Backstepping and sliding modes for observer design of distributed parameter system

2016 ◽  
Vol 40 (2) ◽  
pp. 542-549 ◽  
Author(s):  
Abdessamad Abdelhedi ◽  
Wided Saadi ◽  
Driss Boutat ◽  
Lasaad Sbita
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Exponential Convergence ◽  
Integral Transformation ◽  
Design Method ◽  
Observer Design ◽  
Distributed Parameter ◽  
Sliding Modes ◽  
Partial Differential ◽  
Distributed Parameters

The observer design for partial differential equations has so far been an open problem. In this paper, an observer design for systems with distributed parameters using sliding modes theory and backstepping-like procedure in order to achieve exponential convergence is presented. Such an observer is built using the knowledge available within and throughout an integral transformation of Volterra with the output injection functions. The gains of the observer, which are attained by solving a partial differential equations system with output injection, will guarantee the exponential convergence of the observer. The design method is applied to an epidemic system to consider the sensitive population S.

Gain Scheduling-Inspired Control for Nonlinear Partial Differential Equations

2009 ◽  
Author(s):  
Antranik A. Siranosian ◽  
Miroslav Krstic ◽  
Andrey Smyshlyaev ◽  
Matt Bement
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Design Method ◽  
Nonlinear Partial Differential Equations ◽  
Gain Scheduling ◽  
System Control ◽  
Nonlinear Feedback ◽  
Partial Differential ◽  
System A ◽  
Shear Beam

We present a control design method for nonlinear partial differential equations (PDEs) based on a combination of gain scheduling and backstepping theory for linear PDEs. A benchmark first-order hyperbolic system with a destabilizing in-domain nonlinearity is considered first. For this system a nonlinear feedback law based on gain scheduling is derived explicitly, and a statement of stability is presented for the closed-loop system. Control designs are then presented for a string and shear beam PDE, both with Kelvin-Voigt damping and potentially destabilizing free-end nonlinearities. String and beam simulation results illustrate the merits of the gain scheduling approach over the linearization-based design.

A Lumped-Parameter Modeling Methodology for One-Dimensional Hyperbolic Partial Differential Equations Describing Nonlinear Wave Propagation in Fluids

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
Stephanie Stockar ◽  
Marcello Canova ◽  
Yann Guezennec ◽  
Giorgio Rizzoni
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Compressible Fluid ◽  
Internal Combustion Engines ◽  
System Optimization ◽  
Observer Design ◽  
Nonlinear Wave Propagation ◽  
Partial Differential ◽  
Fluid Systems

Modeling the transient response of compressible fluid systems using dynamic systems theory is relevant to various engineering fields, such as gas pipelines, compressors, or internal combustion engines. Many applications, for instance, real-time simulation tools, system optimization, estimation and control would greatly benefit from the availability of predictive models with high fidelity and low calibration requirements. This paper presents a novel approach for the solution of the nonlinear partial differential equations (PDEs) describing unsteady flows in compressible fluid systems. A systematic methodology is developed to operate model-order reduction of distributed-parameter systems described by hyperbolic PDEs. The result is a low-order dynamic system, in the form of ordinary differential equations (ODEs), which enables one to apply feedback control or observer design techniques. The paper combines an integral representation of the conservation laws with a projection based onto a set of eigenfunctions, which capture and solve the spatially dependent nature of the system separately from its time evolution. The resulting model, being directly derived from the conservation laws, leads to high prediction accuracy and virtually no calibration requirements. The methodology is demonstrated in this paper with reference to classic linear and nonlinear problems for compressible fluids, and validated against analytical solutions.

On Optimum Filtering for a Class of Linear Distributed-Parameter Systems

1969 ◽  
Vol 91 (2) ◽  
pp. 173-178 ◽  
Author(s):  
F. E. Thau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Diffusion Process ◽  
Discrete Time ◽  
Continuous Time ◽  
Time Delays ◽  
Distributed Parameter ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Variance Equation

Filtering equations are derived for processes described by linear partial differential equations with known homogeneous boundary conditions. Both discrete-time and continuous-time measurements are treated. As in the case of linear systems with time delays, the filtering and variance equations become partial differential equations for processes with continuous measurements. A numerical solution to the nonlinear variance equation is obtained for a particular diffusion process.

Solution Non Linear Partial Differential Equations By ZMA Decomposition Method

2021 ◽  
Vol 20 ◽  
pp. 712-716
Author(s):  
Zainab Mohammed Alwan
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Integral Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Adomian Decomposition Method ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Adomian Decomposition

In this survey, viewed integral transformation (IT) combined with Adomian decomposition method (ADM) as ZMA- transform (ZMAT) coupled with (ADM) in which said ZMA decomposition method has been utilized to solve nonlinear partial differential equations (NPDE's).This work is very useful for finding the exact solution of (NPDE's) and this result is more accurate obtained with compared the exact solution obtained in the literature.

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