Nonlinear Time-varying Dynamic System Research Articles

Hybrid Nonlinear Passivity-Based Control Approach to Magnetic-Impulsive Spacecraft Attitude Regulation

This paper proposes two novel passivity-based control design frameworks for hybrid nonlinear time-varying dynamical systems involving an interacting amalgam of continuous-time and discrete-time dynamics whose dynamical properties evolve periodically over time. In this regard, the hybrid Kalman–Yakubovich–Popov conditions are employed in tandem with the passivity theorem to construct a three-step control design algorithm to render closed-loop dynamics stable. The proposed control architectures are subsequently exploited to regulate the attitude motion of spacecraft with magnetic and impulsive modes of operation. Practical considerations involved in implementing the proposed hybrid algorithms are then discussed in detail. Simulation results show significant improvement in the performance of the attitude control system in terms of system response, robustness, and the required magnetic and impulsive control usage as compared to a hybrid linear approach.

Matrix measures, stability and contraction theory for dynamical systems on time scales

<p style='text-indent:20px;'>This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales. Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. Then, we consider nonlinear time-varying dynamical systems on time scales and establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning synchronization of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.</p>

DNN-Based H∞ Control Scheme of Nonlinear Time-Varying Dynamic Systems With External Disturbance and its Application to UAV Tracking Design

The main difficulty in the traditional nonlinear $H_{\infty }$ control design lies in how to solve the nonlinear partial differential Hamilton-Jacobi-Isaacs equation (HJIE), especially for nonlinear time-varying systems. In this study, a novel HJIE-embedded DNN $H_{\infty }$ control scheme is proposed to be efficiently trained for nonlinear $H_{\infty }$ stabilization and tracking control designs of nonlinear dynamic systems with the external disturbance. The proposed DNN-based $H_{\infty }$ control approach not only capitalizes on the availability of theoretical partial differential HJIE but also reduces the amount of empirical data and the complexity to train HJIE-embedded DNN. We have shown that the proposed DNN-based $H_{\infty }$ control scheme can approach the theoretical result of $H_{\infty }$ robust control when the training error approaches zero and the asymptotic stability is also guaranteed if the nonlinear time-varying system is free of external disturbance. The proposed method could be easily extended to DNN-based $H_{\infty }$ reference tracking control of nonlinear systems for more practical applications. Finally, two examples, including $({i})$ an $H_{\infty }$ stabilization of nonlinear time-varying system and $(ii)$ an $H_{\infty }$ unmanned aerial vehicle (UAV) reference tracking control system, are proposed to illustrate the design procedure and to demonstrate the effectiveness of our DNN-based $H_{\infty }$ method.

Model-free data reconstruction of structural response and excitation via sequential broad learning

In this study, a novel sequential broad learning (SBL) approach is proposed to reconstruct the missing signal of damaged sensors in structural health monitoring (SHM) sensory networks. It is capable to reconstruct the structural response and external excitation of linear/nonlinear time-varying dynamical systems under stationary/nonstationary excitation for sensory networks. The proposed approach is a model-free data-driven machine learning methodology and the data reconstruction is executed sequentially with moving time windows. The learning algorithm is developed by adopting the recently developed broad learning system (BLS) (Chen and Liu, 2018). In contrast to deep learning that suffers from excessive computational cost for training the stacks of hierarchical layers, BLS is established with a broadly expandable network and can be modified incrementally based on the inherited results from the previous trained architecture. Therefore, BLS provides a computationally very efficient alternative to deep learning. Taking the benefit of BLS, the proposed SBL approach can efficiently handle the massive data stream generated in long-term monitoring. To demonstrate the efficacy and applicability of the proposed approach, simulated examples that cover linear and nonlinear time-varying dynamical systems subjected to stationary/nonstationary wind-load/base excitation with different types of sensing devices are discussed. Moreover, the SHM database of the field measurement monitored from the MIT Green Building is utilized to examine the performance of the proposed approach in realistic application. It is demonstrated that the proposed approach offers a powerful data reconstruction tool for challenging data missing situations encountered in SHM.

Parameter identification for nonlinear time-varying dynamic system based on the assumption of “short time linearly varying” and global constraint optimization

A new identification approach based on a new assumption of “short time linear varying” is proposed for nonlinear time-varying (NTV) dynamic systems. In the identification procedure, the whole period is divided into a series of shifting windows. In each window, the NTV system model, which is known a priori, can be represented by regression equations and all the time-varying (TV) coefficients are determined by a least squares (LS) algorithm. The proposed approach has better identification precision than the traditional assumption of “short time invariant”. To enhance the robustness and stability, the problem of parameter identification is solved by means of constrained optimization in the global identification strategy when the noise level increases. The validity and accuracy are verified by applying the method to a single degree of freedom (SDOF) numerical example, and a qualitative analysis on the selection of the window size is carried out in this research.

On Some Sufficiency-Type Global Stability Results for Time-Varying Dynamic Systems with State-Dependent Parameterizations

This paper formulates sufficiency-type global stability and asymptotic stability results for, in general, nonlinear time-varying dynamic systems with state-trajectory solution-dependent parameterizations. The stability proofs are based on obtaining sufficiency-type conditions which guarantee that either the norms of the solution trajectory or alternative interval-type integrals of the matrix of dynamics of the higher-order than linear terms do not grow faster than their available supremum on the preceding time intervals. Some extensions are also given based on the use of a truncated Taylor series expansion of chosen truncation order with multiargument integral remainder for the dynamics of the differential system.

Unmanned Powered Paraglider Flight Path Control Based on PID Neutral Network

The unmanned powered paraglider (UPP) is a nonlinear time-varying dynamic system with high coupling degree. In view of how to make the UPP flying along arbitrary predetermined trajectory, the conventional control method is difficult to obtain the control trajectory with a better fitting degree of the presupposition trajectory. To solve this problem, the six degree of freedom (SDOF)dynamic model of the UPP is built and the control characteristics of the UPP is studied. In view of the control characteristics of the UPP, the controller suitable for the UPP is designed and compiled with the PID neural network method (PIDNN), and the simulation system for the flight process is built, which is used to simulate the flight path control. And the results show that the controller realizes the effective flight path control of the UPP.

Optimal Control Strategy of Multi-Stage Pharmaceutical

Abstract The fermentation production process is the basis for ensuring product quality in pharmaceutical production. There are many variable parameters such as temperature, number of revolutions and PH value in the fermentation production process, therefore, it is difficult to describe the complex problems through precise mathematical models. Since the process has to go through four different steps, each of which is a nonlinear time-varying dynamic system, the four steps constitute a large dynamic system, and the global modeling accuracy cannot be guaranteed; and the optimization problem of the nonlinear time-varying dynamic system is also a problem. In order to solve the above problems, the paper proposes to use the BP (Back Propagation）neural network modeling method to screen the input variables of the system with large effect on the basis of the actual production data of a pharmaceutical factory. Then, based on the process parameters of the fermentation and the output of the product quality index, a multi-stage mapping model based on LSSVM (Least Squares Support Vector Machine) was established. Finally, the PSO (Particle Swarm Optimization) algorithm is used to optimize the parameters of the model.The simulation results show that the model obtained by parameter optimization can achieve better prediction results.

Adaptive Recursive Least Squares Algorithm based on Echo State Neural Network

In order to model time-varying dynamical systems with more accuracy, and to further exploit the potential capacities of recurrent neural networks, we propose a novel recursive least square (RLS) algorithm based on echo state network (ESN), and note it as RLSESN in this paper. ESN is a new paradigm for using recurrent neural networks (RNN) with a simpler training method. RLSESN consists of three main components: an ESN, a recursive least square (RLS) algorithm with adaptive forgetting factor and a change detection module. At first, the change detection module modifies the forgetting factor online according to ESN output errors. And then, the RLS algorithm regulates the ESN output connection weights. The simulation experiment results show that the proposed ESN-based filters can model nonlinear time-varying dynamical systems very well; the modeling performances are significantly better than those autoregressive moving average (ARMA) model based filters.

Second-Order Switching Time Optimization for Nonlinear Time-Varying Dynamic Systems

This technical note gives a method for calculating the first and second derivatives of a cost function with respect to switching times for systems with piecewise second-differentiable dynamics. Differential equations governing the linear and bilinear operators required for calculating the derivatives are presented. Example optimizations of linear and nonlinear systems are presented as evidence for the value of second-order optimization methods. One example converges in 33 iterations using second-order methods whereas the first-order algorithm requires over 30 000 iterations.

Parametric Estimation for Delayed Nonlinear Time-Varying Dynamical Systems

A constructive algorithm is proposed and illustrated for parametric estimation in delayed nonlinear time-varying dynamic system models from available data. The algorithm uses Chebyshev spectral collocation and optimization. The problems addressed are estimations with complete state vector and incomplete state vector availability. Using an equivalent algebraic description of dynamical systems by Chebyshev spectral collocation and data, a standard least-squares residual cost function is set up for complete and incomplete information cases. Minimization of this cost yields the unique solution for the unknown parameters for estimation with complete state availability, only owing to the fact that the cost function is quadratic and positive definite. Such arguments cannot be made for estimation with incomplete state availability as the cost function is positive definite albeit a nonlinear function of the unknown parameters and states. All the algorithms are presented stepwise and are illustrated using suitable examples.

A development of signal control system for congestion length along arterials

The signal control system of the congestion length is described by a non-linear time-varying discrete dynamic system and synthesised by using the feedback control based on the volume balance at each signalised junction. Two signal control algorithms are presented to balance congestion lengths which cross each other on a road; one is at single signalised junction, the other along arterials. These control algorithms are such that the sum of congestion lengths along each approach is minimised by using parameter optimisation methods. The signal control system is developed systematically according to the procedure ranging from the road analysis to the simulation. From the comparison of congestion lengths between measurement values controlled by a pattern selection method and simulation values controlled by the proposed signal control algorithms, it is confirmed that the signal control system and its development procedure work effectively to control congestion lengths along an arterial in Hiroshima city, Japan.

Spectral Collocation-Based Optimization in Parameter Estimation for Nonlinear Time-Varying Dynamical Systems

A constructive optimization algorithm using Chebyshev spectral collocation and quadratic programming is proposed for unknown parameter estimation in nonlinear time-varying dynamic system models to be constructed from available data. The parameters to be estimated are assumed to be identifiable from the data, which also implies that the assumed system models with known parameter values have a unique solution corresponding to every initial condition and parameter set. The nonlinear terms in the dynamic system models are assumed to have a known form, and the models are assumed to be parameter affine. Using an equivalent algebraic description of dynamical systems by Chebyshev spectral collocation and data, a residual quadratic cost is set up, which is a function of unknown parameters only. The minimization of this cost yields the unique solution for the unknown parameters since the models are assumed to have a unique solution for a particular parameter set. An efficient algorithm is presented stepwise and is illustrated using suitable examples. The case of parameter estimation with incomplete or partial data availability is also illustrated with an example.

CONTROL OF TRAFFIC FLOW DYNAMICS IN URBAN ROAD NETWORKS

This paper studies a control of traffic flow dynamics in urban road networks. The signal control system of the congestion length is described by a nonlinear time-varying discrete dynamic system and synthesized by using the hierarchical control. A signal control algorithm in which the three signal control parameters consisting of the cycle length, green split and offset are searched systematically so as to minimize the sum of congestion lengths in the urban road network is presented. The dynamic route guidance system is consisted of both the evaluation for the mean OD travel time and the route search algorithm. The traffic flow control system, which is most effective to reduce the congestion lengths in urban road networks, is designed by connecting the signal control system with the dynamic route guidance system in real time.

FORMAL LINEARIZATION OF NONLINEAR TIME-VARYING DYNAMIC SYSTEMS USING CHEBYSHEV AND LAGUERRE POLYNOMIALS

This paper is concerned with a formal linearization problem for a general class of nonlinear time-varying dynamic systems. To a given system, a linearization function is made up of Chebyshev polynomials about its state variables. The nonlinear time-varying system is transformed into a linear time-varying system in terms of the linearization function using Chebyshev interpolation to state variables and Laguerre expansion to time variable. An error bound formula of this linearization which is derived in this paper explains that the accuracy of this algorithm is improved as the order of Chebyshev and Laguerre polynomials increases. As its application, a nonlinear observer is designed to demonstrate the usefulness of this formal linearization approach.

Congestion Control System in Urban Road Networks

This paper studies a signal control system and a signal control algorithm according to the variation of the traffic demand in a two-way traffic network. The signal control system of the congestion length is described by a nonlinear time-varying discrete dynamic system and synthesized by using the feedback control based on the volume balance at each signalized intersection. A network control algorithm in which the three signal control parameters consisting of the cycle length, green split and offset are searched systematically so as to minimize the sum of congestion lengths in the traffic network is presented. From the comparison of congestion lengths and signal control parameters between measurement values and simulation values, it is confirmed that the signal control system and the network control algorithm work effectively to minimize the congestion lengths in the traffic network.

Optimization of Signal Control Parameters for Two-Way Traffic Arterials

This paper studies a signal control method which controls congestion lengths on two-way traffic arterials systematically. The signal control system of the congestion length is described by a nonlinear time-varying discrete dynamic system and synthesized by using the feedback control. A balance control algorithm in which the three signal control parameters consisting of the cycle length, green split and offset are searched systematically so as to minimize the sum of congestion lengths on the arterial is presented. From the comparison between measurement values and simulation values, it is confirmed that the balance control algorithm works effectively to reduce the congestion lengths on the arterial

Nonlinear dynamics of milling processes

In this article, dynamics and stability of milling operations with cylindrical end mills are investigated. A unified–mechanics–based model, which allows for both regenerative effects and loss–of–contact effects, is presented for study of partial–immersion, high–immersion and slotting operations. Reduced–order models that can be used for certain milling operations such as full–immersion operations and finishing cuts are also presented. On the basis of these models, the loss of stability of periodic motions of the workpiece–tool system is assessed by using Poincare sections and the numerical predictions of stable and unstable motions are found to correlate well with the corresponding experimental observations. Bifurcations experienced by periodic motions of the workpiece–tool system with respect to quasi–static variation of parameters such as axial depth of cut are examined and discussed. For partial–immersion operations, consideration of both time–delay effects and loss–of–contact effects is shown to have a significant influence on the structure of the stability boundaries in the space of spindle speed and axial depth of cut. The sensitivity of system dynamics to multiple–regenerative effects, mode–coupling effects and feed rate is also discussed.

10.1162/jmlr.2003.4.7-8.1411

Blind Source Recovery (BSR) denotes recovery of original sources/signals from environments that may include convolution, temporal variation, and even nonlinearity. It also infers the recovery of sources even in the absence of precise environment identifiability. This paper describes, in a comprehensive fashion, a generalized BSR formulation achieved by the application of stochastic optimization principles to the Kullback-Liebler divergence as a performance functional subject to the constraints of the general (i.e., nonlinear and time-varying) state space representation. This technique is used to derive update laws for nonlinear time-varying dynamical systems, which are subsequently specialized to time-invariant and linear systems. Further, the state space demixing network structures have been exploited to develop learning rules, capable of handling most filtering paradigms, which can be conveniently extended to nonlinear models. In the special cases, distinct linear state-space algorithms are presented for the minimum phase and non-minimum phase mixing environment models. Conventional (FIR/IIR) filtering models are subsequently derived from this general structure and are compared with material in the recent literature. Illustrative simulation examples are presented to demonstrate the online adaptation capabilities of the developed algorithms. Some of this reported work has also been implemented in dedicated hardware/software platforms.

Controllability of non-linear systems with distributed delays in control

Relative controllability of a class of non-linear time-varying dynamical systems with distributed delays in the control is considered. Using Schauder's fixed-point theorem, sufficient conditions for global and local relative controllability are formulated. These conditions extend some previous results by considering a more general class of dynamical systems.