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

2007 ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Parameter Estimation ◽  
Dynamical Systems ◽  
Dynamic System ◽  
Unique Solution ◽  
Time Varying ◽  
Unknown Parameters ◽  
Spectral Collocation ◽  
System Models ◽  
Chebyshev Spectral Collocation ◽  
Set Up

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 step-wise and is illustrated using suitable examples.

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

2007 ◽  
Vol 3 (1) ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Parameter Estimation ◽  
Dynamical Systems ◽  
Dynamic System ◽  
Unique Solution ◽  
Data Availability ◽  
Time Varying ◽  
Unknown Parameters ◽  
Spectral Collocation ◽  
System Models ◽  
Chebyshev Spectral Collocation

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.

Parametric Estimation for Delayed Nonlinear Time-Varying Dynamical Systems

2011 ◽  
Vol 6 (4) ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Dynamical Systems ◽  
Cost Function ◽  
State Vector ◽  
Positive Definite ◽  
Parametric Estimation ◽  
Time Varying ◽  
Unknown Parameters ◽  
Chebyshev Spectral Collocation ◽  
Complete State ◽  
The Cost

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.

scholarly journals Using Modified Intelligent Experimental Design in Parameter Estimation of Chaotic Systems

Complexity ◽  
2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Zahra Shourgashti ◽  
Hamid Keshvari ◽  
Shirin Panahi
Keyword(s):  
Parameter Estimation ◽  
System Identification ◽  
Dynamical Systems ◽  
Experimental Design ◽  
Chaotic Systems ◽  
Initial Condition ◽  
Sampled Data ◽  
One Dimensional ◽  
Using Data ◽  
Set Up

Computational modeling plays an important role in prediction and optimization of real systems and processes. Models usually have some parameters which should be set up to the proper value. Therefore, parameter estimation is known as an important part of the modeling and system identification. It usually refers to the process of using sampled data to estimate the optimum values of parameters. The accuracy of model can be increased by adjusting its parameters to the optimum value which need a richer dataset. One simple solution for having a richer dataset is increasing the amount of data, but that can be costly and time consuming. When using data from animals or people, it is especially important to have a proper plan. There are several available methods for parameter estimation in dynamical systems; however there are some basic differences in chaotic systems due to their sensitivity to initial condition (butterfly effect). Accordingly, in this paper, a new cost function which is proper for chaotic systems is applied to the chaotic one-dimensional map. Then the efficiency of a newly introduced intelligent method experimental design in extracting proper data is investigated. The results show the success of the proposed method.

scholarly journals Parameters Identification and Synchronization of Chaotic Delayed Systems Containing Uncertainties and Time-Varying Delay

2010 ◽  
Vol 2010 ◽  
pp. 1-15 ◽  
Author(s):  
Zhongkui Sun ◽  
Xiaoli Yang
Keyword(s):  
Dynamical Systems ◽  
Functional Differential Equations ◽  
Parameters Identification ◽  
Time Varying ◽  
Unknown Parameters ◽  
Delayed Systems ◽  
Adaptive Scheme ◽  
Time Varying Delay ◽  
Functional Differential ◽  
Varying Delay

Time delays are ubiquitous in real world and are often sources of complex behaviors of dynamical systems. This paper addresses the problem of parameters identification and synchronization of uncertain chaotic delayed systems subject to time-varying delay. Firstly, a novel and systematic adaptive scheme of synchronization is proposed for delayed dynamical systems containing uncertainties based on Razumikhin condition and extended invariance principle for functional differential equations. Then, the proposed adaptive scheme is used to estimate the unknown parameters of nonlinear delayed systems from time series, and a sufficient condition is given by virtue of this scheme. The delayed system under consideration is a very generic one that includes almost all well-known delayed systems (neural network, complex networks, etc.). Two classical examples are used to demonstrate the effectiveness of the proposed adaptive scheme.

On the Theory of a Nonlinear Dynamic Circuit Filtering

2020 ◽  
Vol 19 (03) ◽  
pp. 2050022
Author(s):  
Dhruvi S. Bhatt ◽  
Shaival H. Nagarsheth ◽  
Shambhu N. Sharma
Keyword(s):  
Dynamical Systems ◽  
Stochastic Systems ◽  
Observation Equation ◽  
Time Varying ◽  
Physical Systems ◽  
Random Forcing ◽  
Potential Case ◽  
Set Up ◽  
Nonlinear Sampling ◽  
Nonlinear Observation

Stochastic Differential Equations (SDEs) describe physical systems to account for random forcing terms in the evolution of the state trajectory. The noisy sampling mixer, a component of digital wireless communications, can be regarded as a potential case from the dynamical systems’ viewpoint. The universality of the noisy sampling mixer is attributed to the fact that it adopts the structure of a nonlinear SDE and its linearized version becomes a time-varying bilinear SDE. This paper develops a mathematical theory for the nonlinear noisy sampling mixer from the filtering viewpoint. Since the filtering of stochastic systems hinges on the structure of dynamical systems and observation equation set up, we consider three ‘filtering models’. The first model, accounts for a nonlinear SDE coupled with a nonlinear observation equation. In the second model, we consider a bilinear SDE with a linear observation equation to achieve the nonlinear sampling filtering. Note that the bilinear SDE coupled with the linear observation is a consequence of the Carleman linearization to the nonlinear SDE and the nonlinear observation equation. In the third model, we consider a Stratonovich SDE coupled with a nonlinear observation equation. The filtering equation of this paper can be further utilized to guide the design process of the noisy sampling mixer.

scholarly journals Set-membership identifiability of nonlinear models and related parameter estimation properties

2016 ◽  
Vol 26 (4) ◽  
pp. 803-813 ◽  
Author(s):  
Carine Jauberthie ◽  
Louise Travé-MassuyèEs ◽  
Nathalie Verdière
Keyword(s):  
Mathematical Model ◽  
Parameter Estimation ◽  
Dynamic System ◽  
Nonlinear Models ◽  
Differential Algebra ◽  
Related Parameter ◽  
Set Membership ◽  
Estimation Problems ◽  
The Mathematical Model

Abstract Identifiability guarantees that the mathematical model of a dynamic system is well defined in the sense that it maps unambiguously its parameters to the output trajectories. This paper casts identifiability in a set-membership (SM) framework and relates recently introduced properties, namely, SM-identifiability, μ-SM-identifiability, and ε-SM-identifiability, to the properties of parameter estimation problems. Soundness and ε-consistency are proposed to characterize these problems and the solution returned by the algorithm used to solve them. This paper also contributes by carefully motivating and comparing SM-identifiability, μ-SM-identifiability and ε-SM-identifiability with related properties found in the literature, and by providing a method based on differential algebra to check these properties.

scholarly journals Dynamical system of the growth of COVID-19 with controller

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Rabha W. Ibrahim ◽  
Dania Altulea ◽  
Rafida M. Elobaid
Keyword(s):  
Dynamical System ◽  
Optimal Control ◽  
Growth Rate ◽  
Dynamic System ◽  
Unique Solution ◽  
Population Dynamic ◽  
Integral Control ◽  
The Social ◽  
Social Separation ◽  
Strong Control

AbstractRecently, various studied were presented to describe the population dynamic of covid-19. In this effort, we aim to introduce a different vitalization of the growth by using a controller term. Our method is based on the concept of conformable calculus, which involves this term. We investigate a system of coupled differential equations, which contains the dynamics of the diffusion among infected and asymptomatic characters. Strong control is considered due to the social separation. The result is consequently associated with a macroscopic law for the population. This dynamic system is useful to recognize the behavior of the growth rate of the infection and to confirm if its control is correctly functioning. A unique solution is studied under self-mapping properties. The periodicity of the solution is examined by using integral control and the optimal control is discussed in the sequel.

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