THE CONCEPT OF CONTROLLABILITY IN AN E-LEARNING MODEL

Abstract

Three trends concerning applications of Systems Theory in education and e-learning are analysed. The first is the holistic General Systems Theory which studies the structure and the behaviour of systems, and identifying the components and the interconnections within systems. The second is represented by the Dynamical Systems Theory which uses concepts such as attractors, bifurcations or constraints to create a framework for the study of the critical components of the learning process. But being too general, these directions cannot provide concrete mathematical models able to give solutions for the management and improvement of this process. The third direction which tries to solve this problem is developed in this paper, continuing the author's two previous papers. An e-learning innovative model is provided using the tools of the Systems and Control Theory. The inputs, outputs and the states which correspond to the e-learning process are described and the equations of the state-space representation of the model are derived. This model is applied to the foreign languages e-learning. The controllability of the system is analysed by means of a suitable Gramian. A necessary and sufficient condition of controllability is presented and in the case of the controllable systems a formula is given for the optimal control of the problem of minimum energy transfer of states. Two programs are implemented using Control Systems Toolbox of the Matlab software, one which generates the e-learning system and the other which computes the optimal solution of the transfer problem. This solution allows teachers and students to schedule their e-learning tasks in order to acquire a desired level of knowledge.

Headnote

Abstract: Three trends concerning applications of Systems Theory in education and e-learning are analysed. The first is the holistic General Systems Theory which studies the structure and the behaviour of systems, and identifying the components and the interconnections within systems. The second is represented by the Dynamical Systems Theory which uses concepts such as attractors, bifurcations or constraints to create a framework for the study of the critical components of the learning process. But being too general, these directions cannot provide concrete mathematical models able to give solutions for the management and improvement of this process. The third direction which tries to solve this problem is developed in this paper, continuing the author's two previous papers. An e-learning innovative model is provided using the tools of the Systems and Control Theory. The inputs, outputs and the states which correspond to the e-learning process are described and the equations of the state-space representation of the model are derived. This model is applied to the foreign languages e-learning. The controllability of the system is analysed by means of a suitable Gramian. A necessary and sufficient condition of controllability is presented and in the case of the controllable systems a formula is given for the optimal control of the problem of minimum energy transfer of states. Two programs are implemented using Control Systems Toolbox of the Matlab software, one which generates the e-learning system and the other which computes the optimal solution of the transfer problem. This solution allows teachers and students to schedule their e-learning tasks in order to acquire a desired level of knowledge.

Keywords: e-learning; foreign languages e-learning model; systems theory; controllability; optimal control; Matlab software

(ProQuest: ... denotes formulae omitted.)

INTRODUCTION

The last three decades have known a growing presence of the Systems Theory in the literature of education, imposed by the necessity of obtaining a profound understanding of the educational process by using the concepts and the techniques of this interdisciplinary field of science. This approach provides a basis for the improvement of the structure and methods of learning (and particularly of e-learning). The explanation of the possibility of applying Systems Theory in education is the fact that education fulfils the general definition of a system and e-learning is a subsystem of it.

"Systems theory is the study of simple and complex systems, their structure, and their behaviour. It is concerned with identifying the elements and interconnections within systems. It focuses on the interrelationships and interaction of elements through their interconnections" [7]. Thus, the elements of the e-learning systems are their tools, such as writing technologies, communication technologies, visualization and storage, and the participants to this process (trainers and students) interact, use these tools and the inputs of the system, that is the external information, in order to improve the level of knowledge and provide the outputs (applications to their environment).

The novelty of this trend produces a series of different (and sometimes divergent) approaches and we mention three of them.

The first direction in this study of e-learning was to use the holistic General Systems Theory (GST), founded by von Bertalanffy during the 1930s. Bertalanffy mentioned in [2] that the educational demands and the transdisciplinary basic principles are precisely those of the General Systems Theory. Thus GST can be an important tool for interdisciplinary synthesis and integration of education.

There are many researchers who have applied General Systems Theory to education or elearning. For instance John Sterman defined the concept of systems dynamics within the context of learning [16, p.297], Theodore W. Frick [5] has developed an educational systems theory (EST) for providing scientifically based predictions of the outcomes of education systems and Peter Senge has showed that "systems thinking is particularly relevant to education because of the types of problems that are prevalent in school systems"[15]. Donella Meadows has underlined the idea that "systems thinking is a critical tool in addressing the many environmental, political, social, and economic challenges we face around the world" [10].

A second direction in constructing models of learning within the system theory is represented by the Dynamical Systems Theory (DST) approach. One of the promoters of DST is Catherine Ennis who has considered that "a more holistic approach to complexity-described as dynamical system theory-may better explain the integration and connectedness within the learning process" [4]. Her conclusion is that Dynamical Systems Theory, with its concepts such as attractors, constraints, bifurcations etc., can provide a framework for defining and analysing the critical components of the learning process.

B. Nicolescu and T. Petrescu have underlined that within dynamic systems "education as a subsystem of the world-system is already known, defined, and studied with regards to its interactions with the world system". The problem which arises is that the nonlinear differential equations imagined to describe the education system or other dynamic systems remain at an abstract level and cannot provide concrete answers and applicable solutions for their problems. Among the possible conclusions, one quotes the following: "To simplify: there are not established universal rules (written in the objective language of mathematics) that govern dynamic systems attached to education" [11].

In order to answer these questions, another direction was initiated in [13] and [14], namely that of constructing mathematical models for education and especially for e-learning within Linear Systems and Control Theory. Concrete state-space representations were provided, with inputs, outputs and states which characterize the elements and the evolution of the e-learning process, as well as the corresponding model in the frequency-domain approach. We also mention the paper [12] for a presentation of innovative strategies in e-Learning].

The aim of this paper is to use the concept of controllability which is fundamental in Systems and Control Theory to generate an optimal e-learning policy. Chapter I presents and analyses the discrete-time e-learning model and provides its representation by means of the state and the output equations. In Chapter II this innovative model (system) is applied to the foreign languages e-learning with four state variables which correspond to the four macro skills, writing, speaking, reading and listening.

Chapter III is devoted to the study of the concept of controllability. The controllability Gramian is constructed and a criterion of testing the complete controllability of the system using the rank of Gramian is provided. In the case of completely controllable systems the formula of the optimal control is given and it is proved that it allows to obtain desired results in the e-learning activity with a minimum expenditure of energy (and time). An example illustrates these topics in Chapter IV and the Control Systems Toolbox of the Matlab software is used to develop two programs, one which generates the e-learning system and the other which computes the controllability Gramian and the optimal solution of the minimum energy transfer problem.

I.THE STATE-SPACE REPRESENTATION OF THE E-LEARNING MODEL

The necessity of suitable and efficient scientific models is imposed by the increasing importance of e-learning in the 21st century. "The emergence of e-learning with its sustained connectivity has demonstrated that deep and meaningful learning is not limited to the face-to-face classroom experience. E-learning is transformational in how we think about educational experiences in terms of sustained communication and collaboration. The affordance of ubiquitous and powerful communications technologies with their ability to create and sustain communities of learners have brought e-leaming into the mainstream of educational thought and practice. Education is being transformed as a result of pedagogical advances made possible by e-learning. The value of e-learning is as a catalyst to rethink its capacity to stimulate and guide the quest to personally construct meaning and collaboratively confirm knowledge" [6, pp.1-5]. Moreover, the situation of the teaching process in the context of the pandemic has determined the crucial role of e-learning, as a unique means of providing lessons and examinations.

A model of e-learning in the Systems and Control Theory approach was provided in [13]. This model contains a number (denoted m) of input terminals and a number (denotedp) of output terminals. The input terminals correspond to different types of e-learning (see [3, pp. 27-112]), for instance, in the case of high level education, the input terminals can be the following items:

* I1=e-courses (provided by an LMS-Learning Management System, eg. Teams, Zoom, Moodle);

* I2= reading materials for self-study;

* I3=online helpful references (translators, dictionaries, encyclopedias);

* I4=online classrooms/interactive e-lessons;

* I5= audio/video presentations;

* I6=web-based training (using Internet as a platform);

* I7=mobile learning (sending and receiving messages or emails);

* I8=social learning (students discussions for solving tasks on social platforms);

* I9=chat (using public chat (eg. Yahoo!Chat), voice (e.g. Skype) or typed text) ;

* I10= personal sites and blogs (web with journal entries), class blogs;

* I11=Internet-based simulation (of real life situations, e.g. business English, Engineering etc.);

* I12= wiki (web texts edited by any user).

The amount of knowledge provided at time t by the input Ļ, j=1,...,m is denoted by Uj(t), j=1,..., m (here m=12) and it is called input variable.

The different types of examinations denoted by O¿, i=1,..., p are considered output terminals; for instance Oi=progress tests and achievement tests (on Moodle in the form of assignments and quizzes), O2=homework, O3=examination at seminars, O4=partial exam, O5=final exam. The result of the output Oi at time t is denoted by y¿(t), ¿=1,2,...,p and it is called output variable.

The time t=0,1,2,... represents the days, weeks or months, from the beginning of the learning process. It follows that the model is a discrete time system, and we denote it by E.

The basic notion of the model is that of state variables. One considers that the aim of the learning process of a student (or of a group of students) is to acquire a number of skills (or domains of knowledge) Si, i=1,..., n. The amount of Si detained at time t is denoted by Xi(t) and it is called the ith state variable. The number n is called the dimension of E.

The rate of growth of Xi(t) in a unit of time (from t to t+1) due to the amount of the skill Sj is denoted by ay and that due to the input Ij is denoted by by. Similarly, the rate of growth of the output variable y due to the state variable xy is denoted by cy and that due to the input Iy by dy. The values ay, by, cy and dij are called gains (or amplification factors) and they are determined by statistical methods as characteristics of a student or as the mean of the characteristics of students in the case of a group.

The system E described in Figure 1 in which, for the sake of simplicity, one drives only one link of each of the four types of connections (input to skill, skill to skill, skill to output and input to output). The summators which calculate the sums of the quantities (signals) which come at S, or O, are represented by Ф. The skills are represented by yellow half-disks and the amplifiers by blue rectangles with the gains written on them.

Using the definition of gains and Figure 1, one obtains that the state, input and output variables verify the following equalities, called respectively the state and the output equations of the system E:

... (1)

... (2)

One introduces the vectors x(t) = [x₁(t)x₂(t)...x_n(t)]T, u(t) = [u₁(t)u₂(t)...u_m(t)T and y(t) = [y₁(t) y₂(t) ... y_p(t)]T, called respectively the state, input (or control) and output of the system Σ and the matrices A = [aij ]n×n, B = bij]n×m, C = [cij]p×n and D = [dij]n×n, where F = [fij]q×r denotes the matrix with elements fi]·, i = 1,.q, j = 1,.r, i.e. with q rows and r columns. Then we can write the equations (1) and (2) of the system E in the state-space representation

... (3)

where the state x(t) represents the amount of the skills acquired by the student at the moment t, the input (control) u(t) represents the knowledge gained from the used types of learning and the output y(t) represents the results achieved by all types of examinations.

Therefore the e-learning model E is a linear discrete-time control system.

II.CONTROLLABILITY OF THE E-LEARNING MODEL

Using the e-learning model E and equations (3), one can adapt the theorems and techniques of the Systems and Control Theory to obtain information and methods for the learning process. For instance, one can calculate the state x(t), i.e. the level of knowle=dge at any moment t or the output y(t) i.e. the estimation of the examination results. One can also apply the fundamental concepts of the systems theory. For instance, this paper shows how controllability gives the possibility to establish an optimal policy of learning u(k), k=0,1,..., t-1, such that the student's state of knowledge at time t comes to a given final level xf (t).

In the sequel we consider fixed initial and final moments respectively t0 and tj . Denote by [t0, t1] the time set [t0, t1 ] = {t0, t0 +1,..., t1 -1, t1}. The notions defined below refer to this time set.

The formula of the state of the system Σ at the moment t, determined by the initial state x0 and the control u : [t0, t1 -1] → Rm is (see [8, pp. 41-45])

... (4)

If a state x_f verifies x(t₁) = x_f, one says that the control u realizes the transfer of the initial state x0 to the final state xf (x0 → xf).

Definition1. A state x ∈ Rⁿ is controllable if there exists a control u which realizes the transfer of the initial state x to the final state x_f = 0.

The system Σ is completely controllable if any state x ∈ Rⁿ is controllable.

The main device in the study of the controllability of a system is the controllability Gramian. To this aim one assumes in the sequel that the matrix A is non-singular, hence its inverse A-1 exists. We denote by A^-T the transpose of the matrix A and by A^-T the matrix (A^T)^-1.

Definition 2. The controllability Gramian of the system Σ is the symmetrical n × n matrix

... (5)

Remark. One can prove by a straightforward computation that the controllability Gramian C(t₀, t₁) is the solution of the discrete time Lyapunov type equation

P - APA^T = (A^-1)^t-t0 B)^T- BB^T

By adapting [8, p. 106], one obtains the following results.

Theorem 1. The set of the controllable states of the system Σ is the subspace Im C (t0, t1).

By the Definition 1, the system 2 is completely controllable if and only if the subspace of the controllable states coincides with the whole state space R", i.e. Im C(t0, tj) = Rn and this is equivalent to the equality of their dimensions. Since dim Im C(t0, tj )=rank C(t0, tj ) and dimR" =n, we get the following characterization of the controllability concept.

Theorem 2. The system Σ is completely controllable if and only if

... = n. (6)

Actually, one will prove that this condition is equivalent to the fact that, given any states x₀ and x_f , there exists a control u which realizes the transfer x0 → xf.

Definition 3.The energy of the control u(t), t ∈{t0,t0 +1,...,t1 - 1} is the number ... .

Theorem 3. An optimal control which realizes the transfer of the state x0 to the state xf with minimum energy expenditure is the function u(t), t e {t0, t0 +1,., f -1}

... (7)

Proof. We calculate the state determined by the control u (t) and the initial state x0, given by formula (4), taking into account the definition of Gramian. One obtains

...

hence x(t₁) = x_f, i.e. u(t) realizes the transfer x₀ → x_f.

In order to show that u(t) is optimal, let us consider another control u(t) which realizes the same transfer x0 → xf. One denotes щ (t) = u(t) - ~(t) and by (4) it follows that

...

hence ... . We premultiply it by (A-1)t1-t0 to get ... and using (7) it follows that ... . Then ... hence

... . Therefore E_u ≤ E_u for any control u(t) eealizes the transfer x₀ → x_f, i.e. ~(t) is an optimal control.

Using (7) one can prove the following result.

Theorem 4. The minimum energy for the transfer x₀ → X_f is

...

A dual concept of controllability is the observability, both related to minimality [17].

III.THE SYSTEM OF THE FOREIGN LANGUAGES E-LEARNING

The reason to focus on the study of foreign languages is that it is one of the most suitable to the e-learning possibilities. By using e-learning methods teachers and students (and other people) can interact to improve the quality of the pronunciation, the macro and micro skills or grammar and vocabulary knowledge. As it was shown in Chapter I, the system theory model of the foreign languages e-learning can be a system of dimension n greater than 10, with more than twelve inputs and five outputs.

This paper considers for lack of space a simplified model of dimension 4. The four state variables correspond to the four macro skills, writing, speaking, reading and listening which characterize the foreign languages. This choice is determined by the fact that nowadays these four actions are considered skills in their own right, not only support for learning vocabulary or grammar. "Learning and reading tasks furnish rich opportunities for learners to notice grammar in context, as part of a wider skill of making sense of written and spoken discourse" [1, p. 100]. Another argument is that "... the skills-centred approach still approaches the learner as a nser of language rather than as a learner of language.The processes it is concerned with are the processes of language me not of language learning" [9, pp. 70-71].

The number of the inputs is m=3, the three input variables corresponding to the input terminals Moodle, Teams and WBT (Web-based training, which uses the Internet as a platform). There are p=2 output variables, corresponding to exam and job interview simulation.

The gains of the system, i.e. the entries of the matrices A, B and C presented in Chapter I were obtained by a statistical study of a sample of 126 students (the questionnaire was applied to 72 girls and 54 boys) of the Faculty of Applied Sciences and Faculty of Entrepreuneurship, Business Engineering and Management, University Politehnica of Bucharest. The time unit considered is a month and the results are given for t=4, which correspond to one semester.

The result is the state-space representation of the form (3), where the state, input and output are real vectors of dimensions n=4, m=3 andp=2, which have respectively the forms x(t) = [xļ(t) x2(t) ·з(0 x4(t)f, u(t) = \ui(t) u2(t) u3(t)]randy(t) = \yx(t) y2(t)f, t = 1, 2 ,3, 4 where x(t) represents the evaluations of the acquired skills during the semester, u(t) represents the amount of knowledge provided at time t by the three inputs, yi(4) is the grade for the final exam and y2(4) is the grade for the job interview simulation.

MATLAB Program which generates the e-learning control system

The control system ELM of the e-learning model is generated by the following command from Control Systems Toolbox, Matlab software:

A=[1.1 0.1 0.3 0.2;0.1 1.2 0.1 0.4;0.3 0.1 1 0.1;0.1 0.5 0.2 1.2];

B=[0.5 0.2 0.4;0.2 0.2 0.3;0.4 0.1 0.2 ;0.1 0.2 0.3];

C=[0.5 0.3 0.4 0.4;0.5 0.7 0.0 0.0];D=[0 0 0;0 0 0];

ELM=ss(A,B,C,D,1,'inputname',{'md','tm','wt'},...

'statename',{'wr','sp','rd','ls'},'outputname',{'ex','jb'})

This command provides the matrices of the system and indicates the inputs (methods of e-learning) Moodle, Teams and Web Based Training respectively by md, tm and wt, the states (the levels of the four skills) writing, speaking, reading and listening respectively by wr, sp, rd and ls, the outputs (methods of examination). exam and job interview by ex and jb and the corresponding gains.

Matlab returns the following answer, which includes the type of the system (discrete-time), the matrices A, B and C of the system and the indications of the gains (amplification factors) of the states-states, inputs-states and states-outputs connections.

ELM =

A = B = C =

wr sp rd ls md tm wt wr sp rd ls

wr 1.1 0.1 0.3 0.2 wr 0.5 0.2 0.4 ex 0.5 0.3 0.4 0.4

sp 0.1 1.2 0.1 0.4 sp 0.2 0.2 0.3 jb 0.5 0.7 0 0

rd 0.3 0.1 1.0 0.1 re 0.4 0.1 0.2

ls 0.1 0.5 0.2 1.2 li 0.1 0.2 0.3

Discrete-time state-space model.

Remark. In an e-learning model, D must be a null matrix. Otherwise an element dtj Ф 0 would signify a direct connection between the ith learning resource and the jth method of examination, which could mean cheating or plagiarism and this is impossible in an ideal learning process.

The following program was developed using the package Control Systems Toolbox of the Matlab software. It computes the controllability Gramian of the e-learning system, it verifies if the system is completely controllable, it determines the optimal control and it plots the graphs of its components.

MATLAB Program for the Optimal Control

The system 1 is given with the matrices A 4 x 4 and B 4 x 3, hence dim 1 =4.

...

The values of the optimal control ~(t), t = 0,1,2,3,4are the ones indicated in following list and figure:

U = 0 0.1068 0.5344 1.0335

0.0424 0.1201 0.2000 0.2865 0.3828

0 0.0820 0.2584 0.4409 0.6347

The components of the optimal control are plotted in Figure 2, where u\(t) (Moodle) is represented by blue stars, ½(t) (Teams) by red circles and ½(t) (WBT) by green rhombuses.

One can observe that in this model the effort on the three channels must increase in time.

In the first part of the analysed interval the periods of time dedicated to the three methods of elearning are nearly close, that of Moodle being the smallest and Teams the biggest. But finally Moodle becomes the principal tool, indicating the importance of the independent work.

IV.CONCLUSIONS

The fundamental concept of controllability is studied in this paper for an e-learning model developed in a Systems and Control Theory approach. For completely controllable systems a formula was provided which computes the optimal control for the minimum energy transfer. This formula and its Matlab program allow teachers, students or individual learners to schedule their e-learning activities in order to gain a desired level of knowledge.

The results of this paper can be extended to time varying systems by replacing the powers of the drift matrix A by its fundamental matrix. Similar notions and techniques can be developed for a corresponding continuous-time model. Other directions of study can be the analysis of other fundamental concepts of Systems and Control Theory, such as observability, stability, stabilizability of systems by feedback etc.

Acknowledgements

The author is thankful to Dr. Tiberiu Vasilache for the fruitful discussions about systems and his help in the implementation of the Matlab program.

References

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