Backstepping-based output regulation of ordinary differential equations cascaded by wave equation with in-domain anti-damping

Output regulation is considered in this paper for ordinary differential equations cascaded by a wave equation, in which both the body equations and the uncontrolled end are subject to disturbances. The disturbances are generated by an exosystem. A backstepping state-feedback regulator is first designed to force the output to track the reference signal. The design is based on solving cascaded regulator equations, and the solvability condition of the equations is characterized in terms of a transfer function and the eigenvalues of the exosystem. An observer-based output-feedback regulator is then designed to solve the output regulation problem. Finally, the regulator tracking performance is illustrated through numerical simulations.

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Copious Closed Forms of Solutions for the Fractional Nonlinear Longitudinal Strain Wave Equation in Microstructured Solids

Mathematical Problems in Engineering ◽

10.1155/2020/3498796 ◽

2020 ◽

Vol 2020 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Haiyong Qin ◽

Mostafa M. A. Khater ◽

Raghda A. M. Attia

Keyword(s):

Wave Equation ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Longitudinal Strain ◽

Rational Functions ◽

Trigonometric Function ◽

Hyperbolic Function ◽

Strain Wave ◽

Integer Order ◽

Wave Solutions

A computational scheme is employed to investigate various types of the solution of the fractional nonlinear longitudinal strain wave equation. The novelty and advantage of the proposed method are illustrated by applying this model. A new fractional definition is used to convert the fractional formula of these equations into integer-order ordinary differential equations. Soliton, rational functions, the trigonometric function, the hyperbolic function, and many other explicit wave solutions are obtained.

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Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

Journal of Applied Mathematics ◽

10.1155/2012/860482 ◽

2012 ◽

Vol 2012 ◽

pp. 1-19 ◽

Cited By ~ 1

Author(s):

N. Wilson ◽

A. H. Kara

Keyword(s):

Differential Equation ◽

Wave Equation ◽

Partial Differential Equations ◽

Differential Equations ◽

Conservation Laws ◽

Ordinary Differential Equations ◽

Wave Equations ◽

Symmetry Algebra ◽

Lie Point Symmetries ◽

Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

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Performance Output Tracking for a One-Dimensional Wave-Heat Cascade System with Unmatched Disturbance

Complexity ◽

10.1155/2019/1067032 ◽

2019 ◽

Vol 2019 ◽

pp. 1-9 ◽

Cited By ~ 1

Author(s):

Yulong Liu

Keyword(s):

Numerical Simulations ◽

Reference Signal ◽

Output Regulation ◽

Error Signal ◽

Cascade System ◽

One Dimensional ◽

Output Tracking ◽

Control Feedback ◽

Output Regulation Problem ◽

Dimensional Wave

The objective of this paper is to solve the performance output regulation problem for a wave-heat cascade system with unmatched disturbance. Applying the series expansion, the auxiliary trajectory for the cascade system is constructed and the unmatched disturbance is rejected. Meanwhile, the controller and observer only based on error signal are designed, and the performance output regulation problem is solved. Under the control feedback, the performance output can track the reference signal, and the regulation error goes to zero asymptotically. Finally, some numerical simulations are presented for illustration.

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