Application of the iterative learning control of a non-linear MIMO wave equation.☆
Introduction
The wave equation is the most famous hyperbolic PDE. It is used commonly to define waves such as the transmission of electric signals in a cable, the vibrations of a string, water waves, the propagation of electromagnetic and sound waves. The wave equations governed many industrial processes, such as elastic waves, aeroacoustics, acoustic waves (music acoustics), [4], and many other phenomena in engineering and physics. It is generally held that most natural phenomena are non-linear, refer to [1], [20], [28]. Solving the non-linear estimation and control problems for such systems is fundamental in changing their dynamics. Numerous articles discuss this matter. Nevertheless, it is recognized that controlling the non-linear wave equation is ambitious. For the repetitiveness systems, it exists a relatively new, efficient control technique named iterative learning control.
As a branch of intelligent control, the iterative learning control was proposed by Arimoto in 1984. It is a comparatively recent addition to the control engineering toolkit, refer to [2]. The ILC is based on the acquisition of the knowledge in practice. New capacities and skills can be developed by trails, repetition, and correction, which is the learning mechanism usually used by human beings. It was proved to be a decent method to handle the systems with repetitiveness. The ILC has become an alternative method to solve the control problem since it is simple, quick, efficient, and has a perfect tracking performance. Recently, the ILC has become a popular research topic because of its clear engineering structure in practical applications, such as batch processing, chemical plants, and robotic manipulators, see, [6], [8], [23], [27].
The main renowned advantage of the ILC is the need of only the tracking reference and the input-output information to design the controller, and it works even when the system information is unexplored [26], [34]. Many essays have been published on the different applications of this method. Showing its effectiveness in dealing with the non-linear equation [7], [13], [36]. Besides, some contributions consider the use of the ILC to the partial differential equations, such as [9], [12], [15], [16], [17], [18], [19], [21], [24], [25], [32], [38].
Recently, the use of the ILC in the non-linear wave equation sparked the interest of several scholars [16], and [17]. For the boundary control of the non-linear wave equation, the authors in [16] presents a P-type ILC. In [17], to tackle the output tracking problem of the non-linear wave equations, the author introduces an ILC P-type scheme. Nevertheless, the control problem considered in the above literature is boundary control.
As a type of the ILC, the so-called PD-type, it is based on derivative and proportional, it calculates the error and applies a correction. The D-type algorithm is receptive to the high-frequency interference of the error signal due to the differential action, by adding the P-type control algorithm for this problem can be settled. The maximum error, the tracking accuracy can be decreased and improved by a PD-type ILC algorithm. The PD-type ILC applies to control several non-linear systems [3] and [37]. In [3], the application of the PD-type ILC scheme to non-linear time-delay systems with external disturbances. The author in [37] proves the convergence of the open-closed PD-type ILC algorithm for a class of non-linear systems. While PD-ILC has been applied to several systems governed by the non-linear ODEs, it has lately been used in the hyperbolic equation [14], where, the authors called a PD-type ILC with ILS to control a system characterized by a wave equation. Although, in [14], SISO linear equation is considered.
Latterly, some literature discusses the application of the ILC method in the MIMO linear systems [22], [29], MIMO non-linear systems [5], MIMO parabolic PDE with delay [30], MIMO SDPS [10], and for MIMO second-order hyperbolic DPS with uncertainties [31]. Nevertheless, according to the author’s information, there is no study discussing the application of the ILC to the non-linear MIMO hyperbolic system.
In this work, we discuss the control of the non-linear MIMO hyperbolic PDE using the ILC method. Due to its effectiveness in the non-linear systems and its good performance in [14]. The PD-type ILC with initial state learning is applied to this class of equations. The algorithm convergence condition is presented, and the proof of the effectiveness of the algorithm theoretically is given. Finally, we provide a numerical simulation example showing the efficiency of this algorithm when compared to the other ILC scheme. The main benefactions of this paper are:
- 1.
The system dissected in this manuscript is defined by non-linear MIMO hyperbolic PDE (wave equation).
- 2.
The control rule set out in the present work is a combination of the iterative learning control and the PD-type update rule. The conditions of convergence of the tracking error in the repetition domain are addressed.
- 3.
Instead of assuming that the initial value in each repetition is identical to the desired value or it is fixed as in [9], [12], [21], [32]. In this paper, we discuss the problem with an unfixed initial value. Our initial state also has an iterative learning to update law based on the tracking error and the state.
- 4.
Rather than considering the boundary control case as in [16], [17], [19], this paper studies the distributed control of a system described by a non-linear MIMO wave equation.
- 5.
Habitually, to prove the convergence of a control method, we use the system solution. In our work, we prefer to use some fundamental mathematical inequalities, as Gronwall inequality, and that because of the complexity of the solution form.
- 6.
In this paper, we also give a theorem regarding the convergence of the D-type ILC, and the proof is abstracted from the PD-type’s ILC convergence theorem proof.
- 7.
Finally, we provide a small comparison between the PD-type and the D-type schemes, using a numerical example.
The rest of the paper is organized as follows: In Section 2, we provide a presentation to our system, and we describe the ILC update law beside the ILS law. In Section 3, We present the theorem and the proof of the convergence of the PD-scheme, and we give the convergence theorem of the D-type scheme, next in Section 4, a numerical simulation is provided. Finally, Section 5 concludes this work. Notation: Throughout this paper, Let , , and is the closed set of . For a function , take the norm: , and the -norm of the function is given by: .
Section snippets
Problem formulation
Consider the following non-linear MIMO wave equation:Here, , with . represent the space and the time variable respectively. and are positive defined matrix, is a diagonal matrix. and denote the system state and the system input respectively. denote the system output.
Convergence analysis
Lemma 1 Suppose that and are non-negative real sequence satisfying, . If and when , then when .
The proof is easy to complete. Lemma 2 1. If Then,2. If Then, Proof 1. Firstly, we give the proof of the first part of Lemma 2.
Numerical simulation
A specific numerical example is considered to illustrate the effectiveness of the ILC mentioned in this paper, let the following system be,Without losing generation taking: , , , then , and
Here , , and
. For the given desired output with
Conclusion
This paper studies the application of the PD-type ILC to solve the distributed control problem of a non-linear multiple-input multiple-output (MIMO) wave equation. The convergence of the proposed method was proved theoretically, as a particular case (), D-type ILC provided the same theoretical result. The simulation results show that both schemes are effective. However, the PD-type controller resulting in tracking error outperforms the tracking error resulting from the D-type controller.
CRediT authorship contribution statement
Hamidaoui Meryem: . Cheng Shao: Data curation, Writing – original draft.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
References (38)
- et al.
Iterative learning control and repetitive control in hard disk drive industry - a tutorial
Int. J. Adapt. Control Signal Process.
(2008) - et al.
Unified iterative learning control for flexible structures with input constraints
Automatica
(2018) - et al.
Iterative learning control of inhomogeneous distributed parameter systems-frequency domain design and analysis
Syst. Control Lett.
(2014) - et al.
Nonlinear Wave Equations, Formation of Singularities
(1990) - et al.
An iterative learning control based identification for a class of mimo continuous-time systems in the presence of fixed input disturbances and measurement noises
Int. J. Syst. Sci.
(2007) - et al.
Analysis and reduced-order design of quadratic criterion-based iterative learning control using singular value decomposition
Comput. Chem. Eng.
(2000) - et al.
Iterative learning fault-tolerant control for batch processes
Comput. Chem. Eng.
(2006) - et al.
New iterative learning control algorithm using learning gain based on inversion for nonsquare multi-input multi-output systems
Model. Simul. Eng.
(2018) - et al.
Iterative learning control for mimo parabolic partial difference systems with time delay
Adv. Differ. Equ.
(2018/09/25a) Nonlinear Partial Differential Equations in Engineering
(1965)
Bettering operation of robots by learning
J. Field Rob.
Pd-type iterative learning control for nonlinear time-delay system with external disturbance
J. Syst. Eng. Electron.
Formal Proof of a Wave Equation Resolution Scheme: The Method Error Interactive Theorem Proving
Model-free iterative learning control with nonrepetitive trajectories for second-order mimo nonlinear systems- application to a delta robot
IEEE Trans. Ind. Electron.
A survey of the iterative learning control
IEEE Control Syst. Mag.
Model free adaptive iterative learning control for a class of nonlinear systems with randomly varying iteration lengths
J. Franklin Inst.
D-type iterative learning control for a class of parabolic partial difference systems
Trans. Inst. Meas. Control
Iterative learning control for mimo singular distributed parameter systems
IEEE Access
Iterative learning control for nonlinear systems: abounded-error algorithm
Asian J. Control
Cited by (4)
Two-dimensional optimization design of constrained minmax model predictive tolerant-fault control for nonlinear batch processes
2024, Computers and Chemical EngineeringExtension of iterative learning control design for batch processes with time-delay in the input subject to random cycle-varying uncertainties
2023, Journal of the Franklin Institutep-Accelerated normal S-iterative learning control algorithm for linear discrete singular time-delay systems
2024, International Journal of General SystemsIterative learning control for time-delay nonlinear hyperbolic distributed parameter systems
2023, International Journal of Systems Science
- ☆
This work is partly financial supported by the High-tech Research and Development Program of China (2014AA041802).