Uncertain Wave Equation Research Articles

Stabilization involving condition-based adaptive control of an uncertain wave equation for the vibration of a flexible string

This paper investigates the stabilization of an uncertain wave equation which describes the vibration dynamics of a flexible string. All the system parameters are unknown while the disturbance with unknown bound and period is allowed which lead to more serious uncertainties than the related literature, and hence result into the incapability of the traditional schemes. To solve the control problem, a novel condition-based adaptive control framework is developed in the paper. Specifically, as preparation for the adaptive control design, a state-feedback controller is first designed under the assumption that all the system parameters are known. Then, an adaptive controller, together with two continuous adaptive laws and a condition-based updating mechanism for the online tuning of the controller parameters, is proposed for the original uncertain system. Finally, it is proven that the updating of controller parameters under the proposed condition-based updating mechanism stops at a finite time instant, then the proposed controller guarantees the stability of the resulting closed-loop system. The proposed theoretical results are validated by a simulation example.

A numerical method for solving uncertain wave equation

Partial differential equations are usually used to deal with natural or physical problems, and wave equations are an important branch of them. Considering the fluctuation phenomena with uncertain noise in real life, this paper will focus on the study of the uncertain wave equation. The uncertain wave equation is a special kind of partial differential equation excited by the Liu process, which is widely used to describe wave propagation phenomena. However, not all uncertain wave equations can be solved analytically, so numerical solutions are of particular importance. The main objective of this paper is to study the numerical solution of uncertain wave equations. We first introduce the concept of α-path and, based on it, prove an important formula that reveals the connection between the uncertain wave equation and the classical wave equation. Based on this finding, we design a new numerical solution method to solve the uncertain wave equation. To verify the feasibility and accuracy of the method, we conduct numerical experiments. In addition, as an application, we obtain the uncertainty distribution, the inverse uncertainty distribution and the expected value of the solution of the equation based on the important formula. The originality and novelty of the conclusions of this work are demonstrated by comparing the results with those of other researchers.

Existence and Uniqueness Theorem for Uncertain Wave Equation

In the real world, the indeterminate phenomenon and determinate phenomenon are symmetric; however, the indeterminate phenomenon absolutely exists. Hence, the indeterminate dynamic phenomenon is studied in this paper by using uncertainty theory, where the indeterminate dynamic phenomenon is associated with the belief degree and called the uncertain dynamic phenomenon. Based on uncertainty theory, the uncertain wave equation derived by the Liu process is constructed to model the propagation of various types of wave with uncertain disturbance in nature, where the Liu process is Lipschitz-continuous and has stationary and independent increments. First important of all, only the equation has solution which can be used to clearly depict the wave propagation influenced by uncertain disturbance. Therefore, the aims of this paper is to propose and prove a theorem of existence and uniqueness with Lipschitz and linear growth conditions.

Uncertain Wave Equation for Vibrating String

Wave equation is a commonly used tool for describing various kinds of wave phenomena in nature such as sound wave, water wave, electromagnetic wave and string vibration. It is a second-order partial differential equation and describe the wave propagation without noises. However, real world is filled with noises everywhere. So deterministic wave equation is not enough to model some problems with additive noises. As a remedy method, stochastic wave equation driven by Wiener process was presented where the noise is considered random and modeled by using Wiener process. Except for randomness, uncertainty associated belief degrees is another different type of indeterministic phenomenon. For modeling the wave phenomena with uncertain noises, this paper aims at deriving an uncertain wave equation driven by Liu process, which is a type of partial differential equation. Here, Liu process is a Lipschitz continuous uncertain process with stationary and independent increments. Then, we prove the existence and uniqueness of the solution of an uncertain wave equation. Additionally, we give the inverse uncertainty distribution of a solution of an uncertain wave equation.

Stability of solution for uncertain wave equation

The uncertain wave equation is an important type of uncertain partial differential equation driven by Liu process which is a special type of uncertain process with independent and stationary increments. For an uncertain wave equation, it is head for us to get its solution. What is more, even if we get it, we still should to know whether the obtained solution is stable or not. So this paper puts forward the concept of stability of uncertain wave equations in the sense of convergence in uncertain measure. Then we discuss the condition for an uncertain wave equation being stable and prove the stability theorem. In addition, some examples are given to show what is the concept of stability exactly and how to judge an uncertain wave equation being stable.

Uncertain wave equation with infinite half-boundary

Wave equation is a type of second-order and hyperbolic partial differential equation. It is a commonly used tool to model many kinds of wave propagations such as sound wave, electromagnetic wave, water wave and string vibration propagations. Similarly, uncertain wave equation is a type of uncertain partial equation driven by Liu process, which is widely used to model the wave propagation with uncertain noise such as vibrating string in uncertain environment. The existing literature has studied uncertain wave equation with infinite boundary. Since infinite boundary is a much ideal condition, this paper aims at studying the uncertain wave equation with infinite half-boundary.

Distributed Parameter Control Strategies for the Wave Equation with Fractional Order Derivative

In this paper, the control strategies are studied for the fractional order uncertain wave equation subject to persistent external disturbances in Hilbert spaces. The twisting and super-twisting fractional order sliding mode controllers (SMCs) are designed for the infinite dimensional setting and they are applied for addressing the asymptotic state tracking of the fractional order perturbed wave equation. Furthermore, by introducing the adaptive control law to the twisting controller, the bound of the external disturbances which is unknown is dealt with, and for the design of the super-twisting SMC, a fractional order sliding mode manifold is utilized which results in a continuous input control and a chattering free signal. Both of the controllers are associated with the fractional order parameter, which influences the convergence rate of the proposed control algorithms. In addition, the relative theorem involved in the paper for the proof of the stability is proved. Then, the control algorithms are extended to globally asymptotically stabilize the fractional order uncertain wave equation by choosing the appropriate Lyapunov functional. Finally, numerical simulations are presented to verify the viability and efficiency of the proposed fractional order controllers.

On the Lyapunov-based second-order SMC design for some classes of distributed parameter systems

This paper addresses the Lyapunov-based design of second-order sliding mode controllers in the domain of distributed parameter systems (DPSs). To the best of our knowledge, the recent authors’ publications (Orlov et al., 2010, Continuous state-feedback tracking of an uncertain heat diffusion process. Syst. Control Lett., 59, 754–759; Orlov et al., 2011, Exponential stabilization of the uncertain wave equation via distributed dynamic input extension. IEEE Trans. Autom. Control, 56, 212–217; Pisano et al., 2011, Tracking control of the uncertain heat and wave equation via power-fractional and sliding-mode techniques. SIAM J. Control Optim., 49, 363–382) represent the seminal applications of second-order sliding mode control techniques to DPSs. A Lyapunov-based framework of analysis was found to be appropriate in the above publications. While reviewing the main existing results in this new field of investigation, the paper provides the novelty as well and gives several hints and perspectives for the generalization, listing some open problems.

Exponential Stabilization of the Uncertain Wave Equation via Distributed Dynamic Input Extension

The wave equation is studied under the effect of a persistent external perturbation. A dynamical distributed controller is suggested, based on an infinite-dimensional generalization of second-order sliding-mode control techniques, that provides the exponential attainment of a sufficiently smooth arbitrary reference of the state trajectory. The control system comprises of both feedforward and feedback parts, the latter being a discontinuous term directly connected to the plant input through a dynamical filter that augments the system state smoothing out the discontinuity of the feedback control loop. As a result, a continuous input is applied to the plant. A constructive Lyapunov-based proof of convergence of the proposed control algorithm is carried out and supporting numerical results are presented.