Class Of Nonlinear Time-varying Systems Research Articles

Distributed Energy-Based Estimation Over Harvesting-Constrained Sensor Networks.

This article investigates the distributed joint state and fault estimation issue for a class of nonlinear time-varying systems over sensor networks constrained by energy harvesting. It is assumed that data transmission between sensors requires energy consumption, and each sensor can harvest energy from the external environment. A Poisson process models the energy harvested by each sensor, and the sensor's transmission decision depends on its current energy level. One can obtain the sensor transmission probability through a recursive calculation of the probability distribution of the energy level. Under such energy harvesting constraints, the proposed estimator only uses local and neighbor data to simultaneously estimate the system state and the fault, thereby establishing a distributed estimation framework. Moreover, the estimation error covariance is determined to possess an upper bound, which is minimized by devising energy-based filtering parameters. The convergence performance of the proposed estimator is analyzed. Finally, a practical example is presented to verify the usefulness of the main results.

A Novel Approach to Study the Time-Varying Additive Activity during the Electrorefining of Copper

Electrodeposition additives are key components to enhance the electrochemical deposition of metals, either in pure plating applications or during the electrorefining of metals. Many different additives, either organic or inorganic, are applied in industry, depending on the type of process. This work focusses on thiourea and chloride as common additives used for copper electrorefining. This electrochemical industrial process consumes an impure copper anode and produces a pure (99.995 %) copper cathode, while the anode impurities remain in the electrolyte. Additives assist the electroreduction process to produce smooth copper deposits, essential to avoid short circuits and the formation of unwanted cathode morphologies which can entrap impurities.Thiourea is a crucial but complex additive in copper refining. It is known to degenerate into formamidine disulfide (FDS) and both thiourea as FDS can undergo interactions with either copper ions in the solution or the cathode surface. Time dependent additive studies and fundamental knowledge of the effect of aged electrolyte on the cathodic part of the electrorefining process are scarce.The approach applied in this work enables the investigation of the time dependent effect of additives, in this case thiourea and chlorides, assisting the electroplating of copper. Fundamental knowledge of the assisted plating process is extended to create a corner stone for applied research, subsequently, semi-industrial circumstances are used to describe the additive behaviour in industrial relevant conditions. Operando Odd Random Phase Electrochemical Impedance Spectroscopy (ORP-EIS) is used to study the electrochemistry of the additive system [1,2]. Hereby, a multisine input signal is added to the DC current to perform the ORP-EIS experiment simultaneously with the plating process. This measurement technique allows an in situ monitoring of the electrochemical system for hour-long electrodeposition processes. The ORP-EIS technique enables a quantification of the nonlinearities and non-stationarities during the EIS experiment [3]. Additionally, the present non-stationarities are modelled and the nonlinearities are considered as nuisance factor to result in a best linear time-varying approximation [4,5]. An equivalent electrical circuit is used as model to extract the relevant physical parameters from the impedance data. The EIS study is complemented with an ex situ analysis of the thiourea concentration, through ion chromatography, as a function of plating time. Correlations could be made between electrochemical parameters and actual thiourea concentrations [2].[1] T. Collet, et al., An operando ORP-EIS study of the copper reduction reaction supported by thiourea and chlorides as electrorefining additives, Electrochimica Acta (2021) 138762. [2] T. Collet, et al., The time-varying effect of thiourea on the copper electroplating process with industrial copper concentrations, Electrochimica Acta (2022) 141412. [3] Van Gheem, et al., Electrochemical impedance spectroscopy in the presence of non-linear distortions and non-stationary behaviour part i: Theory and validation, Electrochimica Acta 49 (2004) 4753–4762. [4] N. Hallemans, et al., Detection, Classification, and Quantification of Nonlinear Distortions in Time-Varying Frequency Response Function Measurements, IEEE Transactions on Instrumentation and Measurement, 70 (2021) 6500814. [5] N. Hallemans, et al., Best Linear Time-Varying Approximation of a General Class of Nonlinear Time-Varying Systems, IEEE Transactions on Instrumentation and Measurement, 70 (2021) 6503814.

A Nonlinear Filter by Pseudo-Formal Linearization for Nonlinear Time-Varying Systems

This paper is concerned with a nonlinear filter for a general class of nonlinear time-varying systems, by making use of a pseudo-formal linearization method and Kalman filter approach. The given nonlinear time-varying dynamic and measurement equations are piecewisely linearized by the formal linearization method using Taylor expansion so that Kalman filter can be applied for each linearized time-varying equations. Then the resulting filters are smoothly united into a single nonlinear filter by an automatic choosing function. The efficiency of this filter is shown through numerical examples.

Adaptive control of a class of time-varying nonlinear systems via immersion and invariance.

This paper addresses the adaptive control of a class of time-varying nonlinear systems. Under the framework of Immersion and Invariance (I&I) adaptive control, a set of sufficient conditions is obtained to stabilize the concerned time-varying nonlinear systems. It is shown that the presented controller can also be utilized to complete tracking control for a class of nonholonomic constraint systems if the desired trajectories satisfy certain conditions. The effectiveness of the new adaptive controller is demonstrated by some numerical simulations on a nonholonomic mobile robot.

Finite-time stabilization of time-varying nonlinear systems based on a novel differential inequality approach

In this paper, the finite-time stabilization problem is investigated for a class of time-varying nonlinear systems. The classical finite-time differential inequality is extended to time-varying systems and some new lemmas are derived for global finite-time stability (FTS) and local FTS of the corresponding closed-loop systems. Then based on the hybrid control theory and the extended time-varying differential inequality, we present two control schemes including continuous control and hybrid control. It is shown that the continuous control is formulated to make the system converge in finite time, while the impulsive part involved in hybrid control accelerates the stabilization. In addition, the theoretical results are applied to the finite-time synchronization of complex networks with time-varying parametric matrices. Ultimately, two numerical examples are presented to demonstrate the distinctiveness and the effectiveness of our proposed results.

Signal Transmission by Sampled-time Adaptive Synchronization of Time-varying Chaotic Systems

In the paper, the adaptive synchronization problem is studied for a class of time-varying nonlinear systems. The only assumption imposed is the boundedness of the rate of change of the controlled system parameters, while the initial uncertainty can be quite large. The adaptation algorithms in the discrete-time form are derived based on adaptive control of nonlinear time-varying systems with an explicit reference model. As the result, the estimate of the limiting deviation of the solution of the closed-loop system from that of the reference model was obtained. It is shown that with sufficiently slow changes in the parameters and a small initial uncertainty, the limiting error in the system can be made arbitrarily small despite a possibly big range of system parameter values. The systems based on a direct approach are considered. The procedure for the synthesis of a sampled-time adaptive controller and analysis of the synthesized system is illustrated by an example of the signal transmission system. The results obtained allow one to build and analyze a wide class of adaptive systems with reference models under time-varying conditions.

ISS-like properties in Lie-bracket approximations and application to extremum seeking

In this paper, we consider a class of nonlinear time-varying systems that are subject to uncontrolled inputs. Those inputs may represent, for instance, disturbances, time-varying parameters, or model uncertainties. Exploiting the stability properties of an extended system, a so-called Lie-bracket system, we derive stability properties of the original system. Those stability results, in the spirit of input-to-state stability, are then applied to the case where the dynamical system is an extremum seeking system. Two scenarios are analyzed. We first consider that the uncontrolled input is a deterministic noise corrupting the cost measurement. We then examine the case where the uncontrolled input induces time-variations of the cost optimizer.

Finite-time stabilisation of a class of time-varying nonlinear systems by a mixed event-based and continuous-time strategy

In this paper, a mixed event-triggered and continuous-time control method is proposed, which can guarantee finite-time stabilisation of the fixed point in a class of time-varying nonlinear systems. Benefiting from an event-triggered framework, which is constructed based on the indefinite Lyapunov theory, the communication/computation costs in the transient time can be reduced by using the proposed method. In a special case, this method is converted to a fully event-triggered control strategy for asymptotic stabilisation of the fixed point in the considered class of time-varying nonlinear systems. The effectiveness of the proposed method is verified by numerical simulations.

Distributed state and fault estimation over sensor networks with probabilistic quantizations: The dynamic event-triggered case

In this paper, the distributed state and fault estimation problem is discussed for a class of nonlinear time-varying systems with probabilistic quantizations and dynamic event-triggered mechanisms. To reduce resource consumption, a dynamic event-triggered strategy is exploited to schedule the data communication among sensor nodes. In addition, the measurement signals are quantized and then transmitted through the network, where the probabilistic quantizations are taken into consideration. Attention is focused on the problem of constructing a distributed estimator such that both the plant state and the fault signal are estimated simultaneously. By using the matrix difference equation method, certain upper bound on the estimation error covariance is first guaranteed and then minimized at each iteration by properly designing the estimator parameters. Subsequently, for the proposed distributed estimation algorithm, the estimator performance is analyzed and a sufficient condition is established to guarantee that the estimation error is exponentially bounded in mean-square sense. Finally, an illustrative example is provided to verify the usefulness of the developed estimation scheme.

Stable Adaptive Controller Based on Generalized Regression Neural Networks and Sliding Mode Control for a Class of Nonlinear Time-Varying Systems

Finding synergy between a variety of control and estimation approaches can lead to effective solutions for controlling nonlinear dynamic systems in an efficient and systematic manner. In this paper, a novel controller design consisting of generalized regression neural networks (GRNNs) and sliding mode control (SMC) is proposed to control nonlinear multi-input and multi-output (MIMO) dynamic systems. The proposed design transforms GRNN from an offline regression model to an online adaptive controller. The suggested controller does not require any pretraining and it learns quickly from scratch. It uses a low computational complexity algorithm to provide accurate and stable performance. The proposed controller (GRNNSMC) performance is verified with a generic MIMO nonlinear dynamic system and a hexacopter model with a variable center of gravity. The results are compared with the standard PID controller. In addition, the stability of the GRNNSMC controller is verified using the Lyapunov stability method.

Best Linear Time-Varying Approximation of a General Class of Nonlinear Time-Varying Systems

This article presents a method for estimating a linear time-varying approximation of a general class of nonlinear time-varying (NLTV) systems. It starts from noisy measurements of the response of the NLTV system to a special class of periodic excitation signals. These measurements are subject to measurement noise, process noise, and a trend. The proposed method is a two-step procedure. First, the disturbing noise variance is quantified. Next, using this knowledge, the linear time-varying dynamics are estimated together with the NLTV distortions. The latter are split into even and odd contributions. As a result, the signal-to-nonlinear-distortion ratio is quantified. It allows one to decide whether or not a linear approximation is justifiable for the application at hand. The two-step algorithm is fully automatic in the sense that the user only has to choose upper bounds on the number of basis functions used for modeling the response signal. The obtained linear time-varying approximation is the best in the sense that the difference between the actual nonlinear response and the response predicted by the linear approximation is uncorrelated with the input. Therefore, it is called the best linear time-varying approximation (BLTVA). Finally, the theory is validated on a simulation example and illustrated on two measurement examples: the crystallographic pitting corrosion of aluminum and copper electrorefining.

Fault estimation for time-varying systems with Round--Robin protocol

This paper is concerned with the design problem of finite-horizon $H_\infty$ fault estimator for a class of nonlinear time-varying systems with Round-Robin protocol scheduling. The faults are assumed to occur in a random way governed by a Bernoulli distributed white sequence. The communication between the sensor nodes and fault estimators is implemented via a shared network. In order to prevent the data from collisions, a Round-Robin protocol is utilized to orchestrate the transmission of sensor nodes. By means of the stochastic analysis technique and the completing squares method, a necessary and sufficient condition is established for the existence of fault estimator ensuring that the estimation error dynamics satisfies the prescribed $H_\infty$ constraint. The time-varying parameters of fault estimator are obtained by recursively solving a set of coupled backward Riccati difference equations. A simulation example is given to demonstrate the effectiveness of the proposed design scheme of the fault estimator.

Fault estimation for a class of nonlinear time-variant systems through a Krein space–based approach

This paper studies the [Formula: see text] fault estimation problem for a class of discrete-time nonlinear systems subject to time-variant coefficient matrices, online available input, and exogenous disturbances. By assuming that the concerned nonlinearity is continuously differentiable and by using Taylor series expansions, the dynamic system is transferred as a linear time-variant system with modeling uncertainties. A non-conservative but nominal system and its corresponding [Formula: see text] indefinite quadratic performance function are, respectively, given in place of the transferred uncertain system and the conventional performance metric, such that the estimation problem is converted as a two-stage optimization issue. By introducing an auxiliary model in Krein space, the so-called orthogonal projection technique is utilized to search an appropriate choice serving as the estimation of the fault signal. A necessary and sufficient condition on the existence of the fault estimator is given, and a recursive algorithm for computing the gain matrix of the estimator is proposed. The addressed method is applied to an indoor robot localization system to show its effectiveness.

On fixed-time stability of a class of nonlinear time-varying systems

Fixed-time stability of a class of nonlinear time-varying systems is investigated using the introduced notions of sub- and sup-homogeneity. These concepts allow the systems (even if they do not admit homogeneous approximations) to be analyzed using the homogeneity of their extensions. Then, finite-time and fixed-time stability properties can be recovered. The proposed stability conditions are not based on Lyapunov arguments. In the last section, an example illustrates the obtained results.

The eigenvalue product bounds of the Lyapunov matrix differential equation and the stability of a class of time-varying nonlinear system

The Lyapunov matrix differential equation plays an important role in many scientific and engineering fields. In this paper, we first give a class relation between the eigenvalue of functional matrix derivative and the derivative of functional matrix eigenvalue. Using this relation, we convert the Lyapunov matrix differential equation into an eigenvalue differential equation. Further, by the Schur theorem, combining Hölder’s integral inequality with arithmetic–geometric average inequality, we provide several lower and upper bounds on the eigenvalue product for the solution of the Lyapunov matrix differential equation. As an application in control and optimization, we show that our bounds can be used to discuss the stability of a class of time-varying nonlinear systems. Finally, we illustrate the superiority and effectiveness of the derived bounds by a numerical example.

Non-Fragile Distributed Fault Estimation for a Class of Nonlinear Time-Varying Systems Over Sensor Networks: The Finite-Horizon Case

This paper is concerned with a new non-fragile distributed fault estimation problem for a class of time-varying systems through sensor networks over a finite horizon. Both randomly occurring nonlinearities and randomly occurring gain variations, whose occurrences are governed by random variables obeying certain probabilistic distributions on the interval $[0,1]$ are taken into simultaneous consideration. The fault estimation is based on the information not only from the individual sensor, but also from its neighboring ones according to the given topology of the sensor network. The aim of this paper is to design a non-fragile distributed fault estimator such that, in the presence of certain drifts/variations/perturbations of the gain parameters during the implementation, a prescribed average $H_{\infty }$ performance constraint on the estimation error dynamics is satisfied. Based on stochastic analysis techniques, the gain parameters are calculated by solving a series of recursive linear matrix inequalities over a finite horizon. Moreover, a simulation example is provided to show the effectiveness of the proposed fault estimation scheme.

State bounding for nonlinear time-varying systems with delay and disturbance

This paper deals with the problem of state bounding for a class of nonlinear time-varying systems with delay and bounded disturbance. By using a model transformation and an approach developed in positive systems, new delay-dependent explicit conditions have been established to guarantee all the state trajectories of the system converge exponentially within a ball. Two illustrative examples are given to show that the obtained results can be applied to some cases not covered by preceding results.

Finite-time stability analysis of a class of nonlinear time-varying systems: a numerical algorithm

ABSTRACTThe paper investigates the finite-time stability (FTS) analysis of a very general class of nonlinear time-varying systems. The FTS of the considered system, whose vector field consists of a nonlinear part which can be sublinear or superlinear, and a linear part which can be time-varying, has not been fully studied before. By estimating the bound of the norm of the considered system's states with the generalised Gronwall–Bellman inequality, a sufficient criterion is established to guarantee the FTS of the considered system. To facilitate checking the criterion in practice, a novel numerical algorithm is proposed by numerically solving certain differential equations. Therefore, the FTS of the considered class of nonlinear time-varying systems can be easily analysed by the numerical algorithm. Further considering the numerical errors in the practical numerical computation, we strictly prove the credibility and programmability of the numerical algorithm in theory. Finally, three numerical examples are provided to illustrate the effectiveness the proposed results.

Event-based recursive filtering for a class of nonlinear stochastic parameter systems over fading channels

In this paper, the recursive filtering problem is studied for a class of time-varying nonlinear systems with stochastic parameter matrices. The measurement transmission between the sensor and the filter is conducted through a fading channel characterized by the Rice fading model. An event-based transmission mechanism is adopted to decide whether the sensor measurement should be transmitted to the filter. A recursive filter is designed such that, in the simultaneous presence of the stochastic parameter matrices and fading channels, the filtering error covariance is guaranteed to have an upper bound and such an upper bound is then minimized by appropriately choosing filter gain matrix. Finally, a simulation example is presented to demonstrate the effectiveness of the proposed filtering scheme.

Reachable Set Bounding for a Class of Nonlinear Time-Varying Systems with Delay

We investigate the problem of reachable set bounding for a class of continuous-time and discrete-time nonlinear time-varying systems with time-varying delay. Unlike some preceding works, the involved disturbance input and time-varying delay are not assumed to be bounded. By employing an approach which does not involve the conventional Lyapunov-Krasovskii functional, new conditions are proposed such that all the state trajectories of the system converge asymptotically within a ball. Two illustrative examples are also given to show the effectiveness of the obtained results.