Piecewise Linear Dynamical Model Research Articles

Tracking the trajectory of an object in a noisy environment with unknown statistics: A novel robust Kalman filter residue-based approach

A novel robust Kalman filter (KF)-based controller design approach is proposed to accurately track a specified trajectory under unknown stochastic disturbance, known deterministic disturbance, and measurement noise. The system is nonlinear and is approximated by a piecewise-linear dynamic model. The Box–Jenkins model is an augmented model of the signal and the disturbance, is non-controllable and observable, while the signal model is controllable and observable. An emulator-based two-stage identification is employed to obtain an accurate system model needed to design the robust controller. The system and its KF are identified, and the signal and output errors are estimated. From the identified models, the signal, its KF, the disturbance model, and the whitening filter are all obtained using balanced model reduction techniques. It is shown that the signal model is a transfer matrix description relating the system output and KF residual, and the residual is the whitened output error. The disturbance model is identified by inverse filtering. A new combined feedforward–feedback controller is designed and implemented using an internal model of the reference driven by both the error between the reference and the signal estimate, and by the feedforwarded reference signal. The proposed scheme was successfully evaluated on a simulated autonomously guided drone.

A Dynamical Model for Financial Market: Among Common Market Strategies Who and How Moves the Price to Fluctuate, Inflate, and Burst?

A piecewise linear dynamical model is proposed for a stock price. The model considers the price is driven by three rather standard demand components: chartist, fundamental and market makers. The chartist demand component is related to the study of differences between moving averages. This generates a high order system characterized by a piecewise linear map not trivial to study. The model has been studied analytically in its fixed points and dynamics and then numerically. Results are in line with the related literature: the fundamental demand component helps the stability of the system and keeps prices bounded; market makers satisfy their role of restoring stability, while the chartist demand component produces irregularity and chaos. However, in some cases, the chartist demand component assumes the role to compensate the fundamental demand component, felt in an autogenerated loop, and pushes the dynamics to equilibrium. This fact suggests that the instability must not be searched into the nature of the different investment styles rather in the relative proportion of the contribution of market actors.

Dynamic and stability analysis of the vibratory feeder and parts considering interactions in the hop and the hop-sliding regimes

In this paper, the interactions of a translational vibratory feeder and the parts in the hop and the hop-sliding regimes are studied by means of an improved multi-term incremental harmonic balance method. It is an effective approach analyzing the interactions by introducing an analytical model of the motion of the feeding parts to the solution procedure. A generalized time-varying piece-wise linear dynamic model of the vibratory feeder is established to conduct a comprehensive investigation on the interactions, where the friction and the impact from the parts are included. The results indicate the dynamic response of the vibratory feeder affects the motion of the parts largely and the motion of the parts also affects the dynamic response in turn. The influences of the mass of the parts, the vibration angle, the installation angle, and the friction coefficient on the interactions of the vibratory feeder and the parts are discussed. The interactions are very important and not ignored.

Bursting oscillations as well as the mechanism with codimension-1 non-smooth bifurcation

The coupling of different scales in nonlinear systems may lead to some special dynamical phenomena, which always behaves in the combination between large-amplitude oscillations and small-amplitude oscillations, namely bursting oscillations. Up to now, most of therelevant reports have focused on the smooth dynamical systems. However, the coupling of different scales in non-smooth systems may lead to more complicated forms of bursting oscillations because of the existences of different types of non-conventional bifurcations in non-smooth systems. The main purpose of the paper is to explore the coupling effects of multiple scales in non-smooth dynamical systems with non-conventional bifurcations which may occur at the non-smooth boundaries. According to the typical generalized Chua's electrical circuit which contains two non-smooth boundaries, we establish a four-dimensional piecewise-linear dynamical model with different scales in frequency domain. In the model, we introduce a periodically changed current source as well as a capacity for controlling. We select suitable parameter values such that an order gap exists between the exciting frequency and the natural frequency. The state space is divided into several regions in which different types of equilibrium points of the fast sub-system can be observed. By employing the generalized Clarke derivative, different forms of non-smooth bifurcations as well as the conditions are derived when the trajectory passes across the non-smooth boundaries. The case of codimension-1 non-conventional bifurcation is taken for example to investigate the effects of multiple scales on the dynamics of the system. Periodic bursting oscillations can be observed in which codimension-1 bifurcation causes the transitions between the quiescent states and the spiking states. The structure analysis of the attractor points out that the trajectory can be divided into three segments located in different regions. The theoretical period of the movement as well as the amplitudes of the spiking oscillations is derived accordingly, which agrees well with the numerical result. Based on the envelope analysis, the mechanism of the bursting oscillations is presented, which reveals the characteristics of the quiescent states and the repetitive spiking oscillations. Furthermore, unlike the fold bifurcations which may lead to jumping phenomena between two different equilibrium points of the system, the non-smooth fold bifurcation may cause the jumping phenomenon between two equilibrium points located in two regions divided by the non-smooth boundaries. When the trajectory of the system passes across the non-smooth boundaries, non-smooth fold bifurcations may cause the system to tend to different equilibrium points, corresponding to the transitions between quiescent states and spiking states, which may lead to the bursting oscillations.

Piecewise Linear Dynamical Model for Action Clustering from Real-World Deployments of Inertial Body Sensors

Human motion has been reported as having great relevance to various disease, disorder, injuries and emotional state. Therefore, motion assessment using inertial body sensor networks (BSNs) is gaining popularity as an outcome measure in clinical study and neuroscience research. The efficacy of motion assessment heavily relies on the accurate temporal clustering of human motion into actions on various time scales. However, two human factors in real-world deployments of inertial BSNs make such motion assessment challenging: mounting errors (where sensor displacement and orientation do not match what is assumed by processing algorithms) and insecure mounting (where sensors are loosely worn causing them to shake during operations). In order to enhance the robustness of human actions clutsering from real-world BSN data, this work leverages dynamical systems modeling with the considerations of human factors. By proposing a computational body-model framework called the piecewise linear dynamical model (PLDM), we derive a robust method to segment time series data of inertial BSNs in real-world deployment with human factors into motion primitives and actions. We test the proposed method on three different inertial BSN datasets, extract actions on different temporal scales and recognize the actions into clusters. The experimental results demonstrate the effectiveness of our approach.

Stability Analysis of Periodic Orbits in a Class of Duffing-Like Piecewise Linear Vibrators

In this paper, we study the dynamical behavior of a Duffing-like piecewise linear (PWL) spring- mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. From this PWL model, numerical simulations are carried out by computing frequency response and bifurcation diagram under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Fillipov method.

Control and Dynamics of Quarter-Car Models With Dual-Rate Damping

Dynamics of a controlled quarter-car model is investigated. In this model, the damping coefficient of the car suspension is selected in a way that the resulting semiactive system approximates the performance of an active suspension system designed to produce sky-hook damping. According to this control strategy, the damping coefficient switches between two different values, leading to a piecewise linear dynamical model. For this model, the equation of motion is first presented in a general normalized form. Then, an appropriate methodology is applied for obtaining exact periodic motions for the case of forcing resulting from a road with harmonic profile. This methodology is based on employing the exact solution form within response intervals where the damping coefficient remains constant. The unknowns of the problem are then determined by imposing a set of periodicity and matching conditions. The stability analysis of the located motions is also performed by applying a method that is suitable for piecewise linear systems. Next, this analysis is applied and representative numerical results are presented in the form of response diagrams, illustrating the effect of the important system parameters. Finally, results obtained by direct integration of the equation of motion are also presented for transient road excitation. All the results are compared to those obtained for the conventional suspension systems including passive bilinear shock absorbers.

Dynamically scheduled MPC of nonlinear processes using hinging hyperplane models

AbstractThis article deals with the issues associated with designing scheduled model predictive controllers for nonlinear systems within the multiple‐linear‐model‐based control framework. The issues of model set generation from empirical data and closed‐loop application of the generated model set are considered. A method of hinging hyperplanes is proposed as a way to construct a piecewise linear dynamic model, conducive to dynamic scheduling of linear MFC controllers. The design and implementation of dynamically scheduled MFC using a hinge function model are discussed, as well as its advantages. Alternate MFC formulations considered here require more computation, but utilize the hinge function model as a global model. Simulated examples of isothermal CSTRs and a batch fermenter are also presented to illustrate the proposed methodologies.