scholarly journals Boundary Control for a Kind of Coupled PDE-ODE System

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Yuanting Wang ◽  
Fucheng Liao ◽  
Yonglong Liao ◽  
Zhengwei Shen
Keyword(s):  
Differential Equation ◽  
State Feedback ◽  
Coupled System ◽  
Backstepping Method ◽  
Closed Loop System ◽  
Stabilizing Controller ◽  
Varying Coefficients ◽  
Ode System ◽  
Exponentially Stable ◽  
Spatially Varying

A coupled system of an ordinary differential equation (ODE) and a heat partial differential equation (PDE) with spatially varying coefficients is discussed. By using the PDE backstepping method, the state-feedback stabilizing controller is explicitly constructed with the assumptionsλ(x)∈P[x]nandλ(x)∈C[0, l]∞, respectively. The closed-loop system is proved to be exponentially stable by this controller. A simulation example is presented to illustrate the effectiveness of the proposed method.

Observer-based output feedback control design for a coupled system of fractional ordinary and reaction–diffusion equations

2020 ◽  
Author(s):  
Shadi Amiri ◽  
Mohammad Keyanpour ◽  
Asadollah Asaraii
Keyword(s):  
Differential Equation ◽  
Output Feedback ◽  
Closed Loop ◽  
Output Feedback Control ◽  
Reaction Diffusion ◽  
Coupled System ◽  
Reaction Diffusion Equations ◽  
Backstepping Method ◽  
Neumann Type ◽  
Fractional Reaction

Abstract In this paper, we investigate the stabilization problem of a cascade of a fractional ordinary differential equation (FODE) and a fractional reaction–diffusion (FRD) equation where the interconnections are of Neumann type. We exploit the partial differential equation backstepping method for designing a controller, which guarantees the Mittag–Leffler stability of the FODE-FRD cascade. Moreover, we propose an observer that is Mittag–Leffler convergent. Also, we propose an output feedback boundary controller, and we prove that the closed-loop FODE-FRD system is Mittag–Leffler stable in the sense of the corresponding norm. Finally, numerical simulations are presented to verify the results.

Refined Stability Conditions for Linear Uncertain Systems With Multiple Time-Varying Delays

2006 ◽  
Vol 129 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Chih-Peng Huang
Keyword(s):  
Uncertain Systems ◽  
State Feedback ◽  
Multiple Time ◽  
Time Varying ◽  
Closed Loop System ◽  
Stabilizing Controller ◽  
Time Varying Delay ◽  
The Stability ◽  
Varying Delay ◽  
Explicit Criteria

This paper mainly proposes distinct criteria for the stability analysis and stabilization of linear uncertain systems with time-varying delays. Based on the Lyapunov theorem, a sufficient condition of the unforced systems with single time-varying delay is first derived. By involving a memoryless state feedback controller, the condition will be extended to treat with the resulting closed-loop system. These explicit criteria can be reformulated in LMIs forms, so we will readily verify the stability or design a stabilizing controller by the current LMI solver. Furthermore, the considered systems with multiple time-varying delays are similarly addressed. Numerical examples are given to demonstrate that the proposed approach is effective and valid.

scholarly journals Robust Reliable Stabilization of Switched Nonlinear Systems with Time-Varying Delays and Delayed Switching

2010 ◽  
Vol 2010 ◽  
pp. 1-21 ◽  
Author(s):  
Zhengrong Xiang ◽  
Qingwei Chen
Keyword(s):  
Nonlinear Systems ◽  
State Feedback ◽  
Linear Matrix ◽  
Time Varying ◽  
Switched Nonlinear Systems ◽  
Matrix Inequalities ◽  
Closed Loop System ◽  
Parameter Uncertainties ◽  
Actuator Failure ◽  
Exponentially Stable

This paper is concerned with the problem of robust reliable stabilization of switched nonlinear systems with time-varying delays and delayed switching is investigated. The parameter uncertainties are allowed to be norm-bounded. The switching instants of the controller experience delays with respect to those of the system. The purpose of this problem is to design a reliable state feedback controller such that, for all admissible parameter uncertainties and actuator failure, the system state of the closed-loop system is exponentially stable. We show that the addressed problem can be solved by means of algebraic matrix inequalities. The explicit expression of the desired robust controllers is derived in terms of linear matrix inequalities (LMIs).

Quadratic Lyapunov functions for linear systems

1969 ◽  
Vol 66 (1) ◽  
pp. 115-117
Author(s):  
Y. V. Venkatesh
Keyword(s):  
Differential Equation ◽  
Lyapunov Functions ◽  
Coefficient Matrix ◽  
Quadratic Lyapunov Function ◽  
Varying Coefficients ◽  
Quadratic Lyapunov Functions ◽  
System Trajectory ◽  
Exponentially Stable ◽  
Vector Differential Equation ◽  
Stable Linear

AbstractThe paper deals with the existence of a quadratic Lyapunov function V = x′P(t)x for an exponentially stable linear system with varying coefficients described by the vector differential equation The derivative dV/dt is allowed to be strictly semi-(F) and the locus dV/dt = 0 does not contain any arc of the system trajectory. It is then shown that the coefficient matrix A(t) of the exponentially stable system is not identically equal to a unit matrix multiplied by a scalar. The result subsumes that of Lehnigk(1).

Modeling and Boundary Control of a Hanging Cable Immersed in Water

2013 ◽  
Vol 136 (1) ◽  
Author(s):  
Michael Böhm ◽  
Miroslav Krstic ◽  
Sebastian Küchler ◽  
Oliver Sawodny
Keyword(s):  
Differential Equation ◽  
Wave Equations ◽  
Equations Of Motion ◽  
Distributed Parameter ◽  
Stream Velocity ◽  
Water Stream ◽  
Varying Coefficients ◽  
Feedforward Controller ◽  
Spatially Varying ◽  
Two Degree Of Freedom

A nonlinear distributed parameter system model governing the motion of a cable with an attached payload immersed in water is derived. The payload is subject to a drag force due to a constant water stream velocity. Such a system is found, for example, in deep sea oil exploration, where a crane mounted on a ship is used for construction and thus positioning of underwater parts of an offshore drilling platform. The equations of motion are linearized, resulting in two coupled, one-dimensional wave equations with spatially varying coefficients and dynamic boundary conditions of second order in time. The wave equations model the normal and tangential displacements of cable elements, respectively. A two degree of freedom controller is designed for this system with a Dirichlet input at the boundary opposite to the payload. A feedforward controller is designed by inverting the system using a Taylor-series, which is then truncated. The coupling is ignored for the feedback design, allowing for a separate design for each direction of motion. Transformations are introduced, in order to transform the system into a cascade of a partial differential equation (PDE) and an ordinary differential equation (ODE), and PDE backstepping is applied. Closed-loop stability is proven. This is supported by simulation results for different cable lengths and payload masses. These simulations also illustrate the performance of the feedforward controller.

Coupled Bending, Torsion and Axial Vibrations of a Cable-Harnessed Beam With Periodic Wrapping Pattern

2018 ◽  
Author(s):  
Karthik Yerrapragada ◽  
Armaghan Salehian
Keyword(s):  
Coupled System ◽  
Mode Shapes ◽  
Element Analysis ◽  
Varying Coefficients ◽  
Equivalent Continuum Model ◽  
Spatially Varying ◽  
Analysis Models ◽  
Euler Bernoulli Beam ◽  
Equivalent Continuum ◽  
Fixed Boundary Condition

This paper presents an equivalent continuum model to study the bending-torsion-axial coupled vibrations of a cable-harnessed beam. The pre-tensioned cable is wrapped periodically around the beam in a diagonal manner. The host structure is assumed to behave as a Euler-Bernoulli beam. The system is modeled using energy methods. The diagonal wrapping pattern results in variable coefficient strain and kinetic energies. Homogenization technique is used to convert spatially varying coefficients into a constant coefficient one. Coupled partial differential equations representing the bending, torsion and the axial modes are derived using Hamilton’s principle. The free vibration characteristics such as the natural frequencies and the mode shapes of the coupled system are analyzed for a fixed-fixed boundary condition and compared to results from the uncoupled and finite element analysis models.

scholarly journals Boundary Output Feedback Stabilization for a Cascaded-Wave PDE-ODE System with Velocity Recirculation

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Ruicheng Li ◽  
Feng-Fei Jin ◽  
Baoqiang Yan
Keyword(s):  
Output Feedback ◽  
Original System ◽  
Feedback Stabilization ◽  
Feedback Controller ◽  
Target System ◽  
Closed Loop System ◽  
Output Feedback Stabilization ◽  
Ode System ◽  
Exponentially Stable ◽  
Theoretical Results

This paper considers the output feedback stabilization for a cascaded-wave PDE-ODE system with velocity recirculation by boundary control. First, we choose a well-known exponentially stable system as its target system and find a backstepping transformation to design a state feedback controller for the original system. Second, we attempt to give an output feedback controller for the original system by introducing the observer. The resulting closed-loop system admits a unique solution which is proved to be exponentially stable. Finally, we give some numerical examples to prove the validity for the theoretical results.

Output Feedback Stabilization of the Linearized Bilayer Saint-Venant Model

2016 ◽  
Author(s):  
Ababacar Diagne ◽  
Shuxia Tang ◽  
Mamadou Diagne ◽  
Miroslav Krstic
Keyword(s):  
Output Feedback ◽  
Control Method ◽  
Coupled System ◽  
Backstepping Control ◽  
Feedback Stabilization ◽  
Feedback Controller ◽  
Varying Coefficients ◽  
The One ◽  
Spatially Varying ◽  
Constant System

We consider the problem of output feedback exponentially stabilizing the 1-D bilayer Saint-Venant model, which is a coupled system of two rightward and two leftward convecting transport partial differential equations (PDEs). The PDE backstepping control method is employed. Our designed output feedback controller is based on the observer built in this paper and the state feedback controller designed in [1], where the backstepping control design idea can also be referred to [2] and can be treated as a generalization of the result for the system with constant system coefficients [2] to the one with spatially-varying coefficients. Numerical simulations of the bilayer Saint-Venant problem are also provided to verify the result.

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