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This research paper discusses the application of iterative learning control (ILC) with initial state learning (ISL) to a class of nonlinear hyperbolic equations, specifically focusing on distributed control of nonlinear wave equations. The chosen scheme for this study is the P-type ILC. The paper discusses the theoretical convergence of the schemes and provides sufficient conditions for guaranteeing convergence. Both linear and nonlinear cases are investigated in this work.
Control of nonlinear Wave equation, control of linear Wave equation, Iterative Learning Control, P-type learning scheme, Initial State Learning.
The Iterative Learning Control (ILC) method, proposed by Arimoto in 1980 [1], is a powerful control strategy inspired by human learning. It leverages previous tracking errors to improve current error correction. In recent years, ILC has gained significant attention in various applications.
Wave equations are second-order partial differential equations commonly used to describe wave phenomena in diverse fields, such as signal transmission, vibrations, fluid dynamics, and more. They play a crucial role in industrial processes and natural phenomena [2,3,4,5].
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Nonlinearity is inherent in many natural phenomena [6,7,8,9,10], making the control of linear and nonlinear wave equations increasingly important.
Several research works have addressed the control of wave equations, both linear and nonlinear [11,12,13,14,15,16,17,18]. For instance, [14] optimized the control of a one-dimensional nonlinear periodic wave equation using the outer function of the state. [15] developed a pointwise control method for a one-dimensional nonlinear wave equation. [16] proposed a distributed feedback control approach for nonlinear wave localization. [17] discussed forced localization of nonlinear waves by feedback control for nonlinear coupled equations.
[18] extended the space-time finite element method for optimal control of nonlinear hyperbolic equations.
ILC has proven effective for handling nonlinear equations as well [19,20,21,22]. In [19], a new ILC algorithm was introduced for repetitive systems with nonlinear stochastic dynamics. [20] proposed an adaptive ILC for systems with input and state constraints. [21] applied closed-loop D-type ILC to singular systems with one-sided Lipschitz nonlinearity. [22] studied a novel adaptive ILC for systems with randomly varying iteration lengths. Some contributions have also focused on applying ILC to Partial Differential Equations (PDEs) [23,24,25,26,27,28,29,30,31,32]. For example, [23] considered P-type and D-type ILC laws for parabolic PDEs using semi-group theory. [24] applied a D-type anticipatory ILC scheme to boundary control of inhomogeneous heat equations. [25] proposed a P-type ILC law for discrete parabolic distributed parameter systems. [26] investigated ILC for mixed hyperbolic-parabolic distributed parameter systems. Additionally, [27] applied P-type ILC to wave equation boundary control, and [28] presented a general inhomogeneous PDE analysis framework in the frequency domain. Flexible structures and manipulators were also subjects of ILC research [29,30,31,32].
However, there are limited contributions on ILC for nonlinear partial differential equations [33,34]. [33] introduced a Newton-type ILC scheme for nonlinear parabolic equations. [34] discussed ILC's application to optimal boundary control of nonlinear hyperbolic distributed systems. To the best of our knowledge, there is no prior research covering the application of ILC with initial state learning to a nonlinear hyperbolic equation for distributed control.
Motivated by the aforementioned gaps, this paper presents a P-type ILC approach for designing a distributed control law to ensure that the actual system output closely tracks a desired trajectory within a finite time interval. The paper provides theorems regarding the convergence of the proposed method in both linear and nonlinear cases.
The nonlinear wave equation under consideration in this study is a special class of wave equations that describe various physical and engineering processes [35,36,37]. These equations are chosen due to their ability to conserve quantities like energy and entropy and their time-reversal invariance.
The specific nonlinear wave equation for this research is given by:
[u_{tt}(t, x) - c(t)u_{xx}(t, x) + d(t)u(t, x) = f(t, x)]
Where:
The nonlinearity is Lipschitz and continuous with respect to its variables (u) and (u_x), meaning that there exists a constant (L) such that:
[|f(u, u_x)| leq L(1 + |u| + |u_x|)]
The functions (c(t)) and (d(t)) are assumed to be non-negative, i.e., (c(t) geq 0) and (d(t) geq 0).
The following assumptions are made for this study:
The functions (c(t)) and (d(t)) satisfy (c(t) geq 0) and (d(t) geq 0) for all (t geq 0).
For the given trajectory, there exists a unique solution (u(t, x)) for system (1) such that:
[u(t, x) = u_d(t, x)]
For the linear problem, we consider the following iterative learning control scheme:
[u^{(k)}(t, x) = u^{(k-1)}(t, x) + a_k(u_d(t, x) - u^{(k-1)}(t, x))]
Where:
The tracking error is defined as:
[e^{(k)}(t, x) = u_d(t, x) - u^{(k)}(t, x)]
The system output is given by:
[y^{(k)}(t, x) = u^{(k)}(t, x)]
If
[|a_k| leq frac{2}{lambda}]
where (lambda) is a constant, then for all (k) and arbitrary initial input (u^{(0)}(t, x)), the open-loop P-type ILC updating law guarantees that:
[|e^{(k)}(t, x)| leq e^{-(2/lambda)(k-1)}|e^{(0)}(t, x)|]
Suppose that the assumptions are satisfied. If
[|a_k| leq frac{2}{lambda}]
then the iterative process of the system is convergent, i.e.,
[lim_{k to infty} e^{(k)}(t, x) = 0]
For the nonlinear case, the iterative learning control scheme is similar to the linear case, but it accounts for the nonlinearities in the system. The scheme is as follows:
[u^{(k)}(t, x) = u^{(k-1)}(t, x) + a_k(u_d(t, x) - u^{(k-1)}(t, x))]
Where all variables are the same as in the linear case, and (f(u^{(k-1)}, u^{(k-1)}_x)) represents the nonlinear function.
Suppose that the assumptions are satisfied. If
[|a_k| leq frac{2}{lambda}]
then the iterative process of the system is convergent, i.e.,
[lim_{k to infty} e^{(k)}(t, x) = 0]
In this section, a numerical simulation is presented to illustrate the effectiveness of the methods. Detailed numerical results, tables, and figures are available in the full paper.
This research paper introduced the iterative learning control (ILC) method with initial state learning (ISL) for the distributed control of a class of nonlinear wave equations. The P-type ILC scheme was employed, and theoretical convergence was demonstrated in both linear and nonlinear cases. The key contributions of this paper include addressing a class of hyperbolic equations, emphasizing distributed control, and discussing the problem with an unfixed initial value. The use of mathematical inequalities, such as Gronwall's inequality, was crucial in proving the convergence of the control tracking error.
Application of The Iterative Learning Control to Nonlinear Wave Equation. (2024, Jan 06). Retrieved from https://studymoose.com/document/application-of-the-iterative-learning-control-to-nonlinear-wave-equation
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