A CNN-based approach for a class of non-standard hyperbolic partialdifferential equations modeling distributed parameters (nonlinear)control systems

Daniela DanciuDepartment of Automation and Electronics, University of Craiova, A.I. Cuza str., no. 13, RO-200585 Craiova, Romania

a r t i c l e i n f o

Article history:Received 13 January 2014Received in revised form10 December 2014Accepted 14 December 2014Available online 14 March 2015

Keywords:Distributed parameters control systemsHyperbolic partial differential equationsCellular neural networksMethod of linesControl Lyapunov functionalsOverhead crane

a b s t r a c t

The present paper considers a systematic approach within the framework of Neural Mathematics forconstructing a computational procedure. This procedure aims to solve a class of problems arising fromthe control of the systems with distributed parameters; these systems are modeled by second-orderone-dimensional hyperbolic partial differential equations (hPDEs) with non-standard boundary condi-tions. The procedure reveals an explicit algorithmic parallelism and is mainly based on the combinationof two powerful “tools”: a convergent Method of Lines (MoL) and the Cellular Neural Network (CNN)paradigm. The role of the Courant-Isaacson-Rees rule and of the Riemann invariants for a correctapplication of the MoL is emphasized. The procedure is illustrated on a control engineering application –

the overhead crane with flexible cable – within a more general context which includes modeling basedon the generalized Hamilton variational principle, synthesis of a stabilizing controller via the ControlLyapunov Functional (CLF), qualitative analysis, numerical solving using the proposed computationalprocedure, numerical simulations and the evaluation of the performances for the closed loop system.The procedure ensures the convergence of the approximation, preserves the basic properties and theLyapunov stability of the solution of the initial problem and reduces the systematic errors.

& 2015 Elsevier B.V. All rights reserved.

1. Introduction

Beginning with the seminal papers of the 19th century, theimportance and study of distributed parameter systems (DPS)increased exponentially mainly due to their applications in bothengineering and science. The study of DPS encompasses severalareas, such as model development, numerical approximations,control design and experimental implementation. In [1] it isstressed the role of approximations for developing or improvingthe design methods and computational tools for the distributedparameter control systems (DPCS) encountered in engineering aswell as in other areas of science and life. Within this context,taking into account the new concern of using neural computersin solving mathematical problems, a Neural Mathematics (NM)approach is desirable.

Neural mathematics (Neuromathematics) arises as a new inter-disciplinary research direction, being at the confluence of the Computa-tional Mathematics and Neurocomputing fields. As is described in [2], itaims to develop new methods and algorithms for solving both

formalized and non-formalized, or weakly-formalized, problems viathe logical basis of the neural networks. Under this vision, a neuralnetwork algorithm is a computational procedure whose main part canbe implemented using a neural network.

For formalized or weakly-formalized problems which display acertain natural parallelism, the solutions can be computed byusing a neural network which does not need a “learning” proce-dure based on experimental data in order to achieve the weightsof its interconnections. A short list of problems for Neural Mathe-matics includes mathematical problems or tasks arising fromnatural sciences and engineering applications [2]: (a) solvinghigh-dimensional systems of linear or nonlinear algebraic equa-tions and inequalities [3,4], (b) solving optimization and systemsidentification problems [5–7], (c) approximating and extrapolatingvarious functions, (d) solving systems of nonlinear differential[8,9], (e) solving partial differential equations (PDEs) [10–16].

Concerning the solving of PDEs problems by using neuralnetworks, several approaches exist. Most of them consider pro-blems for elliptic or parabolic PDEs. The difficulties in coping withhyperbolic PDEs (hPDEs) are mainly due to the phenomenon ofdiscontinuity propagation; these discontinuities are “ignited” bythe “mismatch” of the boundary and of the initial conditions.

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Neurocomputing

http://dx.doi.org/10.1016/j.neucom.2014.12.0920925-2312/& 2015 Elsevier B.V. All rights reserved.

E-mail address: ddanciu@automation.ucv.ro

Neurocomputing 164 (2015) 56–70

Standard literature examples contain either hPDE problems withknown solutions in close form or with simple e.g. Dirichlet typeboundary conditions (BCs). However, this is not the case of themajority of the problems induced by the DPS of engineering or ofother fields of science.

The dynamics of DPS modeled by hPDEs can be simulated byusing dynamical neural networks, i.e. recurrent neural networks.Among them, Cellular Neural Networks (CNNs), or more general“cell-based neural networks”, are desirable due to the rich dyna-mical behaviors they cover; in [17] the authors show that thedynamics of CNN models is richer than that of PDEs, “eitherbecause they do not approximate any PDE models, or becausethey have a qualitatively different dynamic behavior”

Worth mentioning is that, in all reported applications ofCNNs for solving hPDEs, the main role of CNN structures is toimplement the high-dimensional system of ordinary differen-tial equations (ODEs) approximating the initial hPDE problem[10,11,15,16]. Also, these works mainly focus on the one-to-onemapping of the approximating system of ODEs onto the CNNstructure as well as on the performances of these implementa-tions for the specific applications. Less attention is devoted tothe specific procedure used for deriving the approximatingsystem for the initial hPDE problem. Nevertheless, the role ofthe computational procedure for obtaining a good approxima-tion is equally important, and moreover, a systematic approachfor its construction within the new framework of NeuralMathematics is a necessity.

The structure of the paper is as follows. In Section 2 weformulate the problem of the present work. Section 3 introducesthe computational procedure step-by-step, together with the main“tools” it considers: a convergent Method of Lines (MoL) and theCellular Neural Network paradigm. In Section 4 we consider anengineering application in order to illustrate the entire procedurewithin a more comprehensive context. More precisely, we con-sider a DPS described by hPDEs with non-standard BCs for whichthe mathematical model is deduced. Then, the control is synthe-sized and the mathematical model of the controlled object withdistributed parameters is obtained. Applying next the computa-tional procedure, we derive the approximating system for whichthe simulation results are discussed. In Section 5 we present someconcluding remarks and open directions to be followed.

Notations: Throughout this paper we shall use the standardnotations for: (i) PDEs yttðσ; tÞ≔∂2y=∂t2, yσðσ; tÞ≔∂y=∂σ, ðaðσ; tÞÞσ≔ð∂=∂σÞaðσ; tÞ, (ii) the derivative with respect to time variable_xðtÞ≔ðd=dtÞxðtÞ, (iii) the derivative with respect to the spacevariable x0ðσÞ≔ðd=dσÞxðσÞ. Rn denotes the real n-space, Rþ denotesthe positive real axis, L2ð0;1Þ denotes the space of square-integrable functions on the interval ð0;1Þ, D¼ diagðd1;…; dnÞARn

denotes a diagonal matrix.

2. Problem formulation

Consider the class of, possibly nonlinear, distributed parametersystems modeled by a second-order one-dimensional mixed initialboundary value problem for a hyperbolic partial differentialequation with non-standard boundary conditions which mayinclude control signals on one or both boundaries. More specifi-cally, we consider the BCs of types enumerated in [18] and writtenfor the system in the Friedrichs form (7) as follows:

� Dirichlet BCs

vð0; tÞþβ1wð0; tÞ ¼φ1ðtÞwð0; tÞþβ2vð0; tÞ ¼φ2ðtÞ ð1Þ

� BCs “controlled by a system of ODEs which is itself controlled bythe BCs”

vð0; tÞþβ1wð0; tÞ ¼ c01xðtÞ�β3φðc00xðtÞÞþφ1ðtÞwð0; tÞþβ2vð0; tÞ ¼ c02xðtÞ�β4φðc00xðtÞÞþφ2ðtÞ_xðtÞ ¼ AxðtÞþb11vð0; tÞþb12wð0; tÞþb21vð1; tÞ

þb22wð1; tÞ�b0φðc00xðtÞÞþ f ðtÞ; xð0Þ ¼ x0 ð2Þ

� derivative BCs as considered in [19]

XL10

aj1dj

dtjvð0; tÞþ

XK2

0

aj2dj

dtjwð0; tÞ ¼φ1

XK1

0

bj1dj

dtjvð1; tÞþ

XL20

bj2dj

dtjwð1; tÞ ¼φ2 ð3Þ

� “BCs described by some nonlinear Voltera operators acting on theboundary of the definition domain of PDE [20]”

The specific problems of such DPCS, arising either fromengineering applications or from other fields of science, includemodeling, basic properties of the solution, solution computation,as well as synthesis of the stabilizing controller and evaluation ofthe performances for the closed loop system. Since for a largenumber of DPS the analytical solution is unknown or difficult tocalculate, a good choice is to use the numerical approximatedsolutions. It is thus important that the numerical methods/proce-dures used for their computation ensure the convergence of theapproximating solution to the true one and the preservation of thebasic properties of the solution, i.e. existence, uniqueness, datadependence, stability in the sense of Lyapunov.

This paper has two main objectives. The first one is to propose asystematic approach within the framework of Neural Mathematicsfor numerically solving the aforementioned class of hPDE pro-blems. This is a formalized problem for NM because one can derivean explicit solution algorithm which, at this time, can be at leastpartially solved by using the neural networks. It is the case of anexplicit algorithmic parallelism, where each step can be consid-ered as a sub-problem for which a specific Neurocomputingapproach may be developed. The procedure has to ensure theconvergence of the approximation as well as the preservation ofthe basic properties of the solutions of the initial problem. Also, ithas to ensure the optimum neural network structure for numericalimplementation in order to reduce the systematic errors, storageand computational effort.

The second objective is to illustrate the proposed procedure onan engineering application for DPCS without lossless/distorsion-less phenomena and modeled by coupled hPDEs in the normalform of Riemann invariants – the overhead crane with flexiblecable, a benchmark for modeling and control theory. This studywill give also an overview about the role and the place of theentire procedure for a control engineering application.

3. A Neural Mathematics approach for developing thecomputational procedure

As it has been clear from the previous sections, the paper'sdefining technical feature might be described as deriving a “neuralalgorithm”, based on a convergent Method of Lines and incorporatinga cell-based neural network, for applications described by PDEs ofhyperbolic type.

The outcome is in the line of Neural Mathematics, a newresearch area of Computational Mathematics, for which we intro-duce here the main features, as stated in [2]. NM “provides the

D. Danciu / Neurocomputing 164 (2015) 56–70 57

development of algorithms that are potentially much more paral-lel than any physical implementation”. The algorithms aredesigned “for the solution of a wide class of problems with theuse of neurocomputers”. The suggested approach “for the algo-rithm design includes both well known computational methods,and, knowledge already accumulated in the domain of neural net-works calculations”. Neural algorithms can be developed for “for-malized, weak formalized or non-formalized problems” which “donot require the process of learning based on experimental data”and that are “characterized, first of all, by their pronounced naturalparallelization properties in the performance of signal processing,pattern processing etc.”.

Within this context, it is worth mentioning that the sub-problems (defined in the sequel as steps) of the proposedprocedure may be implemented or have already a neurocomput-ing solution, i.e. a solution based on a neural network logic – see,in [3] and in the references therein, the classes of problems solvedby using NN.

3.1. A convergent Method of Lines for hyperbolic PDEs

The Method of Lines is a computational approach in the study ofPDEs and it relies on the availability of good performance com-mercial software for solving ODEs. Consequently, a discretization isperformed with respect to the space variables only (finite differ-ences, finite elements a.o.) and a large scale system of ODEs isassociated for computational (and other) purposes to the basicboundary value problem for PDEs. Considering large scale systemsof ODEs for computation would require some “optimal” (in factrational) organization of the computational structures to avoidand/or minimize errors (mostly rounding and method ones) inorder to obtain a good numerical stability. This requirement isfulfilled by implementing the repetitive structures of the MoL byCNNs.

On the other hand, the MoL – as any computational approach –

must display some intrinsic features: convergence to the analyticalcomputed solution and preservation of some basic properties ofthis solution e.g. existence, uniqueness, data dependence andstability in the sense of Lyapunov. This aspect will be tackled inbrief in the following.

Convergence of the MoL for hyperbolic PDEs encounters somedifficulties due to a specific phenomenon – propagation of thesingularities: the mismatch of the boundary conditions with theinitial data generates some discontinuities that are propagated intime. Consequently, even the classical solutions are, generallyspeaking, discontinuous.

The applications within the class considered are such that wemay use a convergent Method of Lines for hPDEs, generated by theapproximation of time delays by ODEs. The methodology idea is asfollows. Any hPDE of higher order or system of hyperbolic PDEs inthe symmetric Friedrichs form may be converted into the so-callednormal form, having as variables the Riemann invariants. In thecase of lossless or distortionless propagation, the PDEs of theRiemann invariants are completely decoupled: their couplingtakes place through the boundary conditions only. In this case,integration of the Riemann invariants along the characteristics willallow association of some equations with deviating arguments anda one-to-one correspondence between their solutions and thesolutions of the boundary value problem for hPDEs is established.Basic theory and stability obtained for one object are thusprojected on the other one.

For our case, it is important to point out that these associatedequations contain terms with pure time delay of the type zðt�τÞ:the pure lumped time delay accounts for lossless/distortionlesspropagation and can be approximated by ODEs; the convergenceof this approximation is rigorously proved, including the

convergence rate (the Lemma of [18]). This result was used toapproximate solutions of equations with deviated arguments(Theorems 1, 2 and 3 of [18]) and, when projected back on hPDEs,it turned out to provide a rigorous convergence basis for the MoL.

Regarding the necessary steps of the convergent MoL we used,some remarks must be made. The Riemann invariants form allowsus to get the convergent approximations from the aforementionedtheorems by the space discretization which must be performedforwards for backward waves and backwards for forward waves,following the Courant-Isaacson–Rees rule introduced long ago[21]. This is the main role of the Riemann invariants – a correct,ensuring convergence, application of the MoL. This gives also anew insight on the Friedrichs form as ensuring, by simplediagonalization, the normal form; otherwise, the MoL could beapplied on any form of the PDEs, but there will be no guarantee fora correct discretization ensuring proper convergence of theapproach. And, last but not least, a correct application of theMoL will ensure preservation of the Lyapunov stability.

The aforementioned basics for the MoL holds for the caseof lossless and/or distortionless transmission lines or their analo-gous within hPDE: only in these cases the Riemann invariantsi.e. forward and backward waves can be decoupled by a corre-sponding linear transformation and such functional differentialequations attached that contain pure delay terms zðt�τÞ. In moregeneral cases [20] the functional differential equations associatedto the mixed initial boundary value problems are rather compli-cated. There exists however another way of proving convergenceof the MoL provided one turns to a fundamental text where thebasics for PDEs are considered starting from a computational pointof view [22]. The equations as well as the boundary conditions arediscretized with respect to both space and time variables andthere is proven the convergence of the solutions of the fullydiscretized difference system to a solution of the boundary valueproblem. On the other hand, formal application of the Method ofLines sends to a system of ordinary differential equations, asalready pointed out in the aforementioned development of thispaper. It is not difficult to prove that the fully discretizeddifference system solutions converge also to the solutions of thesystem of ODEs resulting from the MoL, adapting the approach of[22] (Chapter II, §15-§18). Combining the two results will giveconvergence of the Method of Lines to the solution of the mixedproblem for hyperbolic PDE without the lossless/distortionlesspropagation assumption. But this kind of construction gives anadditional insight on the MoL: in comparison to a completediscretization, within the Method of Lines the time discretizationis finer than space discretization and this is particularly importantwhen the analysis aims at dynamics and transients.

3.2. Cell-based neural networks for mapping formalized problems

The most known cell-based neural structure is the Cellular NeuralNetwork [23,24] – a very flexible device belonging to the ArtificialIntelligence field. Due to CNNs' important feature of local interconnec-tions – feature that allows emphasizing the so-called “cloningtemplates”, their applications are wide-spread – from image proces-sing to solving high-dimensional formalized problems displayingsome inherent parallelism and regularities as in the case of the tasksemerging at the solution of PDEs [2,10,11,17,16].

Concerning the qualitative properties of CNNs – as nonlineardynamical systems with, possible, multiple equilibria – a richliterature is devoted for the extensive and intensive study of thoseaspects as limit sets, stability of each equilibrium (Lyapunov,asymptotic or exponential stability), global behavior of the system(dichotomy, global asymptotics and gradient behavior), CNNs withtime-delays etc. – see e.g. [23,25–27], to give only a few examples.

D. Danciu / Neurocomputing 164 (2015) 56–7058

All these facts stand as main reasons which lead us to embedwithin the procedure the CNNs paradigm – as a powerful imple-mentation tool. Our choice is also based on their simplicity, richdynamical behaviors they can model, the possibility of consideringvarious types of inputs and, last but not least, their flexibility inimplementation: CNNs may be software emulated or software/hardware emulated on a digital basis, or may be hardwareimplemented in the VLSI technology. In the sequel we shallintroduce the basics concerning CNNs paradigm.

As already said, the key idea of the CNNs paradigm is therepresentation of the interactions between the cells by cloningtemplates – which can be either translation-invariant or regularlyvarying templates. To cite [10], “the cloning template is theelementary program of the CNNs (an instruction or subroutine)and it can be specified by giving prescribed values, and/orcharacteristics to the template elements”. The dynamics of a cellwithin a 2D array of cells may be described by [11,17]

_xijðtÞ ¼X

k;lANr ðijÞAij;klf ðxklÞþ

Xk;lANr ðijÞ

Bij;kluklþX

k;lANr ðijÞCij;klxijþ Iij ð4Þ

where the notations are as follows: xij – the state variable of theijth cell, ukl – the input variable from the neighboring cells actingon the ijth cell, Nr(ij) – the r-neighborhood of the ijth cell, Aij;kl –

the feedback cloning template, Bij;kl – the control cloning template,Cij;kl – the self-feedback cloning template, Iij – a bias or an externalstimulus.

The nonlinear activation function f : R-R of the standard cellis the unit bipolar ramp function

f ijðxijÞ ¼12ðjxijþ1j � jxij�1j Þ; ð5Þ

hence a continuous, bounded, nondecreasing, piecewise-linearand globally Lipschitzian function, with the Lipschitz constantLij ¼ 1, i.e. it verifies

0o f ijðσÞσ

rLij; f ijð0Þ ¼ 0: ð6Þ

Worth mentioning is that, in the case of problem-orientedapplications, one may encounter mathematical models thatrequires only the linear part of the function, i.e. f ðxijÞ ¼ xij.

3.3. The computational procedure, step-by-step

Consider again, the class of DPCS modeled by second-orderone-dimensional mixed initial boundary value problems for hPDEswith non-standard BCs. Let yðs; tÞ; ytðs; tÞ be the solution of thesystem for the initial conditions (ICs) yðs; t0Þ ¼ y0ðsÞ, _yðs; t0Þ ¼ y1ðsÞ,sA ½0; L�, tA ½t0; T �, t0 ¼ 0. Taking into account the diversity of thehPDE problems within the considered class, the main steps of theproposed procedure are as follows:

Step 1. PreliminariesStep 1.1. Write the model with rated space variable σ ¼ s=L,

σA ½0;1�. This step is not compulsory, but it improves the numer-ical conditioning.

Step 1.2. Synthesize the stabilizing controller and write theclosed loop system: The control law for DPCS may be obtained byusing various approaches and methods.

Step 2. The Friedrichs form: Introduce the new distributedvariables vðσ; tÞ ¼ ytðσ; tÞ and wðσ; tÞ ¼ yσðσ; tÞ and, taking intoaccount that vσðσ; tÞ ¼ ytσðσ; tÞ ¼ yσtðσ; tÞ ¼wtðσ; tÞ, write the sys-tem in the Friedrichs form

Vtðσ; tÞþAðσÞVσðσ; tÞþBðσÞVðσ; tÞ ¼ f ; ð7Þwhere Vðσ; tÞ ¼ ½vðσ; tÞ wðσ; tÞ�T .

Step 3. The Riemann invariantsStep 3.1. Write the system in the normal hyperbolic form:

Introduce the Riemann invariants Rðσ; tÞ ¼ ½r1ðσ; tÞ r2ðσ; tÞ�T , i.e.

find a nonsingular transformation TðσÞ such that Vðσ; tÞ ¼TðσÞRðσ; tÞ and,Rtðσ; tÞþAdðσÞRσðσ; tÞþCðσÞRðσ; tÞ ¼ bðσÞf ð8Þ

with AdðσÞ ¼ diag½λ1ðσÞ; λ2ðσÞ�, where obviously λiðσÞ are theeigenvalues.

Step 3.2. Identify the forward and the backward waves: For thesystem (8), the forward wave corresponds to the variable ri whichis multiplied by the positive eigenvalue and the backward wave isthat corresponding to the negative eigenvalue. Let us consider inthe sequel that the forward wave is r1ðσ; tÞ and the backward waveis r2ðσ; tÞ.

Step 4. Application of the convergent Method of Lines – ourapproach according to the results of [18].

Step 4.0. Discretize the domain of variation for the spacevariable σ, i.e. the interval ½0;1�: As usual, the interval of variationfor σ is divided into a grid with Nþ1 equally spaced points. Denoteby h¼ 1=N the length of an interval and σi ¼ ih, i¼ 0;N .

Step 4.1. Apply the Courant–Isaacson–Rees rule for approximat-ing the derivatives with respect to the space variable σ:

– for the forward wave, use the backward Euler scheme

∂r1∂σ

ðσi; tÞ �1hðr1ðσi; tÞ�r1ðσi�h; tÞÞ; i¼ 1;N ð9Þ

– for the backward wave, use the forward Euler scheme

∂r2∂σ

ðσi; tÞ �1hðr2ðσiþh; tÞ�r2ðσi; tÞÞ; i¼ 0;N�1 ð10Þ

Step 4.2. Write the approximating system of ODEs for thesystem in the normal form (8).

Step 4.2.1. Formally write the approximating system of ODEs forthe system (8): Considering the approximations (9) and (10) andintroducing the average approximation functions ξi1ðtÞ � r1ðσi; tÞ fori¼ 1;N and ξi2ðtÞ � r2ðσi; tÞ for i¼ 0;N�1, we formally write theapproximating system of ODEs in the normal Cauchy form. Remarkthat at this point, we have no approximations for r1ð0; tÞ and r2ðN; tÞ.

Step 4.2.2. Write the boundary conditions of the normal formwith respect to the new functions ξki : Taking into account the BCs,we obtain the equations for ξ01 and ξ2N. According to the features ofthe conditions on the boundaries, it may be possible to addition-ally introduce variables in order to write the entire approximatingsystem in the usual Cauchy form for an ODEs system.

Step 4.2.3. Define the “matching” equations for the boundaryand initial conditions.

Step 5. Determining the approximating system of ODEs for yðσ; tÞat the discrete points of the space interval: Using again the lineartransformation TðσÞ we obtain the approximating functionsνiðtÞ � vðσi; tÞ and further, taking into account the definitions inStep 2, we calculate the approximating system of ODEs for thedisplacement elements γiðtÞ � yðσi; tÞ, i¼ 1;N .

Step 6. Implementation based on Cellular Neural Networks para-digm: According to the application requirements for the imple-mentation purpose (software, software-hardware, or hardware),the structure of the CNN may be different.

Step 6.1. Identify the parallelism and regularities of the newproblem: Considering the flow diagram for the system in thenormal form of Riemann invariants (see Fig. 2) find the optimumarrangement of the ODEs in order to reveal the regularities whichlead to the cloning templates.

Step 6.2. Define the templates for both the inner and theboundary cells: Determine the feedback, control and self-feedback templates as well as the biases or external inputs.

Step 6.3. Define the cloning templates for the system of ODEswritten for the approximating functions of γiðtÞ.

Step 6.4. Implement the entire structure: Usually, a first stage isa software emulation of the cell-based neural network.

D. Danciu / Neurocomputing 164 (2015) 56–70 59

Step 7. Numerical computation of the approximating solution byusing commercial ODE solvers: Determine the approximating solu-tion and evaluate, by numerical simulations, the behavior of thesolution and the performances of the (control) system. Here isworth to mention the importance of “matching” the initial andboundary values in order to avoid the time propagation of thediscontinuities.

It appears as useful to make some remarks on an aspect whichcan be viewed as a drawback of the procedure. The “decomposition”in the forward and backward waves (the Riemann invariants),required by the convergent MoL, imposes also a “reconstruction” ofthe approximating solution from the approximating functions of thetwo waves. This process can be misunderstood as being sequential inthe implementation and computational stages of the procedure, thusintroducing additional rounding errors. This is not the case since, byusing the inverse linear transformation T �1ðσÞ, the ODEs for theapproximating functions of the velocity vðσ; tÞ ¼ ytðσ; tÞ obtained atStep 6.3 can be included within an augmented interconnection(feedback) matrix W of the cell-based structure – see Eq. (56) andthe corresponding block matrix W32 in our example. Consequently,the numerical integration using a ODE solver will be performed onlyonce (i.e. a massively parallel process) for a possibly huge system ofODEs consisting of two subsystems of ODEs: one for the approxi-mated solutions of the Riemann invariants and the other for theapproximated solution of the initial hPDE problem.

According to [2], the proposed procedure represents a neuralnetwork algorithm which solves a formalized problem by “asystematic approach in which the problem solving process isrepresented in the form of some dynamic system functioning intime”. In our case, the explicit algorithmic parallelism consists in theassignment of the different sub-problems (steps) to be solved on aneural network basis (with an appropriate implementation) andthe transfer of the result further to the next step until the entirecomputational process ends. For some steps of the procedure thisis an open direction to be followed even if there exist alreadyencouraging results as those presented in [3] for solving time-varying problems by means of recurrent neural networks. On theother hand, the implementation of the approximating system andthe solution for the aforementioned class of hPDE problems on thebasis of cell-based neural networks reveal a massive parallelprocess. And, last but not least, we recall here from Section 3.1that the proposed procedure ensures from the beginning theconvergence of the approximating solution to the “true” solutionand the preservation of the basic properties of this solution, as isalready proven in [18].

4. Application of the procedure to a benchmark: thecontrolled overhead crane with flexible cable

The purpose of this section is twofold. Firstly, to illustrate theapplicability of the entire procedure to a DPCS modeled by coupledhPDEs in the normal form of the Riemann invariants. As we shallsee, the mathematical model will have space-varying parametersand derivative boundaries conditions including a nonlinear control– the case of BCs (2). Secondly, to give an insight regarding the roleand the place of the procedure within a general approach for acontrol engineering application. Thus, we extend our previouswork [28] and include modeling based on the generalized Hamiltonvariational principle, synthesis of a stabilizing controller via theControl Lyapunov Functional (CLF), qualitative analysis of thesolution, numerical solving using the proposed procedure, numer-ical simulations and the evaluation of the performances for theclosed loop system.

The overhead crane can be viewed as a benchmark for model-ing and control theory – we refer the reader to [29] and thereferences therein. For instance, in [29] it is studied the exponen-tial stabilization of the overhead crane with flexible and non-stretching cable considering the assumption that “the accelerationof the load mass is negligible with respect to the gravitationalacceleration”. Throughout this section we shall overrule thisassumption. More precisely, we shall study the overhead craneassuming: (i) a completely flexible and non-stretching cablehaving constant length and non-negligible distributed mass and(ii) the non-negligible acceleration of the load mass with respectto the gravity acceleration.

4.1. Modeling issues based on the Calculus of Variations methods

The physical system is depicted in Fig. 1, where the notations areas follows [29–31]: M – the mass of the motorized platform movingalong a horizontal bench, YP – the current position of the motorizedplatform of the crane, m – the payload mass, L – the length of theflexible non-stretching cable, ρ – the mass density (per length) of thecrane's cable, u – the control force applied to the motorized platform,s – the curvilinear abscissa of the flexible cable element, yðs; tÞ – thehorizontal displacement at time t of the cable element whose curvi-linear abscissa is s; obviously YPðtÞ � yð0; tÞ.

A suitable approach for modeling this system, as almost allmodels arising from Mechanics, is the application of the general-ized Hamilton principle [32,33]. Firstly, we write explicitly thekinetic Ek and the potential Ep energies, as well as the work Wm

done by the exogenous forces

EkðtÞ ¼12M

ddtyð0; tÞ

� �2

þ12m

ddtyðL; tÞ

� �2

þ12

Z L

0ρðsÞ d

dtyðs; tÞ

� �2

ds

EpðtÞ ¼12

Z L

0TðsÞ d

dsyðs; tÞ

� �2

ds

WmðtÞ ¼ uðtÞyð0; tÞ: ð11ÞNext, we consider the Hamilton functional

Iðt1; t2Þ ¼Z t2

t1ðEkðtÞ�EpðtÞþWmðtÞÞ dt ð12Þ

with arbitrary t1, t2 and apply the standard Euler–Lagrangevariations which correspond to the principle of the virtual dis-placements

yðs; tÞ ¼ yðs; tÞþϵηðs; tÞ ð13Þwhere yðs; tÞ corresponds to the basic trajectory. Let Iϵðt1; t2Þ be theHamilton functional (12) written along (13). Then, the necessaryFig. 1. The overhead crane with flexible cable – suggested by [30].

D. Danciu / Neurocomputing 164 (2015) 56–7060

extremum condition

ddϵ

Iϵðt1; t2Þ����ϵ ¼ 0

¼ 0 ð14Þ

will giveZ t2

t1M

ddtyð0; tÞ

� �ddtηð0; tÞ

� �þm

ddtyðL; tÞ

� �ddtηðL; tÞ

� ��

þZ L

0ρðsÞytðs; tÞηtðs; tÞ ds

�Z L

0TðsÞysðs; tÞηsðs; tÞ dsþuðtÞηð0; tÞ

�dt ¼ 0 ð15Þ

Taking into account the property of the Euler–Lagrange variationsηðs; t1Þ ¼ ηðs; t2Þ and performing integration by parts, we obtainZ t2

t1�M

d2

dt2yð0; tÞþuðtÞþTð0Þysð0; tÞ

" #ηð0; tÞ dt

þZ t2

t1�m

d2

dt2yðL; tÞ�TðLÞysðL; tÞ

" #ηðL; tÞ dt

þZ t2

t1

Z L

0½ðTðsÞysðs; tÞÞs�ρðsÞyttðs; tÞ�ηðs; tÞ ds dt ¼ 0 ð16Þ

which gives, considering the homogeneous cable, i.e. ρðsÞ ¼ ρ40,and the expression of the strain along the cable TðsÞ ¼mgþρsg,the following equations:

�∂2y∂t2

þ ∂∂s

mgρ

þsg� �

∂y∂s

� �¼ 0

Md2

dt2yð0; tÞ ¼mg

∂y∂sð0; tÞþuðtÞ

md2

dt2yðL; tÞ ¼ �ðmgþρLgÞ∂y

∂sðL; tÞ: ð17Þ

As suggested in Step 1.1 of the procedure, in order to improvethe numerical conditioning, we introduce the normalized cablelength σ ¼ s=L, σA ½0;1� and rewrite the system in the standardnotations for PDEs. Thus, we obtain the following second-orderone-dimensional hyperbolic partial differential equation:

Lg� ρLmyttðσ; tÞ� 1þρL

mσ

� �yσðσ; tÞ

� �σ¼ 0; t40; 0oσo1 ð18Þ

with the derivative boundary conditions

Lg€yð0; tÞ ¼m

Myσð0; tÞþ

LMg

uðtÞ; yð0; tÞ ¼ YPðtÞ

Lg€yð1; tÞ ¼ � 1þρL

m

� �yσð1; tÞ: ð19Þ

Let us consider the following initial conditions (ICs):

yðσ;0Þ ¼ 0; σA ½0;1�;_yðσ;0Þ ¼ 0; σAð0;1�; _yð0;0Þ ¼ VP0: ð20ÞIt can be seen that yttðσ; tÞ is the acceleration of the flexible

cable element while, yσðσ; tÞ is the linear approximation of thelocal bending angle of the flexible cable and accounts for themechanical strain.

In order to verify the consistency of the mathematical model(18) and (19), we reduce it to the lumped parameter model byconsidering a standard assumption: the cable mass is negligiblewith respect to the load mass, i.e. ðρLÞ=m{0. The analyticalmanipulations are in the line of [31,34] and we introduce themhere for the sake of completeness as well as in order to show thatour mathematical model (18) and (19) is slightly modified incomparison to that existing in the literature. Nevertheless, ourmodel is in accordance with the physics of the phenomenon, as wewill show in the sequel.

Considering ðρLÞ=m¼ 0, Eqs. (18) and (19) will give for our DPSthe following:

yσσðσ; tÞ ¼ 0; t40; 0oσo1Lg€yð0; tÞ ¼m

Myσð0; tÞþ

LMg

uðtÞ

Lg€yð1; tÞ ¼ �yσð1; tÞ: ð21Þ

The first equation in (21) gives

yðσ; tÞ ¼ σφ1ðtÞþφ0ðtÞ ð22Þwith φi : Rþ-R, i¼ 1;2 some arbitrary time functions. Takinginto account (22), the boundary conditions in (21) lead to

Lg€φ0ðtÞ ¼

mMφ1ðtÞþ

LMg

uðtÞ

Lgð €φ1ðtÞþ €φ0ðtÞÞ ¼ �φ1ðtÞ: ð23Þ

Substituting the former equation in (23) in the latter gives

Lg€φ0ðtÞ ¼

mMφ1þ

LMg

uðtÞ

Lg€φ1ðtÞ ¼ � m

Mþ1

� �φ1�

LMg

uðtÞ ð24Þ

and this implies that the characteristic equation of the autono-mous system

pðsÞ ¼Lgs

2 �mM

0 Lgs

2þmMþ1

������������¼

Lgs2

Lgs2þm

Mþ1

� �¼ 0 ð25Þ

has a double zero root and a pair of roots on the imaginary axis,regardless of the value of m=M.

Thus, we obtained that the uncontrolled system is unstable andthis result is consistent with the physics of the problem. Moreover,our result does not depend on the condition m=Mo1, encoun-tered in other papers (see e.g. [35]), condition which cannot beverified in all practical applications (see the particular case weconsidered for the numerical simulations in Section 4.3.5). This“small” but nevertheless significant modeling error was pointedout by using the variational principle of Hamilton in the modeldeduction.

4.2. Synthesis of the stabilizing controller. The hPDE problemfor the closed loop dynamics

This subsection accounts for the Step 1.2 of the procedure. Ourapproach for system's stabilization and oscillations quenching isthe synthesis of the controller by making use of the ControlLyapunov Functional induced by the energy identity written forthe hPDEs problem (18) and (19). The advantage of using aLyapunov functional obtained in a “natural” way (on the basis ofthe system's energy) is the low complexity of the controller, aswell as of the closed loop control structure.

The first step is to multiply (18) by ytðσ; tÞ and to integrate theproduct on ½0;1� with respect to the space variable σ, i.e.

Lg� ρLm

Z 1

0ytðσ; tÞyttðσ; tÞ dσ

�Z 1

0ytðσ; tÞ 1þρL

mσ

� �yσ ðσ; tÞ

� �σdσ ¼ 0: ð26Þ

Performing integrations by parts and taking into account the BCs(19) we obtain the so-called energy identity

12ddt

Lg

_y2ð1; tÞþMm

_y2ð0; tÞ� �

D. Danciu / Neurocomputing 164 (2015) 56–70 61

þZ 1

0

Lg� ρLmy2t ðσ; tÞþ 1þρL

mσ

� �y2σ ðσ; tÞ

� �dσ

)

¼ Lmg

_y2ð0; tÞuðtÞ: ð27Þ

Hence, the system's energy depends on the velocities of theplatform _Y PðtÞ ¼ _yð0; tÞ, of the payload _yð1; tÞ and of each cableelement ytðσ; tÞ at the normalized curvilinear abscissa σAð0;1Þ, aswell as on the angular displacement with respect to the verticalyσðσ; tÞ for each cable element.

Note that the specific control problem for the overhead cranerequires the platform to attain a reference position Y P . It is thenobvious that the desired state of the closed loop controlled systemto be _yð1; tÞ ¼ _yð0; tÞ ¼ _Y PðtÞ � 0, YPðtÞ ¼ Y P , 8 t40, ytðσ; tÞ ¼yσðσ; tÞ � 0, 8 t40, σAð0;1Þ. Taking into account the errorbetween the actual state and the desired state of the controlledsystem, as well as the energy identity (27), the following candidatefor the Lyapunov functional V : R� R� R� L2ð0;1Þ �L2ð0;1Þ↦Rþ is obtained

VðX;Y ; Z;Φð�Þ;Ψ ð�ÞÞ ¼ 12

LgX2þKðY�Y PÞ2þ

Lg�MmZ2

�

þZ 1

0

Lg� ρLmΦ2ðσÞþ 1þρL

mσ

� �Ψ 2ðσÞ

� �dσ

#

ð28Þwith the restrictions Z ¼Φð0Þ, X ¼Φð1Þ; also K40 is a freeparameter counting for tuning the loop gain. Obviously, since allthe physical parameters are positive, the functional (28) is positivedefinite and written along the solutions of the system (18) and(19) it has the form

Vð _yð1; tÞ;YPðtÞ; _Y PðtÞ; ytðσ; tÞ; yσðσ; tÞÞ

¼ 12

Lg_y2ð1; tÞþKðYPðtÞ�Y PÞ2

�þ Lg�Mm

_Y2PðtÞ

þZ 1

0

Lg� ρLmy2t ðσ; tÞþ 1þρL

mσ

� �y2σðσ; tÞ

� �dσ

#: ð29Þ

Taking into account the energy identity (27), the time deriva-tive of the functional (29) along the solutions of the system (18)and (19), assumed to exist and satisfy corresponding regularityconditions, is then

dVdt

ð _yð1; tÞ;YPðtÞ; _Y PðtÞ; ytðσ; tÞ; yσðσ; tÞÞ

¼ KðYPðtÞ�Y PÞþLmg

uðtÞ� �

_Y PðtÞ: ð30Þ

Thus, the simplest condition for a decreasing energy Lyapunovfunctional might be

KðYPðtÞ�Y PÞþLmg

uðtÞ ¼ � f ð _Y PðtÞÞ ð31Þ

where f ð�Þ is a sector restricted nonlinear (in particular, linear)function verifying

0o f ðσÞσ

o f ; f ð0Þ ¼ 0: ð32Þ

This choice leads to the control law

uðtÞ ¼ �mgLðKðYPðtÞ�Y PÞþ f ð _Y PðtÞÞÞ ð33Þ

which, according to the Lyapunov theory, guaranties the stabilityin the sense of Lyapunov. Indeed

dVdt

ð _yð1; tÞ;YPðtÞ; _Y PðtÞ; ytðσ; tÞ; yσðσ; tÞÞ ¼ _Y PðtÞf ð _Y PðtÞÞr0 ð34Þ

what shows a non-increasing Lyapunov functional along thesystem's solutions. According to the first theorem of Lyapunov

which is generalized to the case of the distributed parameters,Lyapunov stability follows, together with the global boundednessof all state trajectories, since the Lyapunov function definition isglobal, as well as the derivative inequality. In fact, stability andboundedness follow directly from the non-increasing inequality

Vð _yð1; tÞ;YPðtÞ; _Y PðtÞ; ytðσ; tÞ; yσ ðσ; tÞÞrVðy1ð1Þ;YPð0Þ; _Y Pð0Þ; y1ðσÞ; y00ðσÞÞ

and this corresponds to the stability and boundedness in the norminduced by the energy-based Lyapunov functional itself – the so-called “stability in the energy norm”.

While this is outside the main topics of the paper, it is worthcommenting the obtained results from the point of view of thecontrol engineering tasks. First, our controller design belongs tothe class of low complexity and low order controllers – here anonlinear PD (proportional derivative) controller which has aproportional position feedback with respect to the position devia-tion of the platform and a derivative component which introducesan additional nonlinear damping of the platform (at the boundaryσ¼0). The sector restricted nonlinearity of the controller f ð�Þ is anallowed robustness of the controller, in the sense that stability ispreserved for any nonlinear (in particular, linear) function withinthe sector. While there might be more complicated solutions forthe controller, the adopted one has the merit to be simple enoughfor this kind of controlled plants. The free parameters K40 andf 40 (the width of the stability sector) may be considered designparameters for the controller.

For a rigorous discussion of the stability and stabilizationresults, we consider the closed loop equations that are obtainedby replacing in (19) the control u(t) given by (33)

Lg� ρLmyttðσ; tÞ� 1þρL

mσ

� �yσðσ; tÞ

� �σ¼ 0; t40; 0oσo1

Lg€yð0; tÞ ¼m

M½yσð0; tÞ�KðYPðtÞ�Y PÞ� f ð _Y PðtÞÞ�

Lg€yð1; tÞ ¼ � 1þρL

m

� �yσ ð1; tÞ ð35Þ

Since we already showed how Lyapunov stability and globalboundedness are obtained, we continue the discussion by tacklingasymptotic and exponential stability. Asymptotic stability willfollow, as usually in such cases (positive definite non-increasingLyapunov functional) by applying the Barbashin–Krasovskii–LaSalle invariance principle which states that all bounded solutionsapproach asymptotically the largest invariant set contained in theset where the derivative of the Lyapunov functional vanishes (agood reference for the case considered here is the book [36]). Itcan be shown that this set is exactly the unique steady state of thesystem. But this stability is even exponential as follows from a bynow classical result of K.P. Persidski stating that “for linearsystems, global asymptotic stability is always exponential”. Theapproach of Persidskii is valid for systems with sector restrictednonlinearities and distributed parameters also (see the classicalreference of Halanay [37]).

From the simulations, it can be observed in Figs. 9 and 10 agood convergence of the transient controlled processes, as aconsequence of the exponential stability.

4.3. The Method of Lines applied to the controlled overhead crane

4.3.1. The Friedrichs formConsider the distributed parameter control system described

by mixed initial boundary hPDE problem (35) and (20). Weintroduce the new distributed variables

vðσ; tÞ ¼ ytðσ; tÞ

D. Danciu / Neurocomputing 164 (2015) 56–7062

wðσ; tÞ ¼ yσðσ; tÞ ð36Þand, according to the Step 2 of the procedure, we write theFriedrichs form of the system (35)

vtðσ; tÞ�gL

σþmρL

� �wðσ; tÞ

� �σ¼ 0; t40; 0oσo1

wtðσ; tÞ�vσðσ; tÞ ¼ 0 ð37Þand its vectorial form

∂∂t

vðσ; tÞwðσ; tÞ

!þ ∂∂σ

0 �λ2ðσÞ�1 0

!vðσ; tÞwðσ; tÞ

! !¼ 0 ð38Þ

or, more specifically, defining V ðσ; tÞ ¼ ½vðσ; tÞ wðσ; tÞ�T

Vtðσ; tÞþAðσÞVσðσ; tÞþBðσÞVðσ; tÞ ¼ 0; t40; 0oσo1 ð39Þwhere

λðσÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigL

σþmρL

� �s; AðσÞ ¼ 0 �λ2ðσÞ

�1 0

!; BðσÞ ¼ ∂

∂σAðσÞ:

ð40ÞThe boundary conditions in (35) written in the new distributedvariables are

Lg_vð0; tÞ ¼m

M½wð0; tÞ�KðYPðtÞ�Y PÞ� f ð _Y PðtÞÞ� ð41Þ

Lg_vð1; tÞ ¼ � 1þρL

m

� �wð1; tÞ

and the initial conditions in (20) give

vðσ;0Þ ¼ _yðσ;0Þ ¼ 0; σAð0;1�; vð0;0Þ ¼ VP0

wðσ;0Þ ¼ 0; σA ½0;1�: ð42ÞConcerning the equilibria of the closed loop system (37)–(41), it

can be easily seen that, by taking the time derivatives equal tozero, we obtain that the system has a global asymptotic equili-brium at vðσÞ ¼ v � 0, wðσÞ ¼w � 0, Y P for σA ½0;1�; this is inaccordance with the purpose of the stabilizing controller and thequalitative analysis performed in Section 4.2.

4.3.2. The Riemann invariants and the normal formAccording to Step 3.1 we introduce in the sequel the Riemann

invariants r1ðσ; tÞ and r2ðσ; tÞ. After some straightforward buttedious manipulations we obtain the nonsingular linear transfor-mation

TðσÞ ¼�1ffiffiffiffiffiffiffiλðσÞ

p 1ffiffiffiffiffiffiffiλðσÞ

p1

λðσÞffiffiffiffiffiffiffiλðσÞ

p 1λðσÞ

ffiffiffiffiffiffiffiλðσÞ

p

0B@

1CA ð43Þ

which gives the following connections between the pairs of bothold and new variables

vðσ; tÞ ¼ 1ffiffiffiffiffiffiffiffiffiλðσÞ

p ð�r1ðσ; tÞþr2ðσ; tÞÞ

wðσ; tÞ ¼ 1

λðσÞffiffiffiffiffiffiffiffiffiλðσÞ

p ðr1ðσ; tÞþr2ðσ; tÞÞ; ð44Þ

and

r1ðσ; tÞ ¼ffiffiffiffiffiffiffiffiffiλðσÞ

p2

ð�vðσ; tÞþλðσÞwðσ; tÞÞ

r2ðσ; tÞ ¼ffiffiffiffiffiffiffiffiffiλðσÞ

p2

ðvðσ; tÞþλðσÞwðσ; tÞÞ: ð45Þ

As a mention, this sub-problem – of calculating the propertransformation for the diagonalization of the matrix AðσÞ – may becast in an explicit parallel algorithm for implementation on aneural logical basis.

For our system, the componentwise form of (8), written withrespect to the Riemann invariants reads as

∂r1∂t

þλðσÞ � ∂r1∂σ

þ 14λðσÞ �

gLr2ðσ; tÞ ¼ 0

∂r2∂t

�λðσÞ � ∂r2∂σ

� 14λðσÞ �

gLr1ðσ; tÞ ¼ 0 ð46Þ

and one can see that we have a system of coupled hPDEs withspace-varying coefficients. To the system (46) we attach theboundary conditions written with respect to the Riemann invar-iants; taking into account (41) and (44), we deduce

Lg€Y PðtÞ ¼

mM

1

λð0Þffiffiffiffiffiffiffiffiffiλð0Þ

p ðr1ð0; tÞþr2ð0; tÞÞ�KðYPðtÞ�Y PÞ� f ð _Y PðtÞÞ" #

Lgð� _r1ð1; tÞþ _r2ð1; tÞÞ ¼ � 1

λð1Þ 1þρLm

� �ðr1ð1; tÞþr2ð1; tÞÞ

_Y PðtÞ ¼1ffiffiffiffiffiffiffiffiffiλð0Þ

p ð�r1ð0; tÞþr2ð0; tÞÞ: ð47Þ

From (42) and (45) we deduce the initial conditions

r1ðσ; 0Þ ¼ 0; σAð0;1�; r1ð0; 0Þ ¼ �ffiffiffiffiffiffiffiffiffiλð0Þ

p2

VP0

r2ðσ;0Þ ¼ 0; σAð0;1�; r2ð0;0Þ ¼ffiffiffiffiffiffiffiffiffiλð0Þ

p2

VP0 ð48Þ

Next, following the hint of Step 3.2, we identify from (46) theforward wave r1ðσ; tÞ and the backward wave r2ðσ; tÞ.

4.3.3. The Method of Lines and the approximate system of ODEsWe proceed now with the Step 4. Consider the discretization of

the domain of variation for σ, the interval ½0;1�, in Nþ1 equallyspaced points and denote h¼ 1=N the length of an interval, σi ¼ ih,i¼ 0;N .

Taking into account the forward and the backward waves, weuse the Euler schemes (9) and (10) in order to approximate thederivatives with respect to the space variable σ, according to theCourant–Isaacson–Rees rule given in Step 4.1.

Next, we proceed as in Step 4.2 and introduce the approxima-tion functions ξi1ðtÞ � r1ðσi; tÞ for i¼ 1;N and ξi2ðtÞ � r2ðσi; tÞ fori¼ 0;N�1 and formally write for the system (46)

_ξi1ðtÞ ¼ aðσiÞξi�1

1 �aðσiÞξi1�bðσiÞξi2; i¼ 2;N

_ξi2ðtÞ ¼ �aðσiÞξi2þaðσiÞξiþ1

2 þbðσiÞξi1; i¼ 1;N�1 ð49Þ

where the new notations are aðσiÞ ¼NλðσiÞ, bðσiÞ ¼ g=ð4LλðσiÞÞi¼ 0;N with λðσÞ defined in (40).

Now, we complete the system taking into account the bound-ary conditions (47). The first equation suggests us to introduce anadditional variable for the velocity of the platform VPðtÞ ¼_Y PðtÞ ¼ vð0; tÞ, which means _V PðtÞ ¼ _vð0; tÞ. The third equation in(47) gives

ξ01ðtÞ ¼ ξ02ðtÞ�ffiffiffiffiffiffiffiffiffiλð0Þ

qVPðtÞ ð50Þ

which introduced in the first equation gives a new equation for_V PðtÞ. Also, from the second equation in (47), replacing ξ1N from(51), we obtain _ξ

N2 . Finally, we obtain the following systems of

ODEs which embed the boundary conditions

� for the forward wave r1ðσ; tÞ_ξ11ðtÞ ¼ �aðσ1Þξ11þaðσ1Þξ02�aðσ1Þ

ffiffiffiffiffiffiffiffiffiλð0Þ

qVP�bðσ1Þξ12

_ξi1ðtÞ ¼ aðσiÞξi�1

1 �aðσiÞξi1�bðσiÞξi2; i¼ 2;N ð51Þ

D. Danciu / Neurocomputing 164 (2015) 56–70 63

� for the backward wave r2ðσ; tÞ_ξ02ðtÞ ¼ ð�aðσ0Þþbðσ0ÞÞξ02þaðσ0Þξ12�

ffiffiffiffiffiffiffiffiffiλð0Þ

qbðσ0ÞVP

_ξi2ðtÞ ¼ �aðσiÞξi2þaðσiÞξiþ1

2 þbðσiÞξi1 i¼ 1;N�1

_ξN2 ðtÞ ¼ aðσNÞξN�1

1

� aðσNÞþ4 1þρLm

� �bðσNÞ

� �ξN1 �4bðσNÞ

54þρLm

� �ξN2 ð52Þ

� for the equations additionally introduced by the BCs

_Y PðtÞ ¼ VP

_V PðtÞ ¼ 8bðσ0ÞmM

� 1ffiffiffiffiffiffiffiffiffiλð0Þ

p ξ02�KmgLM

YP�4mMbðσ0ÞVP

þKmgLM

YP�mgLM

f ðVPðtÞÞ: ð53Þ

From (48) we deduce the initial conditions

ξi1ð0Þ ¼ 0; i¼ 1;N ; ξ01ð0Þ ¼�

ffiffiffiffiffiffiffiffiffiλð0Þ

p2

VP0

ξi2ð0Þ ¼ 0; i¼ 1;N ; ξ02ð0Þ ¼ffiffiffiffiffiffiffiffiffiλð0Þ

p2

VP0 ð54Þ

which verify the matching condition (50) on the boundary σ¼0.Finally, defining the approximating functions νiðtÞ � vðσi; tÞ and

considering the change of variables (44), we get

νiðtÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiλðσiÞ

p ð�ξi1ðtÞþξi2ðtÞÞ; i¼ 1;N : ð55Þ

Introducing then the approximation functions for the displace-ment of the cable elements γiðtÞ � yðσi; tÞ, i¼ 1;N , and consideringthe initial definitions (36), we obtain vðσi; tÞ ¼ ytðσi; tÞ, and thus

_γ iðtÞ ¼1ffiffiffiffiffiffiffiffiffiffiffiλðσiÞ

p ð�ξi1ðtÞþξi2ðtÞÞ; i¼ 1;N : ð56Þ

Note that for i¼0 we have the equation for the platform displace-ment YP(t), variable additionally introduced together with thevelocity VP(t), as required by the boundary conditions (47).

At this point let us summarize the first part of the procedure.We considered the hyperbolic PDE problem (18)–(20) having theBCs of the type (2). We designed the stabilizing control law u(t)described by (33) and obtained the distributed parameter controlsystem described by (35) and (20). For this system we derived itsFriedrichs form (37) with the BCs (41) and ICs (42). Then, to thenormal form of Riemann invariants (46)–(48) we applied theconvergent MoL thus obtaining a new problem – an initial valueproblem for a large scale system of ODEs (51)–(53) and (56) withthe ICs (54), VPð0Þ≔VP0, γið0Þ ¼ yðσi;0Þ≔γi0 � 0.

Recall that, according to the results of [18] and the discussion inSection 3.1, the solution of this approximating system converges tothe solution of the hPDE problem (35) and (20) and also preservesthe basic properties of this solution – i.e. existence, uniqueness,data dependence and stability in the sense of Lyapunov.

4.3.4. Implementation by means of software emulated CNNsConsidering now Step 6, one can see that the inherent paralle-

lism and regularities, on multiple layers, of the approximatingsystem (51)–(53) and (56) suggest using, as desirable powerfultool for implementation, the cell-based neural networks.

If we consider the CNN paradigm described in Section 3.2 – Eqs.(51)–(53) and (56) suggest a CNN with 4 layers of one-dimensionalarrays of cells. Briefly, for the standard CNN paradigm (4) used in

fast hardware implementations the templates for the inner andboundary cells are as follows:

� Layer 1 – system of ODEs (51)

Cξ1i ¼ �aðσiÞ; Iξ1i ¼ 0; i¼ 1;N ; Bξ1i ¼ 0; i¼ 1;N ;

Aξ1ξ1i f ðξkÞ ¼ aðσiÞξi�11 ; Aξ1ξ2i f ðξkÞ ¼ �bðσiÞξi2; i¼ 2;N ;

Aξ1ξ21 f ðξkÞ ¼ ½aðσ1Þ �bðσ1Þ 0�ξ02ξ12ξ22

2664

3775;

Bξ1P1 uk ¼ ½�aðσ1Þffiffiffiffiffiffiffiffiffiλð0Þ

q0�

VP

YP

" #;

� Layer 2 – system of ODEs (52)

Cξ2i ¼ �aðσiÞ; i¼ 0;N�1; Cξ2N ¼ �4bðσNÞ54þρLm

� �;

Aξ2ξ20 f ðξ02Þ ¼ bðσ0Þξ02; Aξ2ξ1i f ðξkÞ ¼ bðσiÞξi1; i¼ 1;N�1;

Aξ2ξ1N f ðξkÞ ¼ aðσNÞ �aðσNÞ�4 1þρLm

� �bðσNÞ

� � ξN�11

ξN1

24

35;

Bξ2P0 uk ¼ ½�bðσ0Þffiffiffiffiffiffiffiffiffiλð0Þ

q0�

VP

YP

" #; Bξ2i ¼ 0; Iξ2i ¼ 0; i¼ 1;N ;

� Layer 3 – system of ODEs (53)

CPYP

¼ 0; CPPVP

¼ �4mM

bðσ0Þ; IPYP¼ 0; IPVP

¼ KmgLM

YP ;

APPYPf ðVPÞ ¼ VP ; APP

VPf ðxPÞ ¼

�KmgLM

�mgLM

� � f 1ðYPÞf 1ðVPÞ

" #;

APξkYP

f ðξkÞ ¼ 0; APξkVP

f ðξkÞ ¼ 8bðσ0Þm

Mffiffiffiffiffiffiffiffiffiλð0Þ

p ξ02;

with f 1ðYPÞ ¼ YP , and f 2ðVPÞ a sector restricted nonlinear func-tion verifying (32).

� Layer 4 – system of ODEs (56)

Cγii ¼ 0; Bγii ¼ 0; Iγii ¼ 0; i¼ 1;N ;

Aγiξki f ðξikÞ ¼�1ffiffiffiffiffiffiffiffiffiffiffiλðσiÞ

p 1ffiffiffiffiffiffiffiffiffiffiffiλðσiÞ

p" #

ξi1ξi2

24

35:

In the case of a software emulated CNNs, it is more convenientto have a problem-oriented approach which will lead to a morecompact form for the structure of the cellular network and itstemplates.

As already said, the class of problems considered here can bedivided within two subclasses: with and without lossless/distor-sionless phenomena. For the class of hPDEs problems without

Fig. 2. The flow diagram for the case of coupled hPDE in the normal form ofRiemann invariants (51) and (52) allows us to identify the inner cloning templatesfor a cell-based neural network. The notations are ai≔aðσiÞ and bi≔bðσiÞ.

D. Danciu / Neurocomputing 164 (2015) 56–7064

lossless/distorsionless phenomena, the equations in the normalform of Riemann invariants are coupled, i.e. in (8) the matrix CðσÞis non-zero and non-diagonal. This leads to the existence ofinterconnections between the forward and the backward wavesξ1i and ξ2i along the transmission line, as we can see in the flowdiagram presented in Fig. 2; the diagram corresponds to theapproximating system (51) and (52) for the inner points of the

transmission line (the cable, in our case). Certainly, the two wavesare coupled on their boundaries, too.

On the other hand, for hPDEs problems modeling lossless/distorsionless phenomena the two waves are coupled only on theirboundaries. Consequently, in the flow diagram the terms bi vanish:bi¼0, i¼ 1;N�1. In this case, the best CNNs topology for imple-mentation is an array of cells interconnected in a ring shape. This

0 5 10 15 20 25 30 35 40−500

0

500

1000

1500

2000

2500

t

u [N

]

Fig. 3. The boundary control input, u(t).

0

0.5

10 10 20 30 40

−8

−6

−4

−2

0

2

4

6

σ

t

r1

Fig. 4. The space–time evolution of the forward wave, r1ðσ; tÞ.

0

0.5

10 10 20 30 40

−6

−4

−2

0

2

4

6

σ

t

r2

Fig. 5. The space–time evolution of the backward wave, r2ðσ; tÞ.

D. Danciu / Neurocomputing 164 (2015) 56–70 65

arrangement will lead to an interconnection (feedback) matrixwith 2- or 3- diagonals (as resulted from our previous works[38–41]). Studies related to the qualitative behavior of the CNNsrings can be found in [27] and in other works by the same authors.

Returning to our problem, in order to reveal the so-called“cloning templates”, we consider the flow diagram from Fig. 2 as

well as the fact that the control variables YP(t) and VP(t) act on theboundary σ¼0. These aspects suggest the following rearrange-ment of Eqs. (51)–(53) and (56) for building a one-dimensionalarray of 3Nþ4 cells

xT ¼ ½YP VP ξ02 ξ12 ξ11 ξ22 ξ21 ξ32 … ξN�11 ξN2 ξN1 γ1 … γN� ð57Þ

0

0.5

10 10 20 30 40

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

σ

t

v

Fig. 6. The velocity of the cable elements, vðσ; tÞ.

0 5 10 15 20 25 30 35 40−0.2

0

0.2

0.4

0.6

0.8

1

1.2

t

v(0,

t), v

(1, t

)

v(0, t)=VP (t)

v(1,t)

Fig. 7. The transient behavior for the velocities of the platform vð0; tÞ ¼ VP ðtÞ and of the payload vð1; tÞ.

0

0.5

10 10 20 30 40

−0.5

0

0.5

1

1.5

2

2.5

σ

t

y

Fig. 8. The 3D representation of the cable elements displacement, yðσ; tÞ.

D. Danciu / Neurocomputing 164 (2015) 56–7066

This arrangement leads to the following two inner feedbackcloning templates for the two waves

A1j ðiÞ ¼ ½0 bðσiÞ aðσiÞ�; j¼ 3þ i : 2 : 2Nþ2; i¼ 1;N�1

A2j ðiÞ ¼ ½aðσiÞ �bðσiÞ 0�; j¼ 4þ i : 2 : 2Nþ3; i¼ 1;N ð58Þ

For the sake of simplicity, looking ahead to the furthercomputational tasks and taking into account that the nonlinearfunction acts on the boundary σ¼0 as well as the fact that for theinner cells of the forward and the backward waves we havef ðξijÞ ¼ ξij, j¼ 1;2, we will modify the feedback templates Aj

k byincluding the decay term Cξkj ¼ �aðσjÞ, i.e. the inhibitory self-feedback in Layer 1 and Layer 2. This will give the followingmodified inner feedback cloning templates

A1nj ðiÞ ¼ ½�aðσiÞ bðσiÞ aðσiÞ�; j¼ 3þ i : 2 : 2Nþ2; i¼ 1;N�1

A2nj ðiÞ ¼ ½aðσiÞ �bðσiÞ �aðσiÞ�; j¼ 4þ i : 2 : 2Nþ3; i¼ 1;N ð59Þ

and we can see that now we have A2nj ¼ �A1n

j for j¼ 4;2Nþ3.The third feedback cloning template is obtained from (56) and

reads as

A3nj ðiÞ ¼ 1ffiffiffiffiffiffiffiffiffiffiffi

λðσiÞp �1ffiffiffiffiffiffiffiffiffiffiffi

λðσiÞp

" #; j¼ 2Nþ3þ i; i¼ 1;N : ð60Þ

All the inner cells having the cloning templates A1nj ðiÞ, A2n

j ðiÞ andA3nj ðiÞ have zero control templates Bj, zero biases Ij and, in this new

arrangement, Cj are zero, too.The dynamics of these inner cells is described by the following

equations:

_xjðtÞ ¼ A1nj ðiÞ �

xjxjþ1

xjþ2

264

375; j¼ 3þ i : 2 : 2Nþ2;

i¼ 1;N�1;

_xjðtÞ ¼ �A1nj ðiÞ �

xj�2

xj�1

xj

264

375; j¼ 4þ i : 2 : 2Nþ3;

i¼ 1;N ;

_xjðtÞ ¼ A3nj ðiÞ �

x2iþ2

x2iþ3

" #; j¼ 2Nþ3þ i; i¼ 1;N ð61Þ

For the boundary cells we have the following templates:

A1 ¼ ½0 1 0�; C1 ¼ I1 ¼ B1 ¼ 0

C2 ¼�4mbðσ0Þ

M; I2 ¼

KmgLM

YP ; A2 ¼�KmgLM

�mgLM

8mbðσ0ÞM

ffiffiffiffiffiffiffiffiffiλð0Þ

p" #

ð62Þ

with

_x2 ¼�KmgLM

�mgLM

8mbðσ0ÞM

ffiffiffiffiffiffiffiffiffiλð0Þ

p" # f 1ðx1Þ

f 2ðx2Þf 3ðx3Þ

264

375

for f iðxiÞ ¼ xi, i¼ 1;3, and f 2ðx2Þ a sector restricted nonlinearityverifying (32). Further,

A3ð0Þ ¼ ½�bðσ0Þffiffiffiffiffiffiffiffiffiλð0Þ

q�aðσ0Þþbðσ0Þ aðσ0Þ�; C3 ¼ B3 ¼ I3 ¼ 0;

A2Nþ2ðNÞ ¼ aðσNÞ �4bðσNÞ54þρLm

� ��aðσNÞþ4bðσNÞ 1þρL

m

� �� �

C2Nþ2 ¼ B2Nþ2 ¼ I2Nþ2 ¼ 0; ð63Þwith the dynamics described by

_x3ðtÞ ¼ A3ð0Þ �x2x3x4

264

375;

_x2Nþ2ðtÞ ¼ A2Nþ2ðNÞ �x2Nþ1

x2Nþ2

x2Nþ3

264

375: ð64Þ

Note that the hPDE problem considered in this engineeringapplication led to space-varying feedback cloning templates. Con-sidering the arrangement suggested by the state vector (57) aswell as the cloning templates for the inner and boundary cells, weobtain the CNN's interconnection matrix with the followingstructure:

W ¼W11 W12 W13

W21 W22 W23

W31 W32 W33

264

375 ð65Þ

where the block matrices read as

W11 ¼0 1� �

� �; W12 ¼

0 0 … 0� 0 … 0

� �; W13 ¼ 02�ð2Nþ1Þ

W21 ¼�ð2Nþ1Þ�2; W22 ¼ ½A1n; �A1n�ð2Nþ1Þ�ð2Nþ1Þ;

W23 ¼ 0ð2Nþ1Þ�ð2Nþ1Þ

W31 ¼ 0ð2Nþ1Þ�2; W32 ¼ A3nh i

ð2Nþ1Þ�ð2Nþ1Þ;

W33 ¼ 0ð2Nþ1Þ�ð2Nþ1Þ

with the symbol “� ” indicating non-zero entries (of appropriatedimensions). The main advantages of this structure are

0 5 10 15 20 25 30 35 400

0.5

1

1.5

2

2.5

t

y(0,

t)

Fig. 9. Time evolution of the platform displacement, yð0; tÞ ¼ YP ðtÞ.

D. Danciu / Neurocomputing 164 (2015) 56–70 67

� The block matrix ½W11 W12 ; W21 W22�, corresponding toEqs. (51)–(53), is a five-diagonal matrix.

� Since A2nj ¼ �A1n

j , the entire interconnection matrix has onlytwo cloning templates for the inner cells: A1n

j and Aj3n.� In the case of software-emulated CNNs, the sparsity of the

interconnection matrix W allows the storage optimization anda significant reduction of the number of function evaluations(numerical integration).

4.3.5. Simulated experimentsThe numerical simulations were performed for the approxi-

mating system described by Eqs. (51)–(53) and (56) in the casem=M41 (see the discussion concerning the lumped parametermodel, at the end of Section 4.1). The values of the parameters areas follows:

� the overhead crane [42]: L¼8 m, ρ¼3 kg/m, M¼1000 kg,m¼1500 kg,

� the platform: Y P ¼ 2 m, VP0 ¼ 0:2 m=s, YP0 ¼ 0 m,� the PD controller: K¼0.7, f ðVPÞ ¼ 1:8 � tanhðVPÞ – a sigmoidal

function satisfying the conditions (32) with f ¼ 1:8.

The units of measurement for the variables, as resulted from thechoice of the vector state and the parameters of the system:

� the independent variable t: (s),� the independent variable σ: dimensionless� the velocity v, the forward wave r1 and the backward wave

r2: (m/s),� the displacement y: (m).

The initial conditions xk0, k¼ 1;2Nþ1 were chosen as randomnumbers within the interval ½0;0:01�, with initial and boundaryconditions “matched” – in order to avoid the time-propagation ofthe discontinuities. The numerical integrations were performed byusing the MATLAB ode15s solver in order to cope with the stiffnessissues and to exploit the sparsity of the interconnection matrix.

Figs. 4 and 5 show the space–time transient behavior for theforward and backward waves, whereas Fig. 6 depicts the velocityof the cable elements as resulted from the composition of the twowaves, according to Eqs. (44). In Fig. 7 we present the timeevolutions for the velocities of the platform and the payload.

Fig. 8 is a 3D representation of the approximated solution yðσ; tÞfor the hPDE problem (35)–(20) modeling the controlled overheadcrane. Here and in Fig. 9 one can observe the good convergence ofthe transient controlled process to the reference Y P ¼ 2 m.

Analyzing all these representations we can easily conclude thatthe approximations obtained through the proposed procedurefulfill the requirement of stability preserving for the initial system.Also, one can observe that the procedure ensures the convergenceof the approximation for N¼10 points used for the discretizationof the domain of variation for the normalized space coordinate σ.

From the control theory point of view, the controller parametersmay be adjusted to obtain e.g. a lower overshoot and/or a shortersettling time. For instance, the aforementioned PD controller para-meters led to a transient response of the platform displacementcharacterized by an overshoot of 3.5% and a settling time of 6.85 s –

see Fig. 9. Also, for the payload displacement the same qualitativeindices are 8.7% and 8.7 s, respectively – see Fig. 10. Let us remark herethe time delay at the start of the payload displacement which is inaccordance with the physics of the phenomenon – the effect of timepropagation of the commanded movement (see Fig. 3) along the cableelements from σ¼0 to σ¼1.

5. Concluding remarks and further work

The research reported in this paper considers a Neural Mathe-matics approach for the class of distributed parameter systemsmodeled by a mixed initial boundary value problem for a second-order one-dimensional hyperbolic partial differential equationwith possibly non-standard (nonlinear) boundary conditions. Thisis a formalized problem for our systematic approach whichcombines a convergent Method of Lines with the Cellular NeuralNetwork paradigm. It is the case of an explicit algorithmicparallelism, where each step can be considered as a sub-problemfor which a specific Neurocomputing approach already exists ormay be developed.

From the theoretical point of view, the application of the MoLto hPDEs is not a recent idea but it is accompanied by specificdifficulties as convergence and propagation of the discontinu-ities. We have overcame these difficulties by considering classi-cal solutions of the hPDEs and by applying the Courant–Isaacson–Rees rule for discretization with respect to the spacevariables. This way, the prerequisites for the computationalapproach have been ensured. The discretized structure suggestsquite clearly the CNN paradigm with its cloning templates andconnectivity. The cloning templates as well as the optimalstructure of the CNN was derived by means of a flow diagramconstructed for the approximating system in the normal form ofRiemann invariants.

Concerning the generality of the procedure for the class ofproblems it addresses, we mention that it is already successfullyapplied by the author to several DPS modeled by decoupled hPDEsin the normal form of the Riemann invariants:

� The torsional stick slip oscillations of a drilling plant [38]:

GJθxx� Iθtt�βθt ¼ 0; t40; 0oxoL

GJθxð0; tÞ ¼ caddtθð0; tÞ�ΩðtÞ

� �

GJθxðL; tÞþ IBd2

dt2θðL; tÞ ¼ �T

ddtθðL; tÞ

� �ð66Þ

� The controlled flexible arm of an ocean vessel riser [39]:

ρytt�Tyxxþcyt ¼ f ðx; tÞ; t40; 0oxoL

Md2

dt2yð0; tÞþc0

ddtyð0; tÞ�Tyxð0; tÞ ¼ u0ðtÞþd0ðtÞ

md2

dt2yðL; tÞþcL

ddtyðL; tÞþTyxðL; tÞ ¼ uLðtÞþdLðtÞ ð67Þ

� The vibrational percussion system with boundary dissipation atone extremity and a normal compliance contact condition at theother extremity [40]:

ρ €uðt; xÞ�E uxxðt; xÞ ¼ 1Af ðt; xÞ; xA ð0; LÞ; tA ð0; TÞ

AEuxðt;0Þ ¼ k _uðt;0Þ�gðtÞ; tA ð0; TÞEuxðt; LÞ ¼ �pðuðt; LÞ� lÞ; tA ð0; TÞuð0; xÞ ¼ u0ðxÞ; xA ð0; LÞ_uð0; xÞ ¼ u1ðxÞ; xA ð0; LÞ:

8>>>>>><>>>>>>:

ð68Þ

� A unified model from the Contact Mechanics belonging to theclass of longitudinal vibrations of rods [41]:

μττðτ; xÞ�μxxðτ; xÞ ¼1AE

f ðτTc; xÞm0

k0T2c

μττðτ;0Þ

þ c0k0Tc

μτðτ;0Þþμðτ;0Þ�AEk0μxðτ;0Þ

D. Danciu / Neurocomputing 164 (2015) 56–7068

¼ 1k0g0ðτTcÞ

m1

k1T2c

μττðτ;1Þ

þ 1k1p k1

c1k1μτðτ;1Þþuðτ;1Þ�d

� �� �

þAEk1μxðτ;1Þ ¼ 0

μð0; xÞ ¼ u0ðxÞμτð0; xÞ ¼ Tcv0ðxÞ ð69Þ

In this work we considered a more complex application, a DPCSbelonging to the class of problems without lossless/distorsionlessphenomena. In this case, the system is described by hPDEs withspace-varying parameters, controlled derivative nonlinear bound-ary conditions and coupled equations in the normal form of Riemanninvariants. Moreover, we illustrate the procedure within a com-prehensive context in order to give an overview about the role andthe place of the entire procedure for a control engineeringapplication. More precisely, the present study consists of modeldeduction based on the variational principle of Hamilton, controllersynthesis based on a Control Lyapunov Functional suggested by theenergy identity, application of the proposed procedure based onthe convergent Method of Lines and the paradigm of cell-basedstructures, simulated experiments with the evaluation of solution'sconvergence and of the performances for the closed loop system.

The remarkable advantages of the computational procedureemerge from the features of the convergent MoL as well as thoseof the CNN paradigm. These advantages are: (a) the convergence ofthe approximation to the computed (“true”) solution, (b) a goodcoupling of the boundary conditions with the initial conditions onthe boundaries, (c) the preservation of the basic properties of thecomputed solution, i.e. existence, uniqueness, data dependenceand stability in the sense of Lyapunov, (d) increased accuracy ofthe results due to the massive parallel processing feature whichallows avoiding and/or minimize the rounding or method errors,(e) reduced storage capacity and computational effort, and thus,computational time, due to the implementation based on CNNparadigm which exploits the regularities, parallelism and sparsityof the approximating system.

Concluding, the outcome of the entire contribution is not thesum of the results of the two objectives but a “secured andverified” general guide for solving the aforementioned class ofhPDE problems within the framework of Neural Mathematics. Inour opinion, the research is worth to be followed on severaldirections such as application to other benchmarks but also tomore complicated models with distributed parameters with

several space variables – aiming e.g. to improve computationalperformances in a finite element approach. Another open directionto be followed is a Neurocomputing approach for some of the sub-problems (steps) of the proposed procedure.

Acknowledgment

This work was partially supported by the grant number 10C/2014, awarded in the internal grant competition of the Universityof Craiova. The author thanks to the Associate Editor and theanonymous reviewers for the constructive comments whichhelped to improve the presentation of the paper. The authorthanks to Prof. Vladimir Rãsvan for the valuable discussions onthe results of the paper [18] and their implications.

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Daniela Danciu received the engineering Diploma andPhD Cum Laude degree in Automatic Control at Uni-versity of Craiova, Romania. She is currently an Associ-ate Professor with the Department of Automation andElectronics, University of Craiova. Her research interestsinclude dynamical systems, qualitative analysis ofrecurrent neural networks, time-delay systems, sys-tems with multiple equilibria, computational methodsin Artificial Intelligence. She is a Senior member of IEEE,member of SIAM, President of the local division ofSRAIT (Romanian Society of Automation and TechnicalInformatics), founder member of ANSO (Romanianresearch center: Nonlinear Automation, Stability,Oscillations).

D. Danciu / Neurocomputing 164 (2015) 56–7070