Asymptotic evaluation of the state and stiffness of the van der Paul oscillator

2019 ◽  
Vol 33 ◽  
pp. 91-99
Author(s):  
Nadiya Zhogoleva ◽  
Volodymyr Shcherbak
Keyword(s):  
Stable Limit Cycle ◽  
Cardiac Activity ◽  
The State ◽  
The Body ◽  
Nonlinear Observer ◽  
Control Problems ◽  
Analytical Mechanics ◽  
Stable Limit ◽  
State Variables ◽  
Model Equations

In many applications of physics, biology, and other sciences, an approach based on the concept of model equations is used as an approximate model of complex nonlinear processes. The basis of this concept is the provision that a small number of characteristic types movements of simple mathematical models inherent in systems gives the key to understanding and exploring a huge number of different phenomena. With this approach it is a priori assumed that the entire physical diverseness can be represented in the form of fairly simple model equations. It is contributes to a qualitative study of complex systems for various physical nature since basic models individually are well studied, their parameters have a physical interpretation. In particular, it is well known that oscillatory motion of various systems with a stable limit cycle can be modeled by a system consisting of one or more coupled van der Pol oscillators. Such systems are widely represented in various technical devices and in the study and modeling of some biological functions of the body, such as cardiac activity, respiration, locomotor activity, etc. It is considered a typical situation for many practical applications of control theory when the complete state vector of the system is unknown and only some of the functions of the state variables -- the outputs of the system are accessible to measurement. Therefore, the problem of determining in real time the state and parameters of such systems based on the results of measuring the output signals are relevant. One of these inverse control problems, namely, the problem of observability and parameter identification of an model oscillatory system is considered in this article. For observation and identification scheme design the method of invariant relations developed in analytical mechanics is used. Its modification in control problems allows us to synthesize additional relationships between known and unknown quantities of a dynamical system that arise during the observed motion. The method does not involve linearization of the original system and is essentially non-linear. The constructed nonlinear observer provides an asymptotic estimation of unknown parameter and velocity of oscillations.

Finding Alternatives and Reduced Formulations for Process-Based Models

2008 ◽  
Vol 16 (1) ◽  
pp. 63-88 ◽  
Author(s):  
Knut Bernhardt
Keyword(s):  
Differential Equations ◽  
Stable Limit Cycle ◽  
Model Systems ◽  
Model Complexity ◽  
Stable Limit ◽  
State Variables ◽  
Predator Prey ◽  
Oscillatory Dynamics ◽  
Process Based Models ◽  
Simple Models

This paper addresses the problem of model complexity commonly arising in constructing and using process-based models with intricate interactions. Apart from complex process details the dynamic behavior of such systems is often limited to a discrete number of typical states. Thus, models reproducing the system's processes in all details are often too complex and over-parameterized. In order to reduce simulation times and to get a better impression of the important mechanisms, simplified formulations are desirable. In this work a data adaptive model reduction scheme that automatically builds simple models from complex ones is proposed. The method can be applied to the transformation and reduction of systems of ordinary differential equations. It consists of a multistep approach using a low dimensional projection of the model data followed by a Genetic Programming/Genetic Algorithm hybrid to evolve new model systems. As the resulting models again consist of differential equations, their process-based interpretation in terms of new state variables becomes possible. Transformations of two simple models with oscillatory dynamics, simulating a mathematical pendulum and predator-prey interactions respectively, serve as introductory examples of the method's application. The resulting equations of force indicate the predator-prey system's equivalence to a nonlinear oscillator. In contrast to the simple pendulum it contains driving and damping forces that produce a stable limit cycle.

scholarly journals Estimation of oscillation velocities of oscillator network

2018 ◽  
Vol 32 ◽  
pp. 182-189
Author(s):  
Volodymyr Shcherbak ◽  
Iryna Dmytryshyn
Keyword(s):  
Dynamical System ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Dynamical System ◽  
Stable Limit Cycle ◽  
Nonlinear Observer ◽  
Stable Limit ◽  
Nonlinear Dynamical ◽  
Biomechanical Models ◽  
Neural Ensembles

The study of the collective behavior of multiscale dynamic processes is currently one of the most urgent problems of nonlinear dynamics. Such systems arise on modelling of many cyclical biological or physical processes. It is of fundamental importance for understanding the basic laws of synchronous dynamics of distributed active subsystems with oscillations, such as neural ensembles, biomechanical models of cardiac or locomotor activity, models of turbulent media, etc. Since the nonlinear oscillations that are observed in such systems have a stable limit cycle , which does not depend on the initial conditions, then a system of interconnected nonlinear oscillators is usually used as a model of multiscale processes. The equations of Lienar type are often used as the main dynamic model of each of these oscillators. In a number of practical control problems of such interconnected oscillators it is necessary to determine the oscillation velocities by known data. This problem is considered as observation problem for nonlinear dynamical system. A new method – a synthesis of invariant relations is used to design a nonlinear observer. The method allows us to represent unknowns as a function of known quantities. The scheme of the construction of invariant relations consists in the expansion of the original dynamical system by equations of some controlled subsystem (integrator). Control in the additional system is used for the synthesis of some relations that are invariant for the extended system and have the attraction property for all of its trajectories. Such relations are considered in observation problems as additional equations for unknown state vector of initial oscillators ensemble. To design the observer, first we introduce a observer for unique oscillator of Lienar type and prove its exponential convergence. This observer is then extended on several coupled Lienar type oscillators. The performance of the proposed method is investigated by numerical simulations.

scholarly journals Synchronization of oscillations for coupled Van der Pol oscillators by output

2018 ◽  
Vol 32 ◽  
pp. 56-66
Author(s):  
Nadiya Zhogoleva ◽  
Volodymyr Shcherbak
Keyword(s):  
Control Theory ◽  
The State ◽  
Nonlinear Observer ◽  
External Control ◽  
Control Law ◽  
State Variables ◽  
Phase Vector ◽  
Van Der Pol ◽  
Practical Applications ◽  
Synchronization Problem

A number of automatic control tasks, in particular, the synchronization of trajectories, the tracking task, control by a reference system are associated with the synthesis of control algorithms for dynamic cascade systems, which are a set of interconnected active subsystems. In this paper, the oscillation synchronization problem is considered for two Van der Pol coupled oscillators. It is assumed that the driven subsystem depends on the external control action, in addition, the phase vector is not fully known. On the first step the solution of the problem of synchronization in the form of state feedback is written. The aim of the work is to find the synchronizing control in the form of feedback on the state estimation. Such a formulation is relevant, since for many practical applications of control theory, a typical situation is when the complete state vector of the system is unknown and only some of the functions of the state variables - the outputs of the system are accessible to measurement. One can try to use the control law obtained from feedback by replacing the state with its estimate obtained by observer - a special dynamical system whose state eventually approaches (asymptotically or exponentially) to the state of the original system. In this case a question arises whether such control will be solving the synchronization problem. In mathematical control theory, in particular for the stabilization problem of dynamical systems, similar questions constitute the content of the known principle of separation. For the observation problem solving the apparatus of the method of synthesis of auxiliary invariant relations for constructing a nonlinear observer was used. In accordance with this approach a nonlinear observer is constructed for the system under consideration, which ensures the exponential estimates of the phase vector. It is further shown that the use in the control law instead of the state of the system of its evaluation under simultaneously solving the problems of observation and synchronization leads to the local solution of the problem under consideration.

Indirect estimation of state variables of industrial reaction-regeneration systems based on computer simulation

2021 ◽  
pp. 41-50
Author(s):  
Hasan A. Nagiyev ◽  
Nyubar A. Guliyeva
Keyword(s):  
Complex Structure ◽  
Stationary Solutions ◽  
The State ◽  
Object Model ◽  
State Variables ◽  
Regeneration System ◽  
Indirect Estimation ◽  
Direct Measurements ◽  
Model Equations ◽  
And Control

A reaction-regeneration system is described, which is a hardware part of industrial production and is characterized by an exceptional feature – pronounced nonlinearity in the form of a plurality of stationary solutions of model differential equations. This feature forces one to resort to engineering solutions that are alternative to direct measurements. The problem of indirect estimation of the components of the state vector of the reaction-regeneration system is considered. The incorrectness of the indirect assessment of the state of such objects on the basis of the theory of Kalman filters is shown. The incorrectness is due to the ambiguity of the mapping of the state space into the space of vectors tangent to the trajectories. An approach based on synchronous simulation in dynamics is proposed, which consists in comparing two evolutions “object – model” with minimization of the mismatch. A technique based on the inclusion of the second derivatives of the state variables into the mismatch function is presented. The methodology of the sensitivity of indirect estimation systems based on maximizing the similarity of the compared evolutions “object – model” in the regime of strict synchronization with respect to external disturbances and control levers is considered. It is shown that the accuracy of the indirect estimation of physically unmeasurable coordinates is largely determined by the mathematical aspects of minimizing the mismatch function, which, due to the multiplicity of solutions to model equations, has a complex structure of the response surface.

Development of a Robust Observer for Constrained Nonlinear Systems

2007 ◽  
Author(s):  
G. A. Kfoury ◽  
N. G. Chalhoub
Keyword(s):  
Combustion Engine ◽  
General Procedure ◽  
Equations Of Motion ◽  
The State ◽  
Nonlinear Observer ◽  
State Variables ◽  
Connecting Rod ◽  
Algebraic Form ◽  
Highly Nonlinear ◽  
Multi Body

The equations of motion for a constrained multi-body system are usually governed by a set of highly nonlinear differential-algebraic (D-A) equations. For nonlinear complex systems, the substitution method cannot be implemented to eliminate the superfluous coordinates. Thus, the differential-algebraic form of the equations of motion has to be retained. For control purposes, the state variables of the system should be available for the computation of the control signals. The current study presents a general procedure for developing a robust nonlinear observer capable of yielding accurate estimates of the state variables for a complex system whose dynamics are governed by a set of D-A equations. To assess the viability of the proposed approach, the multi-body dynamics of a piston/connecting-rod/crankshaft mechanism for a single cylinder internal combustion engine is considered in this study. The equations of motion account for both the rigid and flexible motions of the crank-slider mechanism. The simulation results demonstrate the capability of the proposed observer in accurately estimating all the state variables of the system including the superfluous ones. They illustrate the robustness of the observer to both structured and unstructured uncertainties. Moreover, they demonstrate that the nominal constraint equations are satisfied by the estimated state variables.

scholarly journals Dynamic Control Applied to a Laboratory Antilock Braking System

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
Cuauhtémoc Acosta Lúa ◽  
Bernardino Castillo Toledo ◽  
Stefano Di Gennaro ◽  
Marcela Martinez-Gardea
Keyword(s):  
Dynamic Control ◽  
Difficult Problem ◽  
The State ◽  
Nonlinear Observer ◽  
State Variables ◽  
Nonlinear Controller ◽  
Braking System ◽  
Laboratory Setup ◽  
Antilock Braking System ◽  
Antilock Braking

The control of an antilock braking system is a difficult problem due to the existence of nonlinear dynamics and uncertainties of its characteristics. To overcome these issues, in this work, a dynamic nonlinear controller is proposed, based on a nonlinear observer. To evaluate its performance, this controller has been implemented on an ABS Laboratory setup, representing a quarter car model. The nonlinear observer reconstructs some of the state variables of the setup, assumed not measurable, to establish a fair benchmark for an ABS system of a real automobile. The dynamic controller ensures exponential convergence of the state estimation, as well as robustness with respect to parameter variations.

scholarly journals Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Yongquan Dai ◽  
Yanping Chen
Keyword(s):  
Optimal Control ◽  
Parabolic Equations ◽  
Optimal Control Problems ◽  
Control Variable ◽  
The State ◽  
Control Problems ◽  
Element Approximation ◽  
Semilinear Parabolic Equations ◽  
State Variables ◽  
Semilinear Parabolic

We will investigate the superconvergence for the semidiscrete finite element approximation of distributed convex optimal control problems governed by semilinear parabolic equations. The state and costate are approximated by the piecewise linear functions and the control is approximated by piecewise constant functions. We present the superconvergence analysis for both the control variable and the state variables.

Development of a Robust Nonlinear Observer for a Single-Link Flexible Manipulator

2004 ◽  
Author(s):  
Nabil G. Chalhoub ◽  
Giscard A. Kfoury
Keyword(s):  
Sliding Mode ◽  
Initial Conditions ◽  
The State ◽  
Observer Design ◽  
Nonlinear Observer ◽  
Flexible Manipulator ◽  
State Variables ◽  
State Equations ◽  
On Line ◽  
Simulation Results

Accurate measurements of all the state variables of a given system are often not available due to the high cost of sensors, the lack of space to mount the transducers or the hostile environment in which the sensors must be located. The purpose of this study is to design a robust sliding mode observer that is capable of accurately estimating the state variables of the system in the presence of disturbances and model uncertainties. It should be emphasized that the proposed observer design can handle state equations expressed in the general form. The performance of the nonlinear observer is assessed herein by examining its capability of predicting the rigid and flexible motions of a compliant beam that is connected to a revolute joint. The simulation results demonstrate the ability of the observer in accurately estimating the state variables of the system in the presence of structured uncertainties and under different initial conditions between the observer and the plant. Moreover, they illustrate the deterioration in the performance of the observer when subjected to unstructured uncertainties of the system. Furthermore, the nonlinear observer was successfully implemented to provide on-line estimates of the state variables for two model-based controllers. The simulation results show minimal deterioration in the closed-loop response of the system stemming from the usage of estimated rather than exact state variables in the computation of the control signals.

Body-fixed SLAM with Local Submaps for Planetary Rover

2019 ◽  
Vol 73 (1) ◽  
pp. 149-171
Author(s):  
Bo Zheng ◽  
Zexu Zhang ◽  
Jing Wang ◽  
Feng Chen ◽  
Xiangquan Wei
Keyword(s):  
Computational Complexity ◽  
The State ◽  
The Body ◽  
Feature Points ◽  
State Variables ◽  
Extended Kalman Filtering ◽  
Exploration Process ◽  
Computational Speed ◽  
Slam Algorithm ◽  
Navigation Accuracy

In traditional Simultaneous Localisation and Mapping (SLAM) algorithms based on Extended Kalman Filtering (EKF-SLAM), the uncertainty of state estimation will increase rapidly with the development of the exploration process and the increase of map area. Likewise, the computational complexity of the EKF-SLAM is proportional to the square of the number of feature points contained in the state variables in a single filtering process. A new SLAM algorithm combining the local submaps and the body-fixed coordinates of the rover is presented in this paper. The algorithm can reduce the computational complexity and enhance computational speed in consideration of the processing capability of the onboard computer. Due to the introduction of local submaps, the algorithm represented in this paper is able to reduce the number of feature points contained in the state variables in each single filtering process. Therefore, the algorithm could reduce the computational complexity and improve the computational speed. In addition, rover body-fixed SLAM could improve the navigation accuracy of a rover and decrease the cumulative linearization error by coordinates transformation during the update process, which is shown in the simulation results.

Uniaxial Cyclic Loading of Elastic-Viscoplastic Materials

1979 ◽  
Vol 46 (4) ◽  
pp. 805-810 ◽  
Author(s):  
S. R. Bodner ◽  
I. Partom ◽  
Y. Partom
Keyword(s):  
Cyclic Loading ◽  
Internal State ◽  
Pure Titanium ◽  
Stable Limit Cycle ◽  
Cyclic Creep ◽  
Cyclic Hardening ◽  
Stable Limit ◽  
Commercially Pure Titanium ◽  
State Variables ◽  
Cyclic Relaxation

Elastic-viscoplastic constitutive equations based on two internal state variables are utilized to determine material response for uniaxial cyclic loading conditions. These equations are capable of representing the principal features of cyclic loading behavior including softening upon stress reversal, cyclic hardening or softening, tendency toward a stable limit cycle, cyclic relaxation, and cyclic creep. Calculations were performed for various stress and strain controlled conditions using material constants intended to represent commercially pure titanium and aluminum and OFHC copper. Capabilities and limitations of the analytical formulations are discussed in relation to computed results and corresponding test data.

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