Parameter Estimation in a Nonlinear Vibrating System Using an Observer for an Extended System

1999 ◽  
Author(s):  
Anindya Chatterjee ◽  
Joseph P. Cusumano
Keyword(s):  
Parameter Estimation ◽  
Time Scale ◽  
Asymptotic Methods ◽  
System Response ◽  
Parameter Estimates ◽  
Fast Time ◽  
State Variables ◽  
Slow Time ◽  
Unknown Parameters ◽  
Lipschitz Continuous

Abstract We present a new observer-based method for parameter estimation for nonlinear oscillatory mechanical systems where the unknown parameters appear linearly (they may each be multiplied by bounded and Lipschitz continuous but otherwise arbitrary, possibly nonlinear, functions of the oscillatory state variables and time). The oscillations in the system may be periodic, quasiperiodic or chaotic. The method is also applicable to systems where the parameters appear nonlinearly, provided a good initial estimate of the parameter is available. The observer requires measurements of displacements. It estimates velocities on a fast time scale, and the unknown parameters on a slow time scale. The fast and slow time scales are governed by a single small parameter ϵ. Using asymptotic methods including the method of averaging, it is shown that the observer’s estimates of the unknown parameters converge like e−ϵt where t is time, provided the system response is such that the coefficient-functions of the unknown parameters are not close to being linearly dependent. It is also shown that the method is robust in that small errors in the model cause small errors in the parameter estimates. A numerical example is provided to demonstrate the effectiveness of the method.

Asymptotic Parameter Estimation via Implicit Averaging on a Nonlinear Extended System

2003 ◽  
Vol 125 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Anindya Chatterjee ◽  
Joseph P. Cusumano
Keyword(s):  
Parameter Estimation ◽  
Time Scale ◽  
Coefficient Matrix ◽  
Parameter Estimates ◽  
Frequency Noise ◽  
Fast Time ◽  
Model Errors ◽  
Unknown Parameters ◽  
Estimation Errors ◽  
Time Scale Separation

We present an observer for parameter estimation in nonlinear oscillating systems (periodic, quasiperiodic or chaotic). The observer requires measurements of generalized displacements. It estimates generalized velocities on a fast time scale and unknown parameters on a slow time scale, with time scale separation specified by a small parameter ε. Parameter estimates converge asymptotically like e−εt where t is time, provided the data is such that a certain averaged coefficient matrix is positive definite. The method is robust: small model errors and noise cause small estimation errors. The effects of zero mean, high frequency noise can be reduced by faster sampling. Several numerical examples show the effectiveness of the method.

Multi-Parameter Optimization of Damped Linear Continuous Systems

1972 ◽  
Vol 94 (1) ◽  
pp. 1-7 ◽  
Author(s):  
O. B. Dale ◽  
R. Cohen
Keyword(s):  
Optimal Parameter ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
System Response ◽  
Continuous Systems ◽  
Frequency Interval ◽  
State Variables ◽  
Initial Value ◽  
Unknown Parameters ◽  
One Dimensional

A method is presented for obtaining and optimizing the frequency response of one-dimensional damped linear continuous systems. The systems considered are assumed to contain unknown constant parameters in the boundary conditions and equations of motion which the designer can vary to obtain a minimum resonant response in some selected frequency interval. The unknown parameters need not be strictly dissipative nor unconstrained. No analytic solutions, either exact or approximate, are required for the system response and only initial value numerical integrations of the state and adjoint differential equations are required to obtain the optimal parameter set. The combinations of state variables comprising the response and the response locations are arbitrary.

A New Model for Long-Term Stochastic Analysis and Prediction—Part I: Theoretical Background

1992 ◽  
Vol 36 (01) ◽  
pp. 1-16
Author(s):  
G. A. Athanassoulis ◽  
P. B. Vranas ◽  
T. H. Soukissian
Keyword(s):  
Time Scale ◽  
Random Function ◽  
Spectral Characteristics ◽  
Time History ◽  
Time Variable ◽  
Sea Surface ◽  
Surface Elevation ◽  
Fast Time ◽  
Slow Time

A new approach for calculating the long-term statistics of sea waves is proposed. A rational long-term stochastic model is introduced which recognizes that the wave climate at a given site in the ocean consists of a random succession of individual sea states, each sea state possessing its own duration and intensity. This model treats the sea-surface elevation as a random function of a "fast" time variable, and the time history of the spectral characteristics of the successive sea states as a random function of a "slow" time variable. By developing an appropriate conceptual framework, it becomes possible to express various probabilistic characteristics of the sea-surface elevation, which are sensible only in the fast-time scale, in terms of the statistics of sea-states duration and intensity, which is meaningful only in the slow-time scale. As an example, we study the random quantity MU(T) = "number of maxima of the sea-surface elevation lying above the level u and occurring during a long-term time period [0,T]." Exploiting the proposed framework, it is shown that, under certain clearly defined assumptions, Mu(T) can be given the structure of a renewal-reward (cumulative) process, whose interarrival times correspond to the duration of successive sea states. Thus, using renewal theory, the complete characterization of the probability structure of MU(T) is obtained. As a consequence, the long-term probability distribution function of the individual wave height is rigorously defined and calculated. The relation of the present results with corresponding ones previously obtained is thoroughly discussed. The proposed model can be extended twofold: either by replacing some of the simplifying assumptions by more realistic ones, or by extending the model for treating the corresponding problems for ship and structures responses.

Integrating Parameter Estimation Solutions From the Time and Time-Scale Domains

2009 ◽  
Author(s):  
James R. McCusker ◽  
Kourosh Danai
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Time Scale ◽  
Prediction Error ◽  
Nonlinear Least Squares ◽  
Integrated Approach ◽  
Parameter Estimates ◽  
Model Parameters ◽  
Single Model ◽  
Iterative Estimation

A method of parameter estimation was recently introduced that separately estimates each parameter of the dynamic model [1]. In this method, regions coined as parameter signatures, are identified in the time-scale domain wherein the prediction error can be attributed to the error of a single model parameter. Based on these single-parameter associations, individual model parameters can then be estimated for iterative estimation. Relative to nonlinear least squares, the proposed Parameter Signature Isolation Method (PARSIM) has two distinct attributes. One attribute of PARSIM is to leave the estimation of a parameter dormant when a parameter signature cannot be extracted for it. Another attribute is independence from the contour of the prediction error. The first attribute could cause erroneous parameter estimates, when the parameters are not adapted continually. The second attribute, on the other hand, can provide a safeguard against local minima entrapments. These attributes motivate integrating PARSIM with a method, like nonlinear least-squares, that is less prone to dormancy of parameter estimates. The paper demonstrates the merit of the proposed integrated approach in application to a difficult estimation problem.

Two-Stage Design of Linear Feedback Controllers for a Proton Exchange Membrane Fuel Cell

2015 ◽  
Author(s):  
Verica Radisavljevic-Gajic ◽  
Patrick Rose ◽  
Garrett M. Clayton
Keyword(s):  
Fuel Cell ◽  
Time Scale ◽  
Proton Exchange Membrane ◽  
Cell Model ◽  
Pem Fuel Cell ◽  
Proton Exchange ◽  
Linear Feedback ◽  
Fast Time ◽  
State Variables ◽  
Exchange Membrane

The paper considers the eighth-order proton exchange membrane (PEM) fuel-cell mathematical model and shows that it has a multi-time scale property, indicating that the dynamics of three model state space variables operate in the slow time scale and the dynamics of five state variables operate in the fast time scale. This multi-scale nature allows independent controllers to be designed in slow and fast time scales using only corresponding reduced-order slow (of dimension three) and fast (of dimension five) sub-models. The presented design facilitates the design of hybrid controllers, for example, the linear-quadratic optimal controller for the slow subsystem and the eigenvalue assignment controller for the fast subsystem. The design efficiency and its high accuracy are demonstrated via simulation on the considered PEM fuel cell model.

DYNAMICS OF COUPLED DAHL TYPE AND NONSMOOTH SYSTEMS AT DIFFERENT SCALES OF TIME

2013 ◽  
Vol 23 (07) ◽  
pp. 1350114 ◽  
Author(s):  
A. TURE SAVADKOOHI ◽  
C.-H. LAMARQUE
Keyword(s):  
Time Scale ◽  
Coupled Oscillators ◽  
Energy Exchange ◽  
Fast Time ◽  
System Behavior ◽  
Slow Time ◽  
Fast Time Scale ◽  
Nonsmooth Systems ◽  
Main System ◽  
Nonsmooth Potential

Vibratory behavior of two coupled oscillators is studied. The main system — Dahl type — is coupled to a very light system with a nonsmooth potential that can be endowed for passively controlling the main system. Invariant manifold of the system at the fast time scale is revealed and the system behavior at slow time scale around the infinity of the fast time scale is detected. This can give us the chance to forecast all possible attractors of the system during energy exchange between the two oscillators.

The Dance of the Interneurons: How Inhibition Facilitates Fast Compressible and Reversible Learning in Hippocampus

2018 ◽  
Author(s):  
Wilten Nicola ◽  
Claudia Clopath
Keyword(s):  
Time Scale ◽  
State Of The Art ◽  
Fast Time ◽  
Slow Time ◽  
Fast Time Scale ◽  
Slow Time Scale ◽  
Spike Sequences ◽  
Incoming Information ◽  
Spiking Networks ◽  
Inhibitory Networks

AbstractThe hippocampus is capable of rapidly learning incoming information, even if that information is only observed once. Further, this information can be replayed in a compressed format in either forward or reversed modes during Sharp Wave Ripples (SPW-R). We leveraged state-of-the-art techniques in training recurrent spiking networks to demonstrate how primarily inhibitory networks of neurons in CA3 and CA1 can: 1) generate internal theta sequences or “time-cells” to bind externally elicited spikes in the presence of septal inhibition, 2) reversibly compress the learned representation in the form of a SPW-R when septal inhibition is removed, 3) generate and refine gamma-assemblies during SPW-R mediated compression, and 4) regulate the inter-ripple-interval timing between SPW-R’s in ripple clusters. From the fast time scale of neurons to the slow time scale of behaviors, inhibitory networks serve as the scaffolding for one-shot learning by replaying, reversing, refining, and regulating spike sequences.

Low-frequency linear response of a cylindrical tokamak with arbitrary cross-section to ‘helical’ perturbations

1980 ◽  
Vol 24 (2) ◽  
pp. 229-236 ◽  
Author(s):  
Torkil H. Jensen ◽  
Ming S. Chu
Keyword(s):  
Cross Section ◽  
Time Scale ◽  
Low Frequency ◽  
Arbitrary Cross Section ◽  
Fast Time ◽  
Tearing Mode ◽  
Slow Time ◽  
Slow Time Scale ◽  
Stable Plasma

Current driven, ‘helical’ (non-axisymmetric) modes of a tokamak with arbitrary cross-section are considered in the straight (cylindrical) geometry approximation. The plasma is considered surrounded by a resistive wall. The plasma may be unstable on a fast time-scale, namely the MHD or tearing mode time-scale, on a slow time-scale given by the wall properties or it may be stable. The formalism given in this paper allows determination of stability by relatively simple numerical means. In the case of instability on the slow time-scale, the formalism allows determination of growth rates and mode structures. Since the formalism is an eigenvalue formalism with orthogonal eigenfunctions, it is well suited for calculation of the effects on a stable plasma of slow error fields caused by externally driven error currents flowing predominantly in the direction of the ignorable co-ordinate.

Phase transitions as a cause of economic development

2018 ◽  
Vol 51 (3) ◽  
pp. 670-686 ◽  
Author(s):  
David Emanuel Andersson ◽  
Åke E. Andersson
Keyword(s):  
Economic Development ◽  
Phase Transitions ◽  
Time Scale ◽  
Structural Changes ◽  
Fast Time ◽  
Slow Time ◽  
Economic Activities ◽  
The Past ◽  
Location Patterns ◽  
Production Techniques

Economic development spans centuries and continents. Underlying infrastructural causes of development, such as institutions and networks, are subject to slow but persistent change. Accumulated infrastructural changes eventually become so substantial that they trigger a phase transition. Such transitions disrupt the prior conditions for economic activities and network interdependencies, requiring radically transformed production techniques, organizations and location patterns. The interplay of economic equilibria and structural changes requires a theoretical integration of the slow time scale of infrastructural change and the fast time scale of market equilibration. This paper presents a theory that encompasses both rapidly and slowly changing variables and illustrates how infrequent phase transitions caused four logistical revolutions in Europe over the past millennium.

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