Special Class Of Nonlinear Systems Research Articles

H∞ Tracking Control of Non-lower Triangular Nonlinear Systems: Coronary Artery System

This work presents a novel control approach that integrates backstepping and dynamic surface control techniques to design an H∞ tracking control system. Our focus is on a particular class of nonlinear systems that do not possess lower triangular structure, with a specific application to the coronary system. By employing dynamic surface control, the issue of differential explosion encountered during the backstepping process is effectively addressed, while also guaranteeing robustness. Finally, simulation of coronary artery system proves the effectiveness of the proposed control strategy.

Provably-Stable Stochastic MPC for a Class of Nonlinear Contractive Systems

We present a model predictive control framework for a class of nonlinear systems affected by additive stochastic disturbances with (possibly) unbounded support. We consider hard input constraints and chance state constraints and we employ the unscented transform method to propagate the disturbances over the nonlinear dynamics in a computationally efficient manner. The main contribution of our work is the establishment of sufficient conditions for stability and recursive feasibility of the closed-loop system, based on the design of a terminal cost and a terminal set. We focus here on a special class of nonlinear systems that exhibit contractive properties in the dynamics. By assuming this property, we propose a novel approach to efficiently compute the terminal conditions without the need of performing any linearization of the dynamics. Finally, we provide an illustrative example to corroborate our theoretical findings.

A moving horizon state and parameter estimation scheme with guaranteed robust convergence

We propose a moving horizon estimation scheme for joint state and parameter estimation for nonlinear uncertain discrete-time systems. We establish robust exponential convergence of the combined estimation error subject to process disturbances and measurement noise. We employ a joint incremental input/output-to-state stability (δ-IOSS) Lyapunov function to characterize nonlinear detectability for the states and (constant) parameters of the system. Sufficient conditions for the construction of a joint (δ-IOSS Lyapunov function are provided for a special class of nonlinear systems using a persistence of excitation condition. The theoretical results are illustrated by a numerical example.

On observer design for a class of Persidskii systems based on steady-state estimation

This work aims to propose conditions for the existence of an observer for a particular class of nonlinear systems, presented as an interconnection of two Persidskii systems. First, we establish analytical expressions for steady-state solutions of an interconnected system. Next, a reduced-order observer for this system is designed, and the stability and boundedness of the error dynamics are proven. An academic example and an example considering Chua's circuits illustrate our results.

Atomic Spin Polarization Controllability Analysis: A Novel Controllability Determination Method for Spin-Exchange Relaxation-Free Co-Magnetometers

This paper investigates the atomic spin polarization controllability of spin-exchange relaxation-free co-magnetometers (SERFCMs). This is the first work in the field of controllability analysis for the atomic spin ensembles systems, whose dynamic behaviors of spin polarization are described by the Bloch equations. Based on the Bloch equations, a state-space model of the atomic spin polarization for SERFCM is first established, which belongs to a particular class of nonlinear systems. For this class of nonlinear systems, a novel determination method for the global state controllability is proposed and proved. Then, this method is implemented in the process of controllability analysis on the atomic spin polarization of an actual SERFCM. Moreover, a theoretically feasible and reasonable solution of the control input is proposed under some physical constraints, with whose limitation of realistic conditions, the controller design can be accomplished more practically and more exactly. Finally, the simulation results demonstrate the feasibility and validation of the proposed controllability determination method.

Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization

Nonlinear dynamical systems with time delay are abundant in applications but are notoriously difficult to analyze and predict because delay-induced effects strongly depend on the form of the nonlinearities involved and on the exact way the delay enters the system. We consider a special class of nonlinear systems with delay obtained by taking a gradient dynamical system with a two-well “potential” function and replacing the argument of the right-hand side function with its delayed version. This choice of the system is motivated by the relative ease of its graphical interpretation and by its relevance to a recent approach to use delay in finding the global minimum of a multi-well function. Here, the simplest type of such systems is explored for which we hypothesize and verify the possibility to qualitatively predict the delay-induced effects, such as a chain of homoclinic bifurcations one by one eliminating local attractors and enabling the phase trajectory to spontaneously visit vicinities of all local minima. The key phenomenon here is delay-induced reorganization of manifolds, which cease to serve as barriers between the local minima after homoclinic bifurcations. Despite the general scenario being quite universal in two-well potentials, the homoclinic bifurcation comes in various versions depending on the fine features of the potential. Our results are a pre-requisite for understanding general highly nonlinear multistable systems with delay. They also reveal the mechanisms behind the possible role of delay in optimization.

Adaptive Kalman filtering for closed-loop systems based on the observation vector covariance

ABSTRACT The Kalman filter is one of the most widely used methods for state estimation and control purposes. However, it requires correct knowledge of noise statistics, which are unknown or not known perfectly in real-life applications and then they need to be identified. Considering such background, this paper introduces a new adaptive Kalman filter algorithm in order to handle the unknown process noise covariance for linear discrete-time closed-loop systems. From the closed-loop joint analysis of the system and the a priori recursive form of the Kalman filter, we adaptively estimate the process noise covariance by relating it to the observation vector covariance. The latter is then obtained from an exponential moving average technique. Lastly, we also extend our adaptive methodology for a special class of nonlinear systems. The performance of the proposed adaptive method is demonstrated through numerical examples and it has been compared to other types of adaptive filtering algorithms.

Two-Phase Attractors for Finite-Duration Consensus of Multiagent Systems

A specific class of nonlinear systems, admitting finite-duration attractors (FDAs), is introduced in this paper, and used for consensus of multiagent systems (MASs). The advantage is that one can obtain a closed-form solution for the settling time function of the presented nonlinear systems. Such a system exhibits a two-phase convergence process, and its settling time function takes values over a finite duration. However, the duration width changes with respect to the initial condition. The FDA-based consensus protocol is shown to make the MAS undertaken achieve finite-duration convergence. With illustrative examples, the perfect consensus property is verified for MASs with or without a leader agent.

Robust Time-Varying Tracking Control of Sandwich Nonlinear Systems with Sandwiched Saturation Nonlinearity

This paper addresses the output-tracking control problem for a class of nonlinear sandwich systems with external disturbances and model uncertainties. Sandwich systems are a special class of nonlinear systems in the cascade structure, which are made up of several distinct subsystems while the non-smooth nonlinear functions are sandwiched among their separate subsystems. This paper concentrates on the sandwich systems consisting of two arbitrary-order nonlinear cascaded subsystems, while the saturation nonlinearity is sandwiched between them. The robust control law is designed such that the output of the sandwich system tracks the desired time-varying reference signal despite the saturation occurrence. Due to the characteristics of the saturation function and also the existence of the external disturbances and model uncertainties, the considered problem is facing some serious challenges. To deal with these difficulties, the design procedure is divided into two phases. In the first phase, the second subsystem and the sandwiched saturation nonlinearity are considered. In this phase, by utilizing the backstepping approach, the desired input signal of the saturation function is designed. In the second phase, the control input of the first subsystem is designed based on the sliding mode technique in a way that the output of the first subsystem tracks the desired signal which is obtained in the first phase. Finally, to verify the effectiveness of the proposed method, it is applied to a practical sandwich system.

Recursive parameter estimation methods and convergence analysis for a special class of nonlinear systems

SummaryThis paper is concerned with the joint estimation of states and parameters of a special class of nonlinear systems, ie, bilinear systems. The key is to investigate new estimation methods for interactive state and parameter estimation of the considered system based on the interactive estimation theory. Because the system states are unknown, a bilinear state observer is established based on the Kalman filtering principle. Then, the unavailable states are updated by the state observer outputs recursively. Once the state estimates are obtained, the bilinear state observer–based hierarchical stochastic gradient algorithm is developed by using the gradient search. For the purpose of improving the convergence rate and the parameter estimation accuracy, a bilinear state observer–based hierarchical multi‐innovation stochastic gradient algorithm is proposed by expanding a scalar innovation to an innovation vector. The convergence analysis indicates that the parameter estimates can converge to their true values. The numerical example illustrates the effectiveness of the proposed algorithms.

Parameter estimation for a special class of nonlinear systems by using the over-parameterisation method and the linear filter

ABSTRACTThis paper studies the parameter estimation issues of a special class of nonlinear systems (i.e. bilinear-in-parameter systems) utilising the measurement input-output data. The estimation idea is based on the data filtering technique and the over-parameterisation method to represent the system as a linearly parameterised form. Then, by means of the filtered model and the noise model, a filtering based over-parameterisation generalised extended gradient iterative (F-O-GEGI) algorithm is developed for estimating all the parameters. For purpose of improving the precision of parameter estimation, a filtering based over-parameterisation generalised extended least squares iterative (F-O-GELSI) algorithm is derived by formulating and minimising two separate criterion functions. By these foundations, the F-O-GEGI algorithm and the F-O-GELSI algorithm with finite measurement data are presented. The simulation example is provided to test and compare the presented approaches.

Unified $\mathcal{H}_{\infty}$ Observer for a Class of Nonlinear Lipschitz Systems: Application to a Real ER Automotive Suspension

This letter presents an extension of the synthesis of a unified $\mathcal {H_{\infty }}$ observer for a specific class of nonlinear systems. The objectives are to decouple the effects of bounded unknown input disturbances and to minimize the effects of measurement noises on the estimation errors of the state variables by using $\mathcal {H_{\infty }}$ criterion, while the nonlinearity is bounded through a Lipschitz condition. This new method is developed to estimate the damping force of an electro-rheological (ER) damper in an automotive suspension system, and is implemented on the INOVE testbench from GIPSA-Lab (1/5-scaled real vehicle) for real-time performance assessment. Both simulation and experimental results demonstrate the effectiveness of the proposed observer to estimate the damper force in realtime, face to measurement noises and road disturbances.

The filtering‐based maximum likelihood iterative estimation algorithms for a special class of nonlinear systems with autoregressive moving average noise using the hierarchical identification principle

SummaryFor a special class of nonlinear systems (ie, bilinear systems) with autoregressive moving average noise, this paper gives the input‐output representation of the bilinear systems through eliminating the state variables in the model. Based on the obtained model and the maximum likelihood principle, a filtering‐based maximum likelihood hierarchical gradient iterative algorithm and a filtering‐based maximum likelihood hierarchical least squares iterative algorithm are developed for identifying the parameters of bilinear systems with colored noises. The original bilinear systems are divided into three subsystems by using the data filtering technique and the hierarchical identification principle, and they are identified respectively. Compared with the gradient‐based iterative algorithm and the multi‐innovation stochastic gradient algorithm, the proposed algorithms have higher computational efficiency and parameter estimation accuracy. The simulation results indicate that the proposed algorithms are effective for identifying bilinear systems.

Hybrid Kalman Filtering Algorithm With Stochastic Nonlinearities and Multiple Missing Measurements

In this paper, the hybrid Kalman filter is designed for a class of special nonlinear systems where the state equation is nonlinear and the measurement equation is linear. The stochastic nonlinearities, which are described by statistical means, are considered in the system model to reflect the multiplicative stochastic disturbances. The phenomenon of multiple missing measurements is depicted by a set of the Bernoulli distributed random variables with known conditional probabilities and the missing rates of every sensor are different. We need to compute the parameters to reduce the effects of the stochastic nonlinearities and the phenomenon of multiple missing measurements. In addition then, based on the recursive projection formula and the unscented transformation approach, a new hybrid Kalman filtering algorithm is proposed such that, for the stochastic nonlinearities and multiple missing measurements, the filtering error is minimized. By solving the recursive matrix equation, the filter gain matrices and the error covariance matrices can be obtained and the proposed results can be easily verified by using the standard numerical software. We finally provide a numerical example to show the performance of the proposed approach.

A Reduced-order Observer-predictor for Nonlinear Systems with Delayed Measurements

The purpose of this work is to propose and characterize reduced-order nonlinear continuous observers coupled with an output predictor for a particular class of nonlinear systems with delayed measurements. Using a small gain approach, we provide sufficient conditions, involving the delay, which ensure an exponential convergence of both observer and predictor. An explicit bound on the maximum allowable delay guaranteeing stability of this observer-predictor is given. A simple procedure design for the observer is described and illustrated on a numerical example.

Recursive generalized extended least squares and RML algorithms for identification of bilinear systems with ARMA noise

Bilinear systems are considered as a particular class of nonlinear systems including the state variables which are typically used for online identification. By using a recursive identification method and the maximum likelihood principle, this paper presents two recursive-based algorithms to identify the parameters of bilinear in parameter systems with ARMA noise. In this regard, recursive generalized extended least squares (RGELS) and recursive Maximum Likelihood (RML) algorithms have been proposed for identification of bilinear systems. These algorithms can be used as an alternative choice in system identification with acceptable performance. The proposed algorithms estimate the correlated noise parameters with high accuracy by making full use of the measurement data. Simulation results indicate that the proposed algorithms are effective for online identification of bilinear in parameter systems with high convergence speed.

Design of robust nonfragile H∞ controller for uncertain nonlinear polynomial systems

SummaryThis paper is concerned with the robust H∞ nonfragile controller design for a particular class of nonlinear systems, namely, the perturbed polynomial systems, which are subject to unstructured bounded uncertainties in both the system model and the control law. Combining the Lyapunov stability theory, properties of linear matrix inequality, and Kronecker product properties, a sufficient condition of robust H∞ nonfragile control design is proposed. More specifically, we propose a robust H∞ controller of nonlinear polynomial systems with additive unstructured uncertainties and variation in the control law itself that guarantee the stability and the attenuation of external perturbations. Two examples are provided to illustrate the effectiveness of the proposed approach.

A novel combined control based on an adaptive generalized back-stepping method for a class of nonlinear systems

This paper was aimed at using an adaptive control to develop a backstepping method for a special class of nonlinear systems, which requires no information on the upper bound of parametric uncertain...

Data-Driven Discovery of Closure Models

Derivation of reduced order representations of dynamical systems requires the modeling of the truncated dynamics on the retained dynamics. In its most general form, this so-called closure model has to account for memory effects. In this work, we present a framework of operator inference to extract the governing dynamics of closure from data in a compact, non-Markovian form. We employ sparse polynomial regression and artificial neural networks to extract the underlying operator. For a special class of non-linear systems, observability of the closure in terms of the resolved dynamics is analyzed and theoretical results are presented on the compactness of the memory. The proposed framework is evaluated on examples consisting of linear to nonlinear systems with and without chaotic dynamics, with an emphasis on predictive performance on unseen data.

Design of robust optimal regulator considering state and control nonlinearities

ABSTRACTMost real systems can be considered as a nonlinear and parameters of the systems may have uncertainties and external disturbances. Hence, in this paper, using the power series algorithm (PSA) based on State Dependent Riccati Equations (SDRE) and considering the proposed conditions that guarantee the asymptotic stability, the optimal regulator problem for a particular class of nonlinear systems is solved. Also, according to the specified pattern in the modified PSA (MPSA) method, weighting matrices are used as a function of state variables to achieve the better regulatory responses. Simulations are carried out on the Lorenz's chaotic system with uncertainties and external disturbances. The efficiency of optimal regulators the PSA and the MPSA methods in eliminating the external disturbances and being robust to uncertainties are compared together. The results show that the values of the performance index and the control cost in the MPSA method are smaller than the PSA method. Then the regulatory response in the MPSA method is more efficient.