observer in order to estimate states and actuator and sensor faults simultaneously under unknown disturbance. Firstly, the approach of the Takagi-Sugeno multi-model is proposed to transform the complex nonlinear model into several simple linear sub-models. However, the states of the considered isotherm CSTR are not completely measurable, so the multi-model is represented with non-measurable premise variables. Then, in order to transform the considered system into a system with an unknown input, a mathematical transformation is introduced to describe the sensor faults as actuator faults. The proposed observer is designed, and the exponential stability conditions are studied with the Lyapunov theory and L2 optimization and formulated in terms of linear matrix inequalities. Finally, to improve the effectiveness of the proposed observer, a numerical simulation is carried out on a CSTR.

Finite-time impulsive observers for nonlinear systems represented by Takagi–Sugeno models: Application to a chaotic system

In this paper, observer design for nonlinear systems represented by Takagi Sugeno models (T-S) is investigated. The first main contribution concerns the finite time convergence of the estimations, ensured by an impulsive observer with state updates. The second contribution, lies with taking into account unmeasurable parameters, using the Differential Mean Value Theorem (DMVT) to express the disturbed error dynamics into a Linear Parameter Varying system. The stability conditions are formulated in terms of Linear Matrix Inequalities (LMI). To prove the efficiency of the proposed procedure, applications are performed on a chaotic system. The obtained results are pretty satisfying.

Sampled-data fuzzy observer design for nonlinear systems with a nonlinear output equation under measurement quantization

In this paper, a sampled-data fuzzy observer (SDFO) of oscillating nonlinear systems with a nonlinear output equation is proposed based on the Takagi–Sugeno (T–S) fuzzy-model-based approach. First, to handle the low transmission capacity of the network, measurements from the system of interest are assumed to be quantized. Next, we employ an exponentially time-varying gain matrix to the SDFO system for enhancing the decay rate performance of the state estimation error dynamics, and these are represented with the T–S fuzzy model. To show better performance on the state estimation, we focus on developing two points: a novel design methodology and a novel looped Lyapunov–Krasovskii functional (LKF). Furthermore, we propose a novel design condition of an SDFO for systems without measurement quantization, which is less conservative than conventional approaches. All of the proposed design conditions are formulated in terms of linear matrix inequalities (LMIs). Finally, appropriate simulation examples are given to validate the effectiveness of the proposed method, and these show better performance compared to the conventional studies.

Functional observer design for nonlinear systems with incremental quadratic constraints

We present a functional observer design method for a class of nonlinear systems with output nonlinearities and incremental quadratic constraints. The system under consideration includes a wider class of nonlinear systems, such as, Lipschitz, one-sided Lipschitz, nondecreasing nonlinearity system as special cases. Based on Moore–Penrose pseudo-inverse and linear matrix inequality theory, the existence conditions of gain matrices of a functional observer are obtained. Moreover, exponential convergence of the estimate errors is achieved. Through the Lorenz system, it is shown that the proposed approach provides a smaller order of observer than that of the conventional approaches, while maintaining satisfactory performances.

Adaptive singularity-free controller design of constrained nonlinear systems with prescribed performance

This paper investigates the controller design for the tracking problem of uncertain nonlinear systems with constraints on input and output. First, to avoid the full-state measurement, the high-gain observer is designed to estimate the unmeasured state. Compared to existing observer design for nonlinear systems, the high-gain observer design only requires a modified version of the Lipschitz condition. The output feedback controller is further presented on this basis. Second, to consider the transient constraints on the tracking performance, a barrier Lyapunov function with the user-defined time-varying performance is developed. In addition, the proposed controller design is shown to be free of control direction singularity. The convergence of the learning scheme and the boundedness of all the closed-loop signals during the learning phase are discussed theoretically. Finally, two simulation examples are conducted to verify the advantage of the proposed adaptive output feedback controller design.

Finite element-based observer design for nonlinear systems with delayed measurements

Abstract An observer for nonlinear systems with delayed measurements is proposed. The design method is based on computation of a certain manifold defined as a solution of a partial differential equation. This equation is solved using finite-element method. Conditions for existence of a solution are given. Viability of the proposed approach is demonstrated by an example.

Observer Design for Nonlinear Systems With Output Transformation

This article addresses the problem of robust observer design for nonlinear systems with transformed output. We first assume that there exists an observer design for a given nonlinear system. The structure of the system output is often crucial for estimating the system states. Therefore, we consider the problem that arises when the output is only available under a nonlinear transformation (affected by noise). This formulation is especially useful in engineering applications, where nonlinear sensors are frequent. We propose a new interconnected observer that estimates both the unavailable output and the system states, and we establish small-gain type conditions ensuring the asymptotic convergence of the overall error dynamics. To explicit our design parameters, we study the particular cases of state-affine systems up to output injection and systems with additive triangular nonlinearities. Finally, we provide corresponding numerical examples and comparisons with known observers.

On the need for switched-gain observers for non-monotonic nonlinear systems

This paper focuses on the challenges in observer design for nonlinear systems which are non-monotonic. A class of nonlinear systems is considered in which the process dynamics and output equations are both composed of nonlinear vector functions of scalar combinations of the states. The nonlinear functions are assumed to be differentiable with bounded derivatives. An observer design algorithm that requires solving just a single linear matrix inequality for exponentially convergent state estimation is developed. The developed algorithm works effectively when the involved nonlinear functions are monotonic. However, it fails when all or even some of the system functions are non-monotonic. Both numerical computation and analytical results show that the observer design LMI has no feasible solutions when either all output functions or all process dynamics functions are non-monotonic. Further, other constant gain LMI-based observer design methods from literature also fail when the involved nonlinear functions are all non-monotonic, no matter how small the Lipschitz constant or the Jacobian bounds of the nonlinearities. This limitation has not previously been recognized in observer design literature. To overcome this limitation, a hybrid observer that switches between multiple constant observer gains is developed that can provide global asymptotic stability for systems with non-monotonic nonlinear functions. Hybrid observers with switched gains enable existing observer design methods to be utilized for non-monotonic nonlinear functions with finite local extrema. The application of the developed hybrid observer to two motion estimation applications, one a vehicle position tracking problem on roads and another a piston position estimation problem for an industrial actuator, are demonstrated.

High-Gain Observer Design for Nonlinear Systems with Delayed Outputs

This paper deals with high-gain nonlinear observer design for a class of triangular systems with delayed output measurements. Based on a recent high-gain like observer design method, called HG/LMI observer, a larger bound of the time-delay is allowed compared to that obtained by using the standard high-gain methodology. Such a HG/LMI observer leads to a significantly lower tuning parameter, which reduces the values of the observer gains and increases the maximum bound of the delay allowed to ensure exponential convergence. Indeed, an explicit relation between the maximum bound of the delay and the observer tuning parameter is inferred by using a Lyapunov-Krasovskii functional jointly with the Halanay inequality. Such a relation shows clearly the superiority of the use of the HG/LMI observer design methodology.

Sampled-data based observer design for nonlinear systems with output distributed delay

Existing observers that accommodate distributed-delays are mainly designed for linear systems with continuous-time output measurements. In this paper, we seek sampled-measurements-based observers for nonlinear systems with strict-feedback globally Lipschitz dynamics and distributed-delay. Invoking the high-gain principle, we design two observers operating with sampled output measurements. In both, the delay effect is compensated for by using distributed output-predictors. The first observer involves an intersample predictor and its exponential convergence is established using a quadratic Lyapunov function. The second observer involves a zero-order-hold predictor and the exponential convergence is proved using a Lyapunov–Krasovskii functional. To the author’s knowledge, it is the first time that an exponentially convergent observer is developed for nonlinear systems with output distributed-delay and sampled output measurements.