Application of the iterative learning control of a non-linear MIMO wave equation.

https://doi.org/10.1016/j.jfranklin.2022.01.009Get rights and content

Highlights

  • The control problem of a non-linear MIMO wave equation is considered.

  • To get the optimal control of the PDE, a repetitive process is used, and the applied method is based on the ILC.

  • Not like almost of the ILC algorithm, the initial condition here also has an iterative learning update law based on the tracking error and the state.

  • The convergence condition is proved using fundamental mathematical properties

  • Two ILC schemes are presented, and a small comparison is given.

Abstract

The control of the MIMO system has become increasingly famous. However, most of the literature considers the linear system. This paper considers a non-linear MIMO wave equation. The control method proposed in this paper is the ILC controller with initial conditions unfixed. The chosen schemes in this paper are the PD-type ILC and the D-type ILC. Moreover, we give sufficient conditions to guarantee the methods’ convergence. Finally, we present a simulation result on a specific numerical example to prove the effectiveness of the proposed control law with a comparison between both schemes.

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 a,bR+, Q=Ω×(0,T)={(x,t);xΩ,0<tT}, ΩR and Q¯ is the closed set of Q. For a function y(x,t):Q¯Rn, take the norm: y(.,t)L2=aby(x,t)Ty(x,t)dx, and the λ-norm of the function is given by: yλ=maxt(0,T)(eλty(.,t)L22).

Section snippets

Problem formulation

Consider the following non-linear MIMO wave equation:{2Ykt2(x,t)A2Ykx2(x,t)=F(Yk(x,t))+BUk(x,t);(x,t)Q;Zk(x,t)=CYk(x,t)+D0tUk(x,τ)dτ(x,t)Q¯Ykx(a,t)=h(t);Ykx(b,t)=H(t);t[0,T]Yk(x,0)=f(x);Ykt(x,0)=g(x);xΩHere, (x,t)Q, with Ω=(a,b). x,t represent the space and the time variable respectively. A,B,C, and D are positive defined matrix, A is a diagonal matrix. Yk(x,t)Rn and Uk(x,t)Rn denote the system state and the system input respectively. Zk(x,t)Rn denote the system output. F(Yk(

Convergence analysis

Lemma 1

Suppose that ak and bk are non-negative real sequence satisfying, ak+1ρak+bk. If 0ρ<1 and bk0 when k, then ak0 when k.

The proof is easy to complete.

Lemma 2

1. If

ICL32<1

Then,limkek(.,0)L2=02. If

ICL3DL22<12

Then,limkekt(.,0)L2=0

Proof

1. Firstly, we give the proof of the first part of Lemma 2.ek+1(x,t)=Zd(x,t)Zk+1(x,t)=ek(x,t)+C(YkYk+1)(x,t)+D0t(UkUk+1)(x,τ)dτek+1(x,0)=ek(x,0)+C(YkYk+1)(x,0)=(ICL3)ek(x,0)ek+1T(x,0)ek+1(x,0)=((ICL3)ek(x,0))T((ICL3)ek(x,0))ek+1(.,0)L22ICL32ek

Numerical simulation

A specific numerical example is considered to illustrate the effectiveness of the ILC mentioned in this paper, let the following system be,{2Y1,kt2(x,t)2Y1,kx2(x,t)=U1,k(x,t)+sin(Y1,k(x,t))2Y2,kt2(x,t)2Y2,kx2(x,t)=U2,k(x,t)+Y2,k(x,t)Zk(x,t)=CYk(x,t)+D0tUk(x,τ)dτWithout losing generation taking: C=D=I, a=0, b=π, then x[0,π],t[0,1], and λ=50

Here Yk=[Y1kY2k], Uk=[U1kU2k], Zk=[Z1kZ2k] and

F(Yk)= [sin(Y1k)Y2k]. For the given desired output Zd=[Z1dZ2d] with

Z1d=x+t1+tcos(t)+1+t2et/2sin(4x)

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 (L1=0), 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.

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  • Cited by (4)

    This work is partly financial supported by the High-tech Research and Development Program of China (2014AA041802).

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