Control Engineering Practice 83 (2019) 165–175

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Control Engineering Practice

journal homepage: www.elsevier.com/locate/conengprac

Modelling, simulation and real-time control of a laboratory tide generationsystemMirco Rampazzo a,∗, Davide Tognin b, Martina Pagan a, Luca Carniello b, Alessandro Beghi a

a Department of Information Engineering, University of Padova, Via Gradenigo 6/B, I-35131, Italyb Department of Civil, Environmental and Architectural Engineering, University of Padova, Via Marzolo 9, I-35131 Padova, Italy

A R T I C L E I N F O

Keywords:Artificial tideSmall-scale experimentsCausal and acausal modellingPIDModel-free controlIntelligent-PID

A B S T R A C T

Small-scale experiments allow to reproduce and understand phenomena and to draw inferences about large-scaleprocesses. In this paper, we consider a peculiar experimental apparatus which is aimed at reproducing a typicallagoonal environment subject to tidal forcings. This apparatus is useful for performing morphometric analysesof synthetic tidal networks. The quality of these kind of experiments strongly depends on the behaviour of theartificial tide that has to exhibit predefined characteristics. To this aim, the height of the artificial water wave iscontrolled in real-time. The experimental apparatus has an intrinsic complexity and it represents an example ofa multi-domain physical system. In order to design and to assess suitable control strategies, we have developed aMatlab-based simulation environment which is able to reproduce the behaviour of the artificial tide generationsystem. The dynamic model is calibrated and validated by using real experimental data and it can be seen asan extremely useful tool in the decision making process of the real control system development. In particular,we have designed, tuned, and tested a model-free control algorithm, that is the intelligent-PI (i-PI). Finally, theproposed controller has been implemented on a real-time hardware and then its performance has been comparedwith that of a standard regulator for different type of experiments.

1. Introduction

Small-scale experiments allow us to reproduce and understand phe-nomena and to draw inferences about large-scale processes, especiallywhen factors that intervene can be hardly controlled. The particularprocess we are interested in is the morphological evolution of la-goonal environments and, to reproduce this phenomenon in scale, wehave designed a peculiar experimental apparatus (Stefanon, Carniello,D’Alpaos, & Lanzoni, 2010). To understand some technical choices andpurposes of the experiment, it may be helpful to briefly report the maincharacteristics of the natural process.

Tidal systems are fragile and complex environments based on adelicate balance between a variety of drivers, such as sediment trans-port, geology, hydrodynamics, vegetation and human actions (Cocoet al., 2013). Understanding the main processes that underlie theformation and development of tidal networks is necessary to addressissues of conservation of these habitats, exposed to the effects of climatechanges and human interference. In a tidal system the primary externalhydrodynamic forcing is represented by the tide. The greatest differencebetween tidal networks and their fluvial counterpart is that they areforced by a bidirectional flux. Indeed, the velocity is direct towardsthe land during the flood and towards the sea during the ebb. As a

∗ Corresponding author.E-mail addresses: mirco.rampazzo@dei.unipd.it (M. Rampazzo), davide.tognin@phd.unipd.it (D. Tognin), martina.pagan@studenti.unipd.it (M. Pagan),

luca.carniello@dicea.unipd.it (L. Carniello), beghi@dei.unipd.it (A. Beghi).

result, sediments are transported in different direction during thesephases, and morphology is shaped according to this actions of erosionand deposition. Moreover, the morphology itself modifies the natureof the flux, because velocity increases in channels and decreases overvegetated shallow areas. Therefore a difference between the velocitycan be experienced during the two phases of the tide and influence themorphology of a tidal environment (Fig. 1a). This difference is referredas tidal asymmetry and can be described using the ratio 𝑝𝑠 between theflood peak and the ebb peak of the cross-sectionally averaged velocity.This parameter provides a distinction of the asymmetrical tidal flow intoflood dominated (𝑝𝑠 > 1) and ebb dominated (𝑝𝑠 < 1). This characteristicof the flow can be easily visualized in the velocity-stage plot (Fig.1b): an ellipse represents a symmetric tide, while increasing asymmetryproduces distorted shapes. What is more, according to Tambroni, Luchi,and Seminara (2017), the nature of the flow field (i.e. flood or ebbdominance) is strictly related with the morphology of the tidal basin,e.g. the position of the point bars relative to the apex of a meanderis strongly affected by the tidal asymmetry (Fig. 2). Once understoodthat the tide cannot simply be modelled with a symmetrical, sinusoidalwave and that particular waveform influences the morphology of tidalbasins, it should be clear that the experimental apparatus has to be able

https://doi.org/10.1016/j.conengprac.2018.10.016Received 10 June 2018; Received in revised form 24 September 2018; Accepted 21 October 2018Available online xxxx0967-0661/© 2018 Elsevier Ltd. All rights reserved.

M. Rampazzo, D. Tognin, M. Pagan et al. Control Engineering Practice 83 (2019) 165–175

Fig. 1. Example of asymmetric tidal wave. Values are scaled in order to be correctlyreproduced in the experimental apparatus.

Fig. 2. Example of tidal meanders in the Venice Lagoon (Italy). The highlighted portionsmark point bars, which, in this case, are shifted seaward with respect to the apex of themeander.

to reproduce wave of any form, period and mean level of propagation,

in particular asymmetrical tides.

The chance to conduct meaningful in-scale experiments relies sig-nificantly on the behaviour of the artificial tide, that has to exhibitpredefined characteristics. To this aim, the height of the artificial waterwave has to be controlled in real-time. To simulate the relevant systemdynamics, for a first assessment of the different control strategies, auseful tool is represented by Computer Aided Control Systems Design(CACSD) softwares (Beghi et al., 2017; Chin, 2017). In order to setupa useful simulation-centric control system design project, there is apreliminary step to be taken into account, namely the derivation of adynamic model (which translates certain interesting properties of thereal system into mathematical equations) of the system to be controlled.In order to develop an effective model we need to knowledge the domainexpert’s (i.e. the actual knowledge of the process and its properties)and the knowledge engineer’s (i.e. how the process functioning and itsproperties can be transferred into a useful model) (Ljung & Glad, 1994).

The considered artificial tide generation apparatus is a non-trivialsystem from a modelling point of view. Basically, it includes a waterpump and a vertical sharp-edge weir, which oscillates vertically thanksto a stepper motor. The artificial apparatus represents an example of amulti-domain physical system (electrical, mechanical, and hydraulic).The system exhibits non-linear behaviours with fast dynamics (e.g. theelectro-mechanical sub-system) and slow dynamics (the hydraulic sub-system), plus dead time (due to the water mass transport). On theother hand, new technologies provide new opportunities for modellingand simulation. In the first part of this paper, we exploit the potentialoffered by a hybrid approach that combine the traditional causal(i.e. block-oriented) modelling approach with the acausal (i.e. declar-ative) one (Kofranek, Matejak, Privitzer, & Tribula, 2008). Beside this,these tools enable physical modelling of multi-domain physical systems.In particular, we have developed a Matlab-based simulation environ-ment for the artificial tide generation system by means of Simulinkcausal block diagram and Simscape acausal components (Dingyu Xue,2013; Ramin S. Esfandiari, 2014). The resulting simulation environmentrepresents an useful tool to design the artificial tide generation appara-tus control system, to which the second part of this paper is devoted.The control system should be able to track the reference wave signal(e.g. with fixed amplitude, period, and mean level of propagation),guaranteeing suitable performance (i.e. stability, robustness, and time-domain performance). In particular, the system should reproduce asym-metrical tides to study the effects on the morphological development ofa lagoonal environment.

As previously mentioned, the system exhibits non-linear behavioursand, moreover, it is expected that certain system configurations andboundary conditions change during new campaigns of experiments. Inthis scenario, traditional model-based control approach may be difficultto use since it is non-trivial to develop and calibrate an effectivesystem dynamic model. Beside these aspects, we must mention thatthe control algorithm has to be designed under some technologicalconstraints, given by the fact that it has to be implemented as an upgradeof a traditional control unit with limited computational and memoryresources.

For these reasons, we want to develop a control algorithm as simpleas possible but able to ensure suitable performance, even when thesystem operating conditions change or if certain system parameters varyover time, avoiding tedious and time-consuming controller retuning.Specifically, we want to exploit the intelligent Proportional–Integral–Derivative (i-PID), that is a step towards a model-free control of plantswith completely or partially unknown dynamics (Fliess & Join, 2009,2013). The i-PID strategy combines a feed-forward control based on theidentification of local models (which represent the plant dynamics overshort periods of time) with conventional PID algorithms. In particular,an i-PI controller is developed and implemented on real hardware. Theexperimental results show the good performance of the i-PI controllerwhich results appropriate to drive the artificial tide generation.

The paper is organized as follows. In Section 2, the experimentalapparatus is depicted. Section 3 is devoted to the system modelling,

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Fig. 3. Experimental set-up.

and to the model calibration and validation. Examples of simulationare shown in Section 4. Section 5 is dedicated to the design of the i-PIcontroller. Results corresponding to the use of standard and intelligentcontrol algorithms are presented in Section 6. Some conclusions andremarks are drawn in Section 7.

2. Experimental apparatus

To reproduce a typical lagoonal environment we have used a largeindoor apparatus, schematically depicted in Fig. 3, which is composedby two adjoining basins representing the sea and a back-barrier lagoon(Fig. 3a, the plant view and Fig. 3b a section).

The lagoon basin is 5.3m long and 4.0m wide, while the much deeperadjacent sea basin is 1.6m long and 4.0m wide. The sea is separatedfrom the lagoon by a barrier of wooden panels, which can be movedto create inlet with different shape and position and a shelf enableus to represent the gentle slope of the sea bed in front of the lagoon.During the experiments, the lagoon is covered with a layer of sedimentsmade of cohesionless plastic grains (Fig. 3c). The tide is generated at thesea by the combined action of a pump and a vertical sharp-edge weir,which is moved by a stepper motor and oscillates vertically. The watercontinuously flowing over the weir is collected in a separate tank, wherethe pump recirculates the flow (Fig. 3d). The apparatus is equippedwith two ultrasonic probes that provide a measurement of the waterlevel in the sea, a potentiometer to measure the position of the weirand a computer driven pantograph to survey the bed elevation withinthe lagoon. It is worth highlighting that, the wave generated at the weirdoes not maintain its form (i.e. it reduces the amplitude and experiencesa time delay) during the propagation to the lagoonal inlet, because ofinertia. This is the reason why the tidal wave cannot be imposed in thesection of the weir but should be reproduced in front of the lagoon tobe sure of the characteristic of the wave.

A block diagram of the system is depicted in Fig. 4. In broad terms,the system can be outlined as two sub-systems that interact with eachother in a structured manner:

• the electro-mechanical sub-system, which includes: the driver, thestepper motor, the worm gear, the lead-screw, and the sharp-edgeweir;

• the hydraulic sub-system, which includes: the water, the sea, theshelf, the lagoon, and the water pump.

In order to conduct meaningful in-scale experiments, which reliessignificantly on the characteristics of tide, the water level at the sea iscontrolled by manipulating the stepper motor position, which in turn,determines the sharp-edge weir height position, while the water pumpis set to a fixed flow rate.

3. Modelling

To preliminary assess different control algorithms, we design amodel-based simulation environment for the artificial tide generationsystem. It is a common practice to build a mathematical model of a com-plex system like the considered experimental apparatus by aggregatingsub-models of its constituent parts, i.e. by using a modular modellingapproach. From this point of view, two options are generally available:causal (or procedural) approach and acausal (or declarative) one. It isworth highlighting that, the causality can explain the evolution whichwas in the past declared as the evolution from block oriented tools intoobject oriented tools.

In the first approach, one assumes that a system can be decomposedinto block diagram structures with causal interactions, the model isdescribed in a form which is close to the numerical solution algorithmand the interaction between the models is formalized in terms of inputand output variables. In this way, it is rather straightforward to simulateelementary and aggregate models. On the other hand, a significanteffort in terms of analysis and analytical transformations is often neededto obtain this system form (e.g. the equations would be converted toordinary differential equations, ODEs, form manually). Furthermore, thecausal model exhibits low re-usability and the corresponding code maybe difficult to read or modified a posteriori. Conversely, in the acausalapproach, the model is described by equations in a context-independentform, without caring about the actual solution algorithm (equations arerepresented as acausal implicit differential algebraic equations, DAEs).It is worth noticing that, in nature real systems are acausal.

Furthermore, the interaction between the models is formalized interms of connection equations without any specification on causality.The acausal approach exhibits both high re-usability and high readabil-ity of the basic models. On the other hand, it is more difficult to go

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Fig. 4. The electro-mechanical sub-system main components are: the driver and the stepper motor, the worm gear with the lead-screw, and the sharp-edge weir. The hydraulic sub-systemincludes mainly: the sea, the lagoon, the water, and the water pump. The water level in the sea 𝑦 is controlled by manipulating the stepper motor position 𝑢. The stepper controller(highlighted by the red coloured dashed-line) is modelled by means the causal approach while the electro-mechanical and hydraulic sub-systems (highlighted by the blue coloureddashed-line) are modelled by using the acausal approach. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

from the mathematical model to the numerical simulation algorithm.It is worth highlighting that, each of the aforementioned approacheshas its intrinsic advantages and disadvantages. In this application, wewant to exploit the advantages provided by both approaches and so amix of causal and acausal models are used. In particular, we developa Matlab-based simulation environment for the artificial tide lagoongeneration system by means of Simulink and Simscape components.In traditional Simulink block-oriented tools (causal), the signals aretransmitted through links between individual blocks and they serve totransfer values of individual variables from the output of one blockto the inputs of other blocks. Input information is processed in theblocks to output information and the interconnection of blocks reflectsboth the structure of the modelled real system and the calculationprocedure (Perelmuter, 2017). Conversely, Simscape lets use and definecomponents as textual files, complete with parametrization, physicalconnections, using DAEs. In the following, the main physical compo-nents of the systems, such as the electro-mechanical sub-system and thehydraulic one are modelled by means of Simscape blocks, while thecontrol systems are modelled by means of traditional Simulink causalblocks.

3.1. Electro-mechanical sub-system

The experimental apparatus is composed by the following mainelectro-mechanical components: a driver, a stepper motor, a worm gearcoupled with a lead-screw, and a sharp edge-weir.

By way of an example, we show the dynamic equations (1) thatreproduce the behaviour of the electric motor, where 𝑒𝐴 and 𝑒𝐵 are theback emfs induced in the A and B phase windings, respectively, 𝑖𝐴 and 𝑖𝐵are the A and B phase winding currents, 𝑣𝐴 and 𝑣𝐵 are the A and B phasewinding voltages, 𝐾𝑚 is the motor torque constant, 𝑁𝑟 is the number ofteeth on each of the two rotor poles. 𝑅 is the winding resistance, 𝐿is the winding inductance, 𝑅𝑚 is the magnetizing resistance, 𝐵 is therotational damping, 𝐽 is the inertia, 𝜔 is the rotor speed, 𝜗 is the rotorangle and 𝑇𝑑 is the detent torque amplitude. The nominal values of themain parameters of the stepper motor are shown in Table 1.

𝑒𝐴 = −𝐾𝑚𝜔 sin(𝑁𝑟𝜗), (1a)

𝑒𝐵 = −𝐾𝑚𝜔 cos(𝑁𝑟𝜗), (1b)𝑑𝑖𝐴𝑑𝑡

=(𝑣𝐴 − 𝑅𝑖𝐴 − 𝑒𝐴)

𝐿, (1c)

𝑑𝑖𝐵𝑑𝑡

=(𝑣𝐵 − 𝑅𝑖𝐵 − 𝑒𝐵)

𝐿, (1d)

𝑇𝑒 = 𝑗 𝑑𝜔𝑑𝑡

+ 𝐵𝜔, (1e)

𝑇𝑒 = −𝐾𝑚(𝑖𝐴 −𝑒𝐴𝑅𝑚

)𝑠𝑖𝑛(𝑁𝑟, 𝜗)

+𝐾𝑚(𝑖𝐴 −𝑒𝐵𝑅𝑚

) cos(𝑁𝑟𝜗) − 𝑇𝑑 sin(4𝑁𝑟𝜗), (1f)

𝑑𝜗𝑑𝑡

= 𝜔. (1g)

Fig. 5 shows different aspects in the electro-mechanical sub-systemSimulink/Simscape modelling. Specifically, Fig. 5a depicts the topology

Table 1Main parameters of the stepper motor.

Parameter Value UoM

Maximum supply voltage 230 VACMotor phase current 2 ArmsNominal torque 4 N mRotor inertia 2.2 kg cm2

Step per revolution 200 stepWinding resistance 1.8 ◦

Current time rise constant 9 m s

and the typology of the hybrid model (causal and acausal). The diagramincludes the causal position controller for the driver stepper motor andthe acausal electro-mechanical sub-system, which comprises the steppermotor driver, the stepper motor, the worm gear, the leadscrew, themass, electrical and mechanical references, and many sensors (e.g. load,position, velocity etc.). In particular, the position controller is developedby means of Simulink causal blocks (Fig. 5b), in which the referenceinput defines the required motor angular position in terms of the stepnumber. The stepper controller provides the pulse direction and thepulse train at the input of the Stepper Motor Driver Simscape block.It is worth highlighting that, the Simulink-PS Converter cyan colouredblocks convert the input Simulink signals into a physical signals and, inthis way, we can build hybrid causal and acausal models. Moreover, byway of an example, Fig. 5c shows the Simscape source code of the idealmechanical translational mass block. It has one mechanical translationalconserving port and the block positive direction is from its port to thereference point. This means that the inertia force is positive if mass isaccelerated in positive direction.

3.2. Hydraulic sub-system

The main hydraulic components of the system (Fig. 6) can bedescribed using the equations that govern the underlying physics. Theexperimental apparatus can be divided into two control volumes: 𝑉𝑐1,which represents the portion of the sea basin between the weir and thedeflector panels, and 𝑉𝑐2, which includes the lagoon and the remainingportion of the sea basin.

For each of these control volumes, we can write a continuityequation. For 𝑉𝑐1 it results as follows:

𝐴𝑐𝑑ℎ𝑐𝑑𝑡

= 𝑄𝑖𝑛 −(

𝑄𝑓 +𝑄𝑠𝑓)

, (2)

where ℎ𝑐 is the water level, 𝐴𝑐 is the area of the control volume, 𝑄𝑖𝑛 isthe pump flow rate, 𝑄𝑓 is the flow rate exchanged with 𝑉𝑐2, and 𝑄𝑠𝑓 isthe overflow rate from the weir; whereas, for the control volume 𝑉𝑐2,the continuity equation results as follows:

𝐴𝑡𝑑ℎ𝑚𝑑𝑡

= 𝑄𝑓 , (3)

where 𝐴𝑡 is the sum of the sea basin area and the lagoon basin area andℎ𝑚 is the water level in this control volume.

We add two equation on the discharge flow

𝑄𝑓 = 𝐴𝑓

√

2𝑔(

ℎ𝑚 − ℎ𝑐)

, (4)

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Fig. 5. Position controller and electro-mechanical sub-systems: hybrid modelling.

𝑄𝑠𝑓 = 𝐶𝑞𝐵√

2𝑔(

ℎ𝑐 − ℎ𝑝)3∕2 , (5)

where 𝐶𝑞 is the discharge coefficient, 𝐵 the width of the weir and ℎ𝑝 theweir crest height.

Fig. 6. Notation used for the hydraulic sub-system modelling.

Fig. 7. Example of an experimental test. The input reference signal is a triangular-wave,the system output is the weir position.

Fig. 8. Model calibration and validation.

Similarly to what has been done for the electro-mechanical sub-

system, the hydraulic model has been implemented in the Matlab-based

simulation environment by using Simulink/Simscape blocks.

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3.3. Models calibration and validation

The aforementioned first-principle models include several parame-ters which can assume different values and so making the shape ofthe equations flexible while maintaining their structure. The nominalmodel parameters are provided from literature and technical data-sheetwhile others parameter values can be guessed by expert knowledgeor can be estimated from real data. Anyway, in order to obtain asuitable representation of the processes of interest that satisfies pre-agreed criteria (e.g. goodness-of-fit or cost function), certain modelparameters need to be adjusted around their nominal values. Here,the calibration procedure is carried out by comparing observed andsimulated data. To this aim, an extensive experimental campaign hasbeen carried out to obtain real data. In Fig. 7 an example of experimentaltest is shown: the considered input to the system is a triangular-wavereference signal to the driver while the considered output is the weirposition. The data gained from the experimental tests have been splitinto a calibration dataset and a validation one. By exploiting the firstdataset, we adjust the values of model parameters by solving offlinean optimization problem that minimize the Root Mean Squared Error(RMSE) between the simulated output and the target one. Finally, thecalibrated model is validated, i.e. it is tested to check its performancesin the validation dataset. It is worth noticing that, validation is alwaysappropriate, in view of the uncertainty affecting the models and for thisreason it is appropriate to use different data for the calibration andvalidation steps. By way of an example, Fig. 8 shows the comparisonbetween target and output data when the calibration of certain electro-mechanical parameters is carried out. Specifically, Fig. 8a refers tothe calibration step: certain electro-mechanical sub-system parameters(e.g. the worm gear friction loss coefficient) have been adjusted byusing an heuristic technique (Beghi, Cecchinato, & Rampazzo, 2011)that minimize the RMSE. Beside this, Fig. 8b is related to the validationstep and we can see that the outputs and the targets are sufficiently closeeach other, indeed the RMSE equals 0.56 mm.

4. Simulation results

We consider here certain relevant simulations, the results of whichare shown in Fig. 9. In particular, the wave height (blue line, Fig.9a) presents a time delay and an amplitude reduction compared withthe position of the sharp edge of the weir (orange line), due to thepropagation processes of the wave. The model output predictions restatethat is necessary to generate the desired wave directly in front of thelagoon in order to be sure of the wave characteristics at the lagoonalinlet. A comparison between the model output predictions with themeasurements of the ultrasonic probe, for the tide (blue markers), andof the potentiometer, for the weir position (orange markers), assessesthe capability of the model to correctly reproduce the real system. Inaddiction to the wave height, the model can also predict the flow rate,hence the velocity values, as presented in Fig. 9b. When the flow rate𝑄𝑓 (blue line) exchanged between the two control volumes 𝑉𝑐1 and 𝑉𝑐2is negative, the velocity is direct seaward and the sediments are erodedand transported from the lagoon to the sea basin.

5. Model-free control

The artificial tide generation system exhibits non-linear behaviourswith both fast dynamics (e.g. the electro-mechanical sub-system) andslow dynamics (e.g. the hydraulic sub-system), plus dead time (due tothe water mass transport). It is also expected that certain system configu-rations, boundary conditions, and some system parameters may changeduring campaigns of experiments. Moreover, the control algorithm forthe artificial tide generation apparatus has to be designed under sometechnological constraints, given by the fact that it has to be implementedas an upgrade of a traditional control unit, with limited computationaland memory resources. From a practical point of view, the use of a

Fig. 9. Hydraulic sub-system: simulation results. (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)

standard regulator sounds good. On the other hand, due to the difficultyin adopting a model-based control approach and/or in setting-up trial-and-error experiments for tuning e.g. PID parameters by inspecting thedynamic behaviour of the process output, we here adopt a model-freecontrol approach, that is the intelligent PID (Bara, Fliess, Join, Day, &Djouadi, 2018; Fliess & Join, 2009, 2013; Join, Robert, & Fliess, 2010;Lafont, Balmat, Pessel, & Fliess, 2015; Rampazzo, Cervato, & Beghi,2017).

This type of control technique is based on an elementary continu-ously update local modelling via the unique knowledge of the systeminput–output behaviour. By way of example, we consider a SISO systemapproximately governed by an unknown finite-dimensional ordinarydifferential equation:

𝐸(

𝑡, 𝑦, ��,… , 𝑦(𝑛), 𝑢, ��,… , 𝑢(𝑛))

= 0, (6)

where 𝑢 and 𝑦 are the input and output variables respectively, whereas𝐸 ∶ R𝑛+1 ×R𝑚+1 → R is assumed to be a sufficiently smooth function ofits arguments. Let us assume that, for integer 𝜈, 0 < 𝜈 ≤ 𝑙:𝜕𝐸𝜕𝑦(𝜈)

≠ 0. (7)

According to the Dini’s implicit function theorem, 𝑦(𝜈) can locally beexpressed as follows:

𝑦(𝜈) = E(

𝑡, 𝑦, ��,… , 𝑦(𝜈−1), 𝑦(𝜈+1),… , 𝑦(𝑙), 𝑢, ��,… , 𝑢(𝑘))

, (8)

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Table 2Correspondence between the gains of sampled classic and intelligent controllers, Fliessand Join (2013).

i-P i-PD i-PI i-PID

PI 𝑘𝑝 −1∕𝛼ℎ𝑘𝑖 𝐾𝑃 ∕𝛼ℎ

PID 𝑘𝑝 𝐾𝐷∕𝛼ℎ𝑘𝑖 𝐾𝑃 ∕𝛼ℎ𝑘𝑑 −1∕𝛼ℎ

PI2 𝑘𝑝 −1∕𝛼ℎ𝑘𝑖 𝐾𝑃 ∕𝛼ℎ𝑘𝑖𝑖 𝐾𝐼∕𝛼ℎ

PI2D 𝑘𝑝 𝐾𝐷∕𝛼ℎ𝑘𝑖 𝐾𝑃 ∕𝛼ℎ𝑘𝑖𝑖 𝐾𝐼∕𝛼ℎ𝑘𝑑 −1∕𝛼ℎ

with, the function E ∶ R × R𝑙 × R𝑘+1 → R. From (8), one can replace(6) by the following phenomenological ultra-local model which is onlyvalid in a very short time interval:

𝑦(𝜈) = 𝐹 + 𝛼𝑢. (9)

In (9), 𝐹 contains all structural information of the process while𝛼 ∈ R is a non-physical parameter, which is typically chosen iteratively,such that 𝐹 and 𝛼𝑢 are of the same magnitude. The derivation order 𝜈is also chosen by experience (typically, 𝜈 is taken equals 1 or 2). If weassume that the measure of the output 𝑦 is available, then 𝐹 could becomputed from (9) by means of 𝑢, 𝛼, and of estimate of 𝑦(𝜈), i.e. ��(𝜈), asfollows:

𝐹 = ��(𝜈) − 𝛼��, (10)

where �� is an approximate value of 𝑢 in order to avoid algebraic loops inthe controller (e.g. �� can be chosen as a past value of the control variable𝑢).

Furthermore, we choose the closed-loop controller such as:

𝑢 =𝑦(𝜈)𝑟 − 𝐹 − 𝒞 (𝑒)

𝛼, (11)

where 𝑦𝑟 is the output reference trajectory and 𝑒 = 𝑦𝑟 − 𝑦 is the trackingerror. The controller 𝒞 should be selected such that a perfect trackingis asymptotically ensured, i.e.:

lim𝑡→+∞

𝑒(𝑡) = 0. (12)

By combining (9) and (11), we obtain:

𝑒(𝜈) + 𝒞 (𝑒) = 0. (13)

It is worth pointing out that 𝐹 does not appear anymore in (13),i.e. the unknown parts and disturbances of the plant vanish. The tuningof the controller 𝒞 becomes, therefore, straightforward for obtaining agood tracking of the reference. This is a major benefit when comparedto the tuning of conventional standard regulator. Indeed, the intelligentcontroller can be regarded as a feed-forward control based on the localplant model in combination with a conventional controller (Fig. 10).

By setting 𝜈 value and by choosing a standard regulator as 𝒞 ,which ensures the desired behaviour, we could obtain different typeof intelligent controllers as follows:

• 𝜈 = 1, 𝒞 = 𝐾𝑃 𝑒(𝑡) → i-P• 𝜈 = 2, 𝒞 = 𝐾𝑃 𝑒(𝑡) +𝐾𝐷 ��(𝑡) → i-PD• 𝜈 = 1, 𝒞 = 𝐾𝑃 𝑒(𝑡) +𝐾𝐼 ∫ 𝑒(𝜏)𝑑𝜏 → i-PI• 𝜈 = 2, 𝒞 = 𝐾𝑃 𝑒(𝑡) +𝐾𝐼 ∫ 𝑒(𝜏)𝑑𝜏 +𝐾𝐷 ��(𝑡) → i-PID

The connection between the intelligent controllers and the conven-tional PIDs was shown by D’Andréa-Novel, Fliess, Join, Mounier, andSteux (2010). In particular, the proof relies on a crude time-sampling ofboth types of regulators and it shows that the gains in a classic PI or PID

Fig. 10. Model-free control architecture; 𝑟 is the reference signal, 𝑦 is the process output,𝐹 is an estimation of the structural information of the process, and 𝑢 is the actuating signalgenerated by the controller.

take into account, if they are properly tuned, the estimated structuralpart of the intelligent controllers. Table 2 depicts the correspondencebetween the gains of sampled classic and intelligent controllers, where𝑘𝑝, 𝑘𝑖, and 𝑘𝑑 refer to standard regulators tuning gains, while 𝐾𝑃 , 𝐾𝐼 ,and 𝐾𝐷 correspond to intelligent controllers tuning parameters; ℎ is thesampling time interval. Moreover, the PI2 and PI2D result as follows:

PI2 → 𝑘𝑝𝑒(𝑡) + 𝑘𝑖 ∫ 𝑒(𝜏)𝑑𝜏 + 𝑘𝑖𝑖 ∬ 𝑒𝑑𝜏𝜎, (14)

PI2D → 𝑘𝑝𝑒(𝑡) + 𝑘𝑖 ∫ 𝑒(𝜏)𝑑𝜏 + 𝑘𝑖𝑖 ∬ 𝑒𝑑𝜏𝜎 + 𝑘𝑑 ��(𝑡). (15)

The double integrals in (14) and (15) seem to be quite uncommonin control engineering, but may result useful in certain applications.

Last but not least, the intelligent controller may bring with it intrinsicadvantages, such as those highlighted by Fliess and Join (2013):

• questions on the structure and on the parameter identificationof systems might lose their importance if the need of a ‘‘good’’mathematical modelling is diminishing;

• many effort on robustness issues with respect to a ‘‘poor’’ mod-elling and/or to disturbances may be viewed less important. As amatter of fact those issues disappear to a large extent thanks to thecontinuously updated numerical values of 𝐹 in (9).

5.1. Estimation of the process structural information

In order to estimate the structural information of the process 𝐹 from(10), it is crucial to have available a good estimate of the differentiationof the output 𝑦 with respect to time. For example, if we consider thederivation order 𝜈 equal to 1, �� can by inferred by using a peculiarfeedback loop where the output of an integrator (i.e. ‘the plant’) hasto track the reference 𝑦, Fig. 11. As a consequence, the integratorinput can be regarded as an estimate for �� (Horn & Reichhartinger,2009). In particular, a robust exact differentiator scheme is used (Levant,1998), where the controller 𝐶 implements the so-called super-twistingalgorithm (Levant, 2007):

𝑦 = 𝑧 − 𝜑√

|𝜉| sign(𝜉), (16a)

�� = − 𝑘 sign(𝜉), (16b)

and where 𝑧 is an auxiliary variable, whereas 𝜑 and 𝑘 are positiveconstants. It is worth highlighting that, thanks to this approach, theerror 𝜉 as well as its first time-derivative are forced to zero in finitetime (Levant, 2003).

Remark on stability margins and model-free control: stabilitymargins, e.g. gain and phase margins (Astrom & Murray, 2008), arequite often utilized in order to check the control design of plants or,more exactly, of their mathematical models. Fliess and Join (2014)have extended calculations related to stability margins to the model-free techniques. In order to ensure satisfactory performances, it would

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Fig. 11. Robust exact differentiator scheme.

Fig. 12. Comparison between the results obtained with a standard PID and a i-PIcontroller, tested on a sinusoidal wave. (For interpretation of the references to colourin this figure legend, the reader is referred to the web version of this article.)

Table 3Controllers parameters.

PI i-PI

𝑘𝑝 = 150 ⋅ 103 [step cm−1] 𝐾𝑃 = 175 [step cm−1]𝑡𝑖 = 𝑘𝑝∕𝑘𝑖 = 7 [s] 𝑇𝐼 = 𝐾𝑃 ∕𝐾𝐼 = 1 [s]

𝛼 = 5 ⋅ 10−3 [cm s−1 step−1]

seem that to obtain a ‘‘good’’ estimate of 𝐹 is the key point. Beside this,as discussed in Fliess and Join (2013), delays remain one of the mostirritating questions in the model-free setting. Further investigations arerequired on these subjects.

6. Experimental results

Among the aforementioned intelligent controllers, we have cho-sen the i-PI configuration. This is based on the rationale that inthe considered application the output reference trajectory usually in-creases/decrease its value over the time and, for example, it could belocally approximated by a ramp. Given a ramp reference trajectory, acontroller with a double integrator is potentially able to avoid perma-nent tracking errors. Due to correspondence between the PI2 and thei-PI (Table 2), we have selected the intelligent-proportional–integralconfiguration. In particular, by exploiting the simulation environment,we have designed, tuned, and tested the i-PI. Along with the simulationexperience, we have moved on to the control system real implementa-tion.

6.1. Sinusoidal tide reference

By way of an example, we consider as tide reference a sinusoidalsignal with amplitude equal to 0.75 [cm] and period equals 8 [min].In particular, we compare the performance of a standard PI regula-tor (Aström & Hägglund, 2006; Visioli, 2006) with that of an intelligent-PI controller. The controllers gains have been initially set by usingstandard tuning rules (O’Dwyer, 2009; Skogestad, 2001; Ziegler &Nichols, 1942) and then they have been adjusted by means of trial-and-error procedures. The controllers parameters are shown in Table 3.

In Fig. 12a we can see the reference signal (the dashed black line),the output provided by the standard regulator (the blue line), and theoutput provided by the i-PI controller (the orange line). The intelligentPI is able to satisfactory tracks the reference, on the contrary by usingthe standard regulator the output is lagging behind the reference. Thetracking error 𝑒 and the controller output 𝑢, for both controllers, aredepicted in Figs. 12b and 12c, respectively. It is worth highlighting that,the PI controller exhibits an ITSE (Integral Time Squared Error) equalto 55060 [cm2s] while the i-PI controller entails an ITSE equals 9980[cm2s], that is about 5 times smaller.

6.2. Ebb-dominated and flood-dominated tide references

In these experimental examples we consider more realistic referencesfor the tide and the considered on-board controller is the i-PI one.Fig. 13 depicts some results concerning the ebb-dominated reference,while Fig. 14 refers to the flood-dominated case. In particular, Figs. 13aand 14a show the height of the tide provided by the ultrasonic probe(the blue line) and the sharp-edge weir vertical position provided bythe potentiometer (the orange line). For the ebb-dominated tide, theoutput, which follows the reference with a small error (less or equalthen about 0.1 [cm], Fig. 13b) and the ITSE, that is equal to 10 874 [cm2

s], confirm the satisfactory performance of the i-PI controller. The samecan be said for the flood-dominated wave, which presents an ITSE equalto 10 867 [cm2 s] and an error again less the about 0.1 [cm] (Fig. 14b).Furthermore, it is worth highlighting that the sharp-edge weir verticalposition signal (the orange line) presents an advance and an highervalue of amplitude with respect to the tidal wave effectively generated

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Fig. 13. Results obtained with the generation of an ebb-dominated tidal wave. (Forinterpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

(the blue line), due to the fact that the wave changes its form duringthe propagation from the weir to the lagoonal inlet. This observationrestates the necessity of generate the desired wave directly in front ofthe lagoon in order to avoid unwanted transformation, that results inthe impossibility of ensuring certain tidal forcing characteristics at the

Fig. 14. Results obtained with the generation of an flood-dominated tidal wave. (Forinterpretation of the references to colour in this figure legend, the reader is referred to theweb version of this article.)

lagoonal inlet. Figs. 13c and 14c show the value of the manipulatedvariable 𝑢, calculated by the i-PI as input for the stepper-motor driver,for the ebb-dominated and the flood-dominated tide, respectively. Itis perhaps worth stressing that smooth profiles of 𝑢 avoid unwantedadditional stresses on the stepper-motor and contribute to a longer life of

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Fig. 15. Results obtained with the generation of an ebb-dominated tidal wave.

the mechanical components. The ebb-dominated tidal wave height andvelocity evolution over time are shown in Figs. 15a and 15b respectively,while Fig. 15c depicts the velocity-stage plot. Similar information arepresented in Figs. 16a, 16b, and 16c for the flood-dominated tidal wave.

Fig. 16. Results obtained with the generation of an flood-dominated tidal wave.

7. Conclusion

CACSD software tools are very useful to simulate relevant systemdynamics for a first assessment of the different control strategies. We

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have developed and tested a Matlab-based simulation environment forthe multi-domain artificial tidal wave generation system. In particular,we have used Simulink traditional block diagrams and Simscape com-ponents to integrate different causal and acausal sub-models in order tomake them transparent to the users, to preserve their testability, and togain the necessary user confidence. A comparison between simulationsand real experiments shows that the simulation tool is able to correctlyreproduce the relevant system dynamics. Once tested different controlstrategies with the simulation tool, we have compared the performanceof a standard regulator with those of an i-PI on the real experimentalapparatus. It is worth noticing that, the intelligent PI can be regarded asa feed-forward control based on the local plant models in combinationwith a conventional PI algorithm. For these reasons the i-PI reaches asatisfactory balance between performance and architectural complexity.From a practical point of view, we have performed various tests on theexperimental apparatus (e.g. by reproducing flood dominated and ebbdominated tidal flows). From the experimental tests, we can concludethat the performance of the i-PI model-free controller is ever better thanthat of the standard regulator. In particular, the use of this model-freeapproach allows to adequately control the system without resorting totedious and time-consuming controller retuning when changes of thesystem operating conditions or time-varying parameters arise.

References

Aström, K. J., & Hägglund, T. (2006). Advanced PID control. Advanced PID control: ISA.Astrom, K. J., & Murray, R. M. (2008). Feedback systems: An introduction for scientists and

engineers. Princeton, NJ, USA: Princeton University Press.Bara, O., Fliess, M., Join, C., Day, J., & Djouadi, S. M. (2018). Toward a model-free

feedback control synthesis for treating acute inflammation. Journal of TheoreticalBiology, 448, 26–37.

Beghi, A., Cecchinato, L., & Rampazzo, M. (2011). A multi-phase genetic algorithm forthe efficient management of multi-chiller systems. Energy Conversion and Management,52(3), 1650–1661.

Beghi, A., Franceschetti, P., Rampazzo, M., Sisti, E., Algarvia, C. A., & Lionello, M.(2017). Modelling and simulation of a convective low temperature sludge dryer withmultilayer belt. In 2017 IEEE 3rd international forum on research and technologies forsociety and industry (RTSI) (pp. 1–6).

Chin, C. S. (2017). Computer-aided control systems design: Practical applications usingMATLAB and simulink. CRC Press Taylor and Francis Group.

Coco, G., Zhou, Z., van Maanen, B., Olabarrieta, M., Tinoco, R., & Townend, I. (2013).Morphodynamics of tidal networks: Advances and challenges. Marine Geology, 346,1–16.

D’Andréa-Novel, B., Fliess, M., Join, C., Mounier, H., & Steux, B. (2010). A mathematicalexplanation via ‘‘intelligent’’ PID controllers of the strange ubiquity of PIDs. In Controlautomation (MED), 2010 18th mediterranean conference on (pp. 395–400).

Dingyu Xue, Y. C. (2013). System simulation techniques with MATLAB and simulink. Wiley& Sons.

Fliess, M., & Join, C. (2009). Model-free control and intelligent pid controllers: Towards apossible trivialization of nonlinear control? IFAC Proceedings Volumes, 42(10), 1531–1550.

Fliess, M., & Join, C. (2013). Model-free control. International Journal of Control, 86(12),2228–2252.

Fliess, M., & Join, C. (2014). Stability margins and model-free control: A first look. In 2014european control conference (ECC) (pp. 454–459).

Horn, M., & Reichhartinger, M. (2009). Model-free control of a thermal plant. In Controland automation, 2009. ICCA 2009. IEEE international conference on (pp. 1839–1843).IEEE.

Join, C., Robert, G., & Fliess, M. (2010). Model-free based water level control forhydroelectric power plants. IFAC Proceedings Volumes, 43(1), 134–139, 1st IFACConference on Control Methodologies and Technology for Energy Efficiency.

Kofranek, J., Matejak, M., Privitzer, P., & Tribula, M. (2008). Causal or acausal modelling:labour for humans or labour for machines. In Technical computing prague (pp. 124–140).

Lafont, F., Balmat, J. -F., Pessel, N., & Fliess, M. (2015). A model-free control strategy foran experimental greenhouse with an application to fault accommodation. Computersand Electronics in Agriculture, 110, 139–149.

Levant, A. (1998). Robust exact differentiation via sliding mode technique. Automatica,34(3), 379–384.

Levant, A. (2003). Higher-order sliding modes, differentiation and output-feedback con-trol. International Journal of Control, 76(9–10), 924–941.

Levant, A. (2007). Principles of 2-sliding mode design. Automatica, 43(4), 576–586.Ljung, L., & Glad, T. (1994). Modeling of dynamic systems. Upper Saddle River, NJ, USA:

Prentice-Hall, Inc..O’Dwyer, A. (2009). Handbook of PI and PID controller tuning rules (3rd ed.). Imperial

College Press.Perelmuter, V. (2017). Electrotechnical systems: Simulation with simulink 1 and simpower-

systbemsTM. CRC Press,.Ramin S. Esfandiari, B. L. (2014). Modeling and analysis of dynamic systems (2nd ed.). CRC

Press Taylor and Francis Group.Rampazzo, M., Cervato, A., & Beghi, A. (2017). Remote refrigeration system experiments

for control engineering education. Computer Applications in Engineering Education,25(3), 430–440.

Skogestad, S. (2001). Probably the best simple pid tuning rules in the world. Journal ofProcess Control.

Stefanon, L., Carniello, L., D’Alpaos, A., & Lanzoni, S. (2010). Experimental analysis oftidal network growth and development. Continental Shelf Research, 30(8), 950–962.

Tambroni, N., Luchi, R., & Seminara, G. (2017). Can tide dominance be inferred from thepoint bar pattern of tidal meandering channels? Journal of Geophysical Research: EarthSurface, 122(2), 492–512.

Visioli, A. (2006). Practical PID control. Springer-Verlag.Ziegler, J., & Nichols, N. (1942). Optimum settings for automatic controllers. Transactions

of the ASME, 64, 759–768.

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