Uniform exponential stability of semi-discrete scheme for observer-based control of 1-D wave equation

Ren, Han‐Jing; Guo, Bao‐Zhu

doi:10.1016/j.sysconle.2022.105346

Cited by 7 publications

(1 citation statement)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is worthwhile to mention that an alternate semi-discretized Finite-Difference based model reduction, based on reducing the order of the system (1) in time and space variables, is reported [14,22]. This type of model reduction, similar to the one by mixed-finite elements [2,3,16], does not need any numerical filtering.…”

Section: Introductionmentioning

confidence: 99%

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Ozer,

Emran

2023

Preprint

View full text Add to dashboard Cite

The one-dimensional PDE model of the wave equation with a state feedback controller at its boundary, which describes wave dynamics of a wide-range of controlled mechanical systems, has exponentially stable solutions. However, it is known that the reduced models of the wave equation by the standard Finite Differences and Finite Elements suffer from the lack of exponential stability (and exact observability without a state feedback controller) uniformly as the discretization parameter tends to zero. This is due to the loss of uniform gap among the high-frequency eigenvalues as the discretization parameter tends to zero. One common remedy to overcome this discrepancy is the direct Fourier filtering of the reduced models, where the high-frequency spurious eigenvalues are filtered out. After filtering, besides from the strong convergency, the exponential decay rate, mimicking the one for the partial differential equation counterpart, can be retained uniformly. However, the existing results in the literature are solely based on an observability inequality of the control-free model, to which the filtering is implemented. Moreover, the decay rate as a function of the filtering parameter is implicit. In this paper, exponential stability results for both filtered Finite Difference and Finite Element reduced models are established directly by a Lyapunov-based approach and a thorough eigenvalue estimation.The maximal decay rate is explicitly provided as a function of the feedback gain and filtering parameter. Our results, expectedly, mimic the ones of the PDE counterpart uniformly as the discretization parameter tends to zero. Several numerical tests are provided to support our results. Mathematics Subject Classification (2010). 35Q60, 35Q93; 74F15; 35Q74, 93B52.

show abstract

Section: Introductionmentioning

confidence: 99%

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Ozer,

Emran

2023

Preprint

View full text Add to dashboard Cite

show abstract

Uniformly exponential stability of semi-discrete scheme for a vibration cable with a tip mass under observer-based feedback control

Guo¹,

Zhao²

2023

ESAIM: COCV

View full text Add to dashboard Cite

In this paper, we consider uniformly exponential approximation for a vibrating cable with tip mass under a non-collocated output stabilizing feedback control. By designing an observer-based output feedback control, the closed-loop system is composed of the coupled same type of PDEs and ODEs. By order reduction method, we find a global Lyapunov functional for the closed-loop system. The closed-loop system is then semi-discretized by the finite difference method. For the discrete systems, we also construct the Lyapunov functions. The uniform exponential stability of the semi-discretized systems is then established analogously as the proof for the continuous counterpart via an indirect Lyapunov functional approach.

show abstract

A novel sensor design for a cantilevered Mead-Marcus-type sandwich beam model by the order-reduction technique

Özer,

Aydin

2023

ESAIM: COCV

View full text Add to dashboard Cite

A novel space-discretized Finite Differences-based model reduction introduced in \cite{Guo3} is extended to the partial differential equations (PDE) model of a multi- layer Mead-Marcus-type sandwich beam with clamped-free boundary conditions. The PDE model describes transverse vibrations for a sandwich beam whose alternating outer elastic layers constrain viscoelastic core layers, which allow transverse shear. The major goal of this project is to design a single tip velocity sensor to control the overall dynamics on the beam. Since the spectrum of the PDE can not be constructed analytically, the so-called multipliers approach is adopted to prove that the PDE model is exactly observable with sub-optimal observation time. Next, the PDE model is reduced by the "order-reduced'' Finite-Differences technique. This method does not require any type of filtering though the exact observability as $h\to 0$ is achieved by a constraint on the material constants. The main challenge here is the strong coupling of the shear dynamics of the middle layer with overall bending dynamics. This complicates the absorption of coupling terms in the discrete energy estimates. This is sharply different from a single-layer (Euler-Bernoulli) beam.

show abstract

Uniform exponential stability of semi-discrete scheme for observer-based control of 1-D wave equation

Cited by 7 publications

References 19 publications

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Revisiting the Direct Fourier Filtering Technique for the Maximal Decay Rate of Boundary-damped Wave Equation by Finite Differences and Finite Elements

Uniformly exponential stability of semi-discrete scheme for a vibration cable with a tip mass under observer-based feedback control

A novel sensor design for a cantilevered Mead-Marcus-type sandwich beam model by the order-reduction technique

Contact Info

Product

Resources

About