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.

The Use of Integral Adaptation Principle to Synthesize Robust Control of Electric Vehicle Wheel Slip

2019 ◽  
Vol 20 (7) ◽  
pp. 412-416 ◽  
Author(s):  
A. A. Kolesnikov ◽  
A. A. Kuz’menko
Keyword(s):  
Robust Control ◽  
Control Theory ◽  
Electric Vehicle ◽  
Estimation Methods ◽  
Control Law ◽  
State Variables ◽  
Wheel Slip ◽  
Braking System ◽  
Antilock Braking System ◽  
Linear And Nonlinear

The usage of "motor-wheel" systems requires the electric vehicle control system improvement by using the characteristics of the wheel adhesion to the road surface. One of the aspects of such improvement is the enhancement of the algorithms for the functioning of the antilock braking system (ABS). In developing the ABS control algorithms, various approaches and methods of modern control theory are used, including methods based on the estimation of wheel slip, traction force, wheel friction coefficient using linear and nonlinear estimation methods, linear and nonlinear regulators. This work illustrates the application of the principle of high order integral adaptation (PIA) of Synergetic Control Theory (SCT) for constructing a robust control law for an electric vehicle wheel slip. The main features of the SCT contain: firstly, a fundamental change in the goals of the behavior of the synthesized systems; secondly, direct consideration of the natural properties of nonlinear objects; thirdly, the formation of an analytical mechanism for generating feedbacks, i.e. control laws. PIA consists in introducing nonlinear integrators into the control law that compensate for disturbances without their immediate estimation. The obtained in this work control law has a fairly simple structure, is focused on using physically accessible state variables of the braking system, and its implementation does not require immediate estimation of disturbances or building a complex neural network to calculate disturbances. The results of computer simulations of the synthesized robust control law for ABS indicate its effectiveness in functioning under conditions of external environment uncertainty.

When Is a Linear Control System Optimal?

1964 ◽  
Vol 86 (1) ◽  
pp. 51-60 ◽  
Author(s):  
R. E. Kalman
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Control Variable ◽  
Point Of View ◽  
Complete Analysis ◽  
Control Law ◽  
Quadratic Loss ◽  
Mathematical Form ◽  
State Variables

The purpose of this paper is to formulate, study, and (in certain cases) resolve the Inverse Problem of Optimal Control Theory, which is the following: Given a control law, find all performance indices for which this control law is optimal. Under the assumptions of (a) linear constant plant, (b) linear constant control law, (c) measurable state variables, (d) quadratic loss functions with constant coefficients, (e) single control variable, we give a complete analysis of this problem and obtain various explicit conditions for the optimality of a given control law. An interesting feature of the analysis is the central role of frequency-domain concepts, which have been ignored in optimal control theory until very recently. The discussion is presented in rigorous mathematical form. The central conclusion is the following (Theorem 6): A stable control law is optimal if and only if the absolute value of the corresponding return difference is at least equal to one at all frequencies. This provides a beautifully simple connecting link between modern control theory and the classical point of view which regards feedback as a means of reducing component variations.

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.

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.

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.

Development of a Robust Controller and Observer for the Control of a Single-Link Flexible Robotic Manipulator

2005 ◽  
Author(s):  
N. G. Chalhoub ◽  
G. A. Kfoury ◽  
B. A. Bazzi
Keyword(s):  
Sliding Mode ◽  
Angular Displacement ◽  
Robotic Manipulator ◽  
The State ◽  
Nonlinear Observer ◽  
State Variables ◽  
Elastic Mode ◽  
Single Link ◽  
Elastic Modes ◽  
Flexible Robotic Manipulator

A fuzzy-sliding mode controller (FSMC) has been developed in this study to control the rigid and flexible motions of a single-link robotic manipulator. Only the angular displacement at the base joint of the beam is assumed to be measured. Therefore, a robust nonlinear observer has been designed, based on the sliding mode methodology, to accurately estimate the state variables in the presence of both structured and unstructured uncertainties. Both the controller and the observer account for the first elastic mode of the beam in their design. The dynamic model, used in assessing the performance of the closed-loop system, considers the first two elastic modes of the beam. The second elastic mode is included in order to investigate the effects of the higher order dynamics on the overall performance of the system. The digital simulations demonstrate the capability of the observer in yielding accurate estimates of the state variables in the presence of modeling inaccuracies. Furthermore, they serve to prove the viability of using the observer to provide on-line estimates of the state variables for the computation of the control signals. The simulation results illustrate robust performances of the controller and the observer in controlling the rigid and flexible motions of the single-link robot in the presence of both structured and unstructured uncertainties.

Non Linear State and Parameters Estimation in Chemical Processes: Analysis and Improvement of Three Estimation Structures Applied to a CSTR

2011 ◽  
Vol 9 (1) ◽  
Author(s):  
Héctor Botero ◽  
Hernán Álvarez
Keyword(s):  
Parameter Estimation ◽  
Chemical Process ◽  
Chemical Processes ◽  
The State ◽  
Convergence Speed ◽  
Parameters Estimation ◽  
State Variables ◽  
Input Output ◽  
On Line ◽  
Linear State

This paper proposes a new composite observer capable of estimating the states and unknown (or changing) parameters of a chemical process, using some input-output measurements, the phenomenological based model and other available knowledge about the process. The proposed composite observer contains a classic observer (CO) to estimate the state variables, an observer-based estimator (OBE) to obtain the actual values of the unknown or changing parameters needed to tune the CO, and an asymptotic observer (AO) to estimate the states needed as input to the OBE. The proposed structure was applied to a CSTR model with three state variables. With the proposed structure, the concentration of reactants and other CSTR parameters can be estimated on-line if the reactor and jacket temperatures are known. The procedure for the design of the proposed structure is simple and guarantees observer convergence. In addition, the convergence speed of state and parameter estimation can be adjusted independently.

scholarly journals A Robust Observer—Based Adaptive Control of Second—Order Systems with Input Saturation via Dead-Zone Lyapunov Functions

Computation ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 82
Author(s):  
Alejandro Rincón ◽  
Gloria M. Restrepo ◽  
Fredy E. Hoyos
Keyword(s):  
Lyapunov Functions ◽  
Dead Zone ◽  
Tracking Error ◽  
Input Saturation ◽  
Chemical Reactor ◽  
Second Order ◽  
Control Law ◽  
Compact Set ◽  
State Variables ◽  
Robust Observer

In this study, a novel robust observer-based adaptive controller was formulated for systems represented by second-order input–output dynamics with unknown second state, and it was applied to concentration tracking in a chemical reactor. By using dead-zone Lyapunov functions and adaptive backstepping method, an improved control law was derived, exhibiting faster response to changes in the output tracking error while avoiding input chattering and providing robustness to uncertain model terms. Moreover, a state observer was formulated for estimating the unknown state. The main contributions with respect to closely related designs are (i) the control law, the update law and the observer equations involve no discontinuous signals; (ii) it is guaranteed that the developed controller leads to the convergence of the tracking error to a compact set whose width is user-defined, and it does not depend on upper bounds of model terms, state variables or disturbances; and (iii) the control law exhibits a fast response to changes in the tracking error, whereas the control effort can be reduced through the controller parameters. Finally, the effectiveness of the developed controller is illustrated by the simulation of concentration tracking in a stirred chemical reactor.

scholarly journals Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs

2021 ◽  
Vol 11 (4) ◽  
pp. 1717
Author(s):  
Gilberto Gonzalez Avalos ◽  
Noe Barrera Gallegos ◽  
Gerardo Ayala-Jaimes ◽  
Aaron Padilla Garcia
Keyword(s):  
Steady State ◽  
Linear Time ◽  
Direct Determination ◽  
The State ◽  
Steady State Response ◽  
State Variables ◽  
Linear Time Invariant ◽  
State Response ◽  
Time Invariant

The direct determination of the steady state response for linear time invariant (LTI) systems modeled by multibond graphs is presented. Firstly, a multiport junction structure of a multibond graph in an integral causality assignment (MBGI) to get the state space of the system is introduced. By assigning a derivative causality to the multiport storage elements, the multibond graph in a derivative causality (MBGD) is proposed. Based on this MBGD, a theorem to obtain the steady state response is presented. Two case studies to get the steady state of the state variables are applied. Both cases are modeled by multibond graphs, and the symbolic determination of the steady state is obtained. The simulation results using the 20-SIM software are numerically verified.

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