Conical tank level control using fractional order PID controllers:a simulated and experimental study

Cristian J’AUREGUI,Manuel A.DUARTE-MERMOUD ?,Rodrigo OR’OSTICA ,Juan Carlos TRAVIESO-TORRES,Orlando BEYT’IA

1.Electrical Engineering Department and Advanced Mining Technology Center,Universidad de Chile,Av.Tupper 2007,Casilla 412-3,Santiago,Chile;

2.Industrial Technologies Department,Technological Faculty,Universidad de Santiago de Chile,Av.El Belloto 3735,Santiago,Chile

1 Introduction

In the control literature there exist numerous works addressing the level control in conical tanks.This has been addressed from simulation as well as from experimental view points and several control techniques have been em ployed.Results ranging from basic PI/PID control strategies[1,2]to IMC(internal model control)[3]have been informed,having as disadvantage the use of linear control strategies for a nonlinear system and performing the tuning process based on phenomenological model linearizations.Even more advanced strategies such as MPC(model predictive control)[4,5]or PBC(passivity based control)[6]have been reported.How-ever,it is used a model of the system based on a FOPDT(first order p lus death time)identification around operating points.Facing this problem experimentally represents a challenge,due to im plementation problem s,physical limitations of actuators and sensing devices and sensing devicesand the presence of noise at different parts of the plant.It is worth to m ention some experimental works on conical tank system s app lying IMC based PID strategies[7],variable parameters strategies as LPV(linear parameter varying)[8]and model reference based fuzzy adapted PI controllers[9].How ever,the aforementioned works(both in Simulations and experim ents)consider for the parameters tuning a system linearization or a first order plant around certain operating points,neglecting the plant nonlinear behavior.

In this work levelcontrolin a conicaltank is addressed from simulated and experimental points of view.To this extent,a nonlinear model representing the plant behavior in an accurate w ay is first developed.Based on this model representation of the plant,and for comparative purposes only,a classic integer order PI control strategy is developed whose parameters are tuned using the root locus(RL)method and considering a linearized modelat three differentoperating points,similar to what was done in some preliminary works,both,from simulation[10]and experimental[11]view points.Then,integer order(IO)PI/PID control strategies and their fractional order versions,fractional order PI(FOPI)and fractional order PID(FOPID)are designed,where the controller parameters are tuned using Ziegler&Nichols(Z&N),and particle swarm optimization(PSO)tools,in this last case considering all the dynamical information from the nonlinear model,without approximations around operating points as previously cited works.

This work is organized as follow s:Section 2 includes general concepts about fractional calculus,a brief summary on integer and fractional order tuning methods for PID controllers(RL and Z&N)and the optimization method PSO used in this study.Section 3 presents the description of the conical tank system and the derivation of the mathematical model of the plant based on physicalconsiderations.Section 4 presents theprocedures for tuning the controller parameters of the PID/FOPID controllers using the RL method and the Z&N techniques,whose controller parameters are optimized by PSO.Sections 5 and 6 show the simulation and experimental results when applying the aforementioned control techniques to the mathematical model of the plant and on the real installation it self,respectively.Details of the im plementation and comparisons amongst the control techniques studied are also discussed.In Section 7,the main conclusions of the w ork are draw n.

2 General concep ts

In this section,some general concepts about the techniques used in this w ork are presented.The basic definition of fractional operators is first introduced,followed by the structure of the classic PID controller and its extension to the fractional order case(FOPID).Root locus and Z&N tuning techniques are then briefly recalled.Finally,some generals concepts about the optimization technique PSO are described.

2.1 Fundamentals of fractional calculus

Fractional calculus is the extension of the known integral and derivative operators from the integer order case to the case where the order is any real(or even complex)number.The integral FO operator is an extension of the Cauchy Formula for the IO integral of ordernof functionf(·),which is given by the following equation:

Then,the extension to an arbitrary order α is the Riemman-Liouville integral defined as[12,13]

where Γ(·)is the Gamma function[12].

There are several definitions of the fractional order derivativeofa function,however in thisstudyweuse the Caputo’s fractional derivative definition[12,13],since it is the m ost comm only used in engineering applications.Its definition is given by

wheren=m in{k∈N/k＞ α}and α∈R+.Its principal advantage is that the Lap lace Transform uses the initial conditions off(·)and its IO derivatives,which have a physical meaning.

2.2 Fractional PID controller

In general,FO controllers are extensions of IO controllers including advanced IO controllers such as adaptive controllers[14]and sliding mode controllers[15].One of the m ost interesting extension is that of the classical PID controller,which in its IO version is defined,in the Laplace domain,as[16]

The FO counterpart of the PID controller can be formally defined,in Laplace domain for λ,μ ∈ R+,as[17,18]

The first evident difference between PID and FOPID controllers is that the former has only three tuning parameterswhereas the latter has five.This gives more flexibility in designing FOPID controllers.

2.3 Root locus method

A classic strategy of tuning PID controllers corresponds to the root locus technique,which considers a closed-loop control whose overall transfer function(TF)has the form

whereYandRcorrespond to the Laplace transform of the output and the reference system signals,respectively.In this case,it is consideredG(s)as the TF formed by the controller and the process plant,w hileH(s)represents the TF of the measuring device.

This tuning technique seeks to determine the poles,zeros and a high frequency gain of the controller,through the fulfillment of two conditions on the system.The first is that the open-loop TF has unity gain:

Magnitude condition

The second condition is that the angle of open-loop TF be an odd multiple of π:

Angle condition

In order for the previous algebraic conditions form an equations system with a unique solution,specifications of the transient response are im posed,such as a settling timetsand a maximumovershoot MOV,or a dam ping ratio ξ and a natural frequency ωn.

2.4 Ziegler-Nichols method

The problem of tuning parameters of PID controllers is when the parameters defining the dynamics of the system are unknown.To deal with this problem,several strategies have been developed.The most common is the Ziegler and Nichols tuning method,which has also been extended to the case of tuning FOPID controllers.

There are two variants of the method,which are based on a particular dynamic behavior of the plant under specific conditions.This information can be obtained either from the mathematical model and/or experimentally.

2.4.1Step response

The step response of the open-loop TF is first obtained and this method is based on the assumption that the plant can be suitably represented by a TF of first order plus a time delay(FOTFPTD),described as

The system response when it is excited with a step input is assumed to have a similar response as show n in the Fig.1.

From the information contained in Fig.1,parametersK,LandTare obtained.Then the parameters of the PID controller can be determined as indicated in Table 1.

Fig.1 Typical step response for a FOTFPTD for Z&N method.

Table 1 Tuning rules for Z&N step response method.

In this case,a structure for the PID controller of the following type is considered:

Nevertheless,in practical term s,this form is equivalent to the characterization shown in(4).

2.4.2Stability limit

Regarding the stability limit method,the procedure consists in generating a closed-loop control,considering only a proportional gain.This gain is changed until an oscillatory behavior of the system is found,such as the one shown in Fig.2.

From this behavior,the critical period of oscillationPcand the corresponding critical gainKcare determined.Based on these two parameters the gains of the PID controller are determined through the rules presented in Table 2,for a structure of the PID given by(10).

Fig.2 Stability limit for Z&N method.

Table 2 Tuning rules for Z&N stability limit method.

2.4.3Fractional order case

The application of the aforementioned Z&N methods for the PID has also been modified for fractional order PID based on the same time curves m entioned above.How ever,to determine the parameters of the controllers,the method is based on specifications in the frequency domain.

Considering a closed-loop control system whereC(s)represents the controller TF andG(s)is the plant TF,the following conditions are defined for the system under control.

1)The open-loop system must have a cut-off frequency ωcgsatisfying

2)The phase m argin φmmust have the value

3)The closed-loop system must be robust under high frequency noise,therefore its magnitude for a certain high frequency ωhmust be less than a specified valueH,

4)The sensitivity function should have a small magni magnitude at low frequencies to eliminate output disturbances and to converge to the reference.Thus,to a certain low frequency ωlits magnitude must be less than a specified valueN,

5)To ensure robustness under parametric variations of the plant,the open-loop phase of the system must be roughly constant in a neighborhood of the cut-off frequency ωcg,

These five specifications lead to an optimization problem where it ispossible to determine the value of the five parameters of the PIDOF controller.However,the computational effort to perform the minim ization is quite high,so easier alternative applications have been proposed.The ideas in[19,20]seek to generate similar rules to those for the integer order PID tuning.To this extent a plant described by a FOTFPTD(9)is considered.Then specifications in the frequency domain contained in Table 3 are proposed.

Table 3 Frequency domain specifications for FO Z&N method.

Finally,based on the parameters obtained for each test in Figs.1 and 2,the parameters of the FOPID controller are obtained form Tables 4 and 5 for the step response Z&N method and from Tables 6 and 7 for the stability limit Z&N method.

Table 4 FOPID parameters for Z&N step response method.Case 0.1≤T≤5 and L≤2.

Table 5 FOPID parameters for Z&N step response method.Case 5≤T≤50 and L≤2.

Table 6 FOPID parameters for Z&N stability limit method.Case K c P c≤64.

Table 7 FOPID parameters for Z&N stability limit method.Case 64≤K c P c≤640.

2.5 Particle swarm optimization

PSO is an heuristic global optimization technique[21]that belongs to the category of swarm intelligence,which is a sub-category of evolutionary com putation.This technique allow s solving optimization problem s by the use of clusters or swarm s of particles,which computationally simulate the behavior of social groups in nature(flocks,banks,crowds,etc.)in its process of searching for a common benefit[22].

There exist several versions of the PSO algorithm,however in this work we use the original version.In this algorithm thesparticles of the swarm,of dimension λ which are possible solutions of the optimization problem,move along the search space with the aim of minim izing the fitness functionF.Each particle has a position vectorxi∈Rλwhich represent a possible solution of the optimization problem,with its corresponding velocityvi∈ Rλ.The particles social interaction is related with the best position reached by one of the particles.In thekth iteration particles move in the search space iteratively according to the following equations:

wherei∈{1,2,...,s}.pi(k)andg(k)are vectors with the best position of each particle and the best global position,respectively.The constantsc1andc2are the cognitive acceleration and social coefficients,respectively,which determine the influence of individual and collective experiences on the performance of each particle.The term sr1(k)andr2(k)are random numbers uniform ly distributed in the interval[0,1],representing the randomness of any social group.

For the problem studied in this work,PSO is implemented through the use of a MATLAB-SIMULINK toolbox[23],defining the number of particles of the population(P),the number of iterations or generations(G)that determines the ending condition of the algorithm,and an objective functionor fitness function(F),which in this case it is chosen the performance index given by the integral of the absolute error(IAE)defined in the expression(18)fromt=0 until the finite final timeTchosen for the simulation or the experiment.Thus,problem s of the index tending to infinite are avoided[24].

However,it is important to m ention that without loss of generality any other index(or a proper combination of several indexes)may be used.For instance,the integral of the square input(ISI)defined as

3 Model of the conical tank system

The layout of the conical tank system is shown in Fig.3.The water is pumped from the bottom of the recirculating tank to the upper part of the conical tank by means of a pump driven by an induction motor of variable speed driven by a variable frequency drive.

Fig.3 Conical tank system.

The water flows from the bottom of the conical tank to the input of the recirculating tank by gravity through a pipe with a valve whose opening can be fixed manually.The water inside the conical tank can be modeled using the Principle of m ass conservation.Considering homogeneous and constant density hypotheses,the mass of water inside the conical tank is given by

w here ρ is the water density in g/cm3,Vis the volum e of water inside the conical tank in cm3,and ρFin,ρFoutrepresent the mass inflow and massout flow of the liquid expressed in g/s,respectively.Fig.4 shows the relevant variables to derive the conical tank nonlinear model.

The water volum e inside the conical tank can be calculated using the volume cone formula considering a cone of radiusRand heightH,as a function of variableh,according to

However,formula(21)does not represent accurately the water volume inside the conical tank due to the tank mounting imperfections(e.g.,small angle deviation of the cone height with respect to the vertical line).Thus,some experimental tests w ere perform ed in order to built a more accurate volume representation.In[10]it is shown in detail the derivation of the experimental test performed to relate the volumeVinside the conical tank with the water levelh.This relationship turned out to be

which will be used in(20)to derive the nonlinear model.

The inflow and outflow are shown in Fig.4.The inflowFin(t),is a consequence of pump rotation velocity which is directly related with the electrical frequency imposed by the variable frequency drive.Then,it will be assumed a linear dependence between the frequency and the inflow according to

wherefis the electrical network frequency expressed in%as a percentage of the nominal frequency(50Hz)and ranges in the interval 0%-100%.Fin(t)is expressed in cm3/s.

Fig.4 Variables of the conical tank for the nonlinear model.

Parameters α1,α2and β are constants that depend on the geometry of the tank that were experimentally determined through tests specially designed.These are explained in detail in[10].The resulting parameters values are shown in Table 8.The outflowFoutit is produced just for gravity action and depends on the pressure produced by the water in the bottom of the conical tank.Using the Torricelli’s law it can be modeled by

Parameter β is a constant depending on the geometry of the tank that was also experimentally determined through special tests[10].

Table 8 Values of inflow parameters.

Taking into account equations(20)and(22)-(24),together with the values in Table 8,the nonlinear model for the conical tank becomes

wherefis the input,his the output andg(h,f)is a nonlinear function of these two variables show ing clearly the nonlinearities of the conical tank system.

4 Controllers tuning methodologies

This section describes the methodologies used to tune the the controllers em ployed in this study.First the RL methodology is app lied for tuning the integer order PI controllers for three different ranges for the water level(low,medium,high),which will be used for comparison purposes.Next,the tuning procedure for PID/FOPID controllers using Z&N,both step response and stability limit methods in the integer and fractional order cases,and PSO optimization are presented.

4.1 PI tuning using RL method

Using the procedure described in Section 2.3,the following specifications are considered;the settling timets=150s and maxim umovershoot MOV=10%.It is considered first order Taylor approximations of model(25),around different operating points.These approximations are given by

To determine the operating points the whole operation range 15 cm-60 cm is divided into three segments,as illustrated in Table 9.Thehopvalues were selected as the center point of each interval,and thefopvalues are obtained by solving the algebraic equationg(hop,fop)=0.Finally,Table 9 also show s the resulting values forAandBfor each interval of the linear models.

Table 9 Operating points and parameters of the linearized models.

By app lying the tuning procedure described in Section 2.3 the controllers gain values turn out to be those shown in Table 10.

Table 10 PI controller’s parameters tuned by RL method.

In order to test the controllers designed by RL method from simulated and experimental view points,a standard reference signal is defined(See Section 5).Its principal characteristic is that includes all the plant working levels.The results are presented and discussed in Sections5 and 6.

4.2 PI tuning using Z&N method

In this section,the three Z&N methodologies described in Section 2.4 are used to tune the PID/FOPID controllers.

4.2.1Step response

First,the step response of the system is determined using a computational implementation in MATLABSIMULINK of the mathematical model(25).The resulting step response is shown in Fig.5 which allow s to determine parametersK,LandT.

The general step response is shown in the upper part of Fig.5.A detailed view of the initial step response is shown in the lower part of Fig.5,which can be used to compute parametersK,LandT.In blue it is show n the system output,in green it is draw n the tangent at the inflection point and in magenta it is shown the step input.From a graphical analysis the numerical values of parametersK,LandTare determined and they are show n in Table 11.

Fig.5 Simulated system step response for the Z&N method.

Table 11 Parameters for the Z&N step response method.

It is interesting to point out that the delay between the output(water level)and the input(frequency of pump drive)can be experimentally measured in the laboratory plant.The experimental value was found to be approximately 2s,which coincides with the value determined from the simulated step response based on the plant model(25).

Param eter tuning of the integer order PID controller is performed according to the information presented in Section 2.4.In particular numerical values are determined from the values obtained in Table 11 and using the relationships given in Table 1.The final values are show n in Table 12.

Table 12 Controller parameters tuned by the Z&N step response method.

>are tuned based on the discussion are tuned based on the discussion in Section 2.4.3.In principle,considering the step response of Fig.5 it is concluded that parameterT≥5.By using the rules described in Table 5,the resultant controller is not feasible since the derivation order μ results negative.Due to this fact the values of the FOPID are obtained from Table4 given for the caseT≤5.with this relaxation a feasible controller is obtained which is detailed in Table 12.

4.2.2Stability limit

For this method a critical proportional feedback control of gainKcis employed,as stated in Section 2.4.2.App lying this proportional controller to the mathematical model of the plant(25)the response shown in Fig.6 is obtained for a critical gainKc=42.

Fig.6 Simulated stability limit response for Z&N method.

Considering the graphical information contained in Fig.6 the critical periodPcof the system oscillations is determined.See Table 13.

Table 13 Controller’s parameters of the Z&N stability limit method.

following the procedure explained in Section 2.4,the parameters of the PID and FOPID were determined.For the PID controller,rules indicated in Table 2 were used,whereas for the FOPID information contained in Table 7 was used,since in this case it is satisfied thatKc·Pc=80.64.The resulting parameters values are presented in Table 14.

Table 14 Controller parameters values tuned by Z&N stability limit method.

4.3 PID/FOPID tuning using PSO

In this section,the general PSO methodology described in Section 2.5 is used here foroptimum tuning of the controllers.Tw o kind of controllers are studied:IO controllers and FO controllers,including PI/PID strategies and FOPI/FOPID strategies.

The optimization process is performed through simulations using the closed control loop illustrated in Fig.7,where the nonlinear model of the conical tank(25)is utilized.The process considers the minim ization of the IAE index,which uses the tracking error to quantify the controller performance.As a result of the optimization process the best gains for the controllers are obtained.

A general reference signal ranging over the whole operation interval is designed for the optimization process and can be observed in Fig.8.These best values are later used in order to get the simulation and experimental results in Sections5 and 6.

Fig.7 Closed-loop control for optimization process.

Fig.8 Reference signal for optimization process.

For the setup of the PSO optimization process a number of generationsG=50 was chosen,since with this number a com p lete convergence is achieved.Furthermore,according to[25]and the dimensionality of the problem,a number of particlesP=30 was established,and the IAE index was considered as fitness function.Besides,in order to guide the search procedure an initial range for each controller parameter is defined.For the gains,a flexible initial range is defined,that is to say the gains can get values outside the range along the optimization process but its initial value lies in the prespecified interval.The only constraint is that they cannot be less or equal than zero.The initial ranges for choosing the gains are defined byKp∈[50,100]andKi,Kd∈[0,5].In the case of the integration and derivative orders for the FOPI/FOPID controllers,a strict range is defined for λ ∈ [0.75,1]and μ ∈ [0,0.25]to avoid signals divergence related to fractional order operators implementation through Ninteger toolbox[26].The whole optimization process was carried out using the PSO toolbox[23]for MATLAB-SIMULINK.

As a result of app lying the PSO method previously described and illustrated in Fig.7,the optimal values of controller’s parameters were obtained for PI and PID,given by(4)and for FOPI and FOPID,described by(5).These optimal values are shown in Table 15.Besides,the value of the fitness function(IAE Index)is computed for each case.From Table 15 it is seen that the best performance corresponds to the FOPI controller.

Table 15 Optimal values of controllers’parameters from PSO method and fitness function values(IAE).

5 Sim u lation results

In this section,simulation results corresponding to the controllers previously described in Section 2 and tuned using the methodology in Section 4 are presented and discussed.

In order to analyze and com pare the performance of each controller a standard reference signalr(t)is applied,consisting of steps with different am plitudes.This signal moves in the whole range of water level in the tank and it is described in Table 16.

Table 16 Standard reference signal used to evaluate the controller performance.

The simulations w ere carried out using MATLABSIMULINK and for the FO controllers the Toolbox Ninteger was em ployed[26].It is important to point out that implementation of FO operators is a critical point since they are approximated by finite series and approximation problem s may occur[27,28].The results are show n in Fig.9 for the case of using the three different PI controllers tuned for each operating point defined in Tables 9 and 10,using the RL method.In the upper part of Fig.9 is shown the water level evolution and the reference signal,whereas in the low er part the control signal(f)is p lotted in each case.

Fig.9 Simulation results using PI controllers tuned using RL method.

From Fig.9 overshoots are appreciated at each charging process,as well as undershoots at each drainage process.The performance indexes for this case are included in Table 17.

The simulation results using the PID and FOPID controllers tuned with the step response Z&Nmethod under the standard reference are illustrated in Fig.10.In the upper partis plotted the water leveland in the low erpart is the control signal(frequencyf).The corresponding performance indexes are listed in Table 18.

Fig.10 Simulation results for PID/FOPID tuned using Z&N step response method.

Table 18 performance indexes for controllers tuned by Z&N step response method(simulation results).

From the simulation results obtained using Z&N step response method,some comments can be made.The performance of the FOPID controller is better than the PID.This is supported qualitatively from Fig.10 and numerically from Table 18.FOPID has lesser IAE and ISI indexes,together with better characteristics for the transient response.Nevertheless,both controllers are not as good as the PI controller tuned using RL method,as observed from Table 18 and Table 17.Thus,in this case the Z&N step response method could be used as starting point of more elaborated tuning methods,e.g.,rule based heuristic methodologies.

On the other hand,when using the Z&N stability limit method,only the PID controller was able to be determined since in the case of the FOPID the derivative and integral orders w ere such that an unstable behavior was observed.For this reason the analysis of this controller is not included here and only the results for the PID controller are shown in Fig.11.

The indexes and characteristics of the PID controller are summarized in Table 19.It is interesting to point out that in the IO case the PID controller tuned by Z&N stability limit method presents some advantages as compared with the PID tuned by Z&N step response method,since it has better IAE and MOV indexes but a worse ISI.

Fig.12 show s the simulation results for each controller em ploying PSO to tune their parameters and using the standard reference signal described in Table 16.The left part of Fig.12 shows the global evolution of the water level whereas in the right part a zoom of the steady state response(for each step in the test)is plotted,where a slow convergence is observed.Generally speaking FOPI exhibits the best performance of all controllers studied.

The control action obtained in each case can be seen in Fig.13,while Table 20 sum arizes the indexes for all controllers used in Fig.12 for comparison purposes.

Fig.11 Simulation results for PID tuned using Z&N(stability limit).

Table 19 Performance indexes for controller tuned by Z&N stability limit method(simulation results).

Fig.12 Simulation results using PI/PID/FOPI/FOPID tuned by PSO.

Fig.13 Manipulated variable for controllers tuned using PSO.

Table 20 Performance indexes for controllers tuned byPSO method(simulation results).

6 experimental results

The conical tank system corresponds to that existing at the Automatic Control Laboratory of Electrical Engineering Department at Universidad de Chile.Fig.14 show s the real installation.The manual valve opening was fixed at 45°angle in all the experimental trials.

Fig.14 Conical tank system.

The right part of Fig.15 show s the position of the pressure sensor to measure the level inside the conical tank,whereas on the left side it is shown the manual valve.

Fig.15 Manual valve and pressure sensor.

In Fig.16,it is shown the water pump(left side)together with its frequency control system(right side).

All the sensor signals and the control actions of the variable frequency drive go through the input/output modules to the Opto 22 controller illustrated at the left side in Fig.17.The signals are sam p led every 0.05 s and they are transmitted via LAN protocol to the Lab PC specially dedicated to the plant.Using MATLAB-SIMULINK environm ent it is possible to im p lem ent any control strategy over the system,as it can be observed on the right side of Fig.17.

Fig.16 Pump and frequency control.

Fig.17 Opto 22 interface.

The experimental results using the PI controllers designed by RL can be observed in Fig.18,where in the uppergraph is shown the reference tracking for the three controllers designed,and the lower graph show s the associated control action for linearized controllers.

Fig.18 Control results for PI controllers tuned by RL method.

The experimental control loop is depicted in Fig.19.

Fig.19 Closed-loop control for experimental control.

Table21 shows a summary of the experimental results for RL tuned PI controllers.The IAE index is calculated considering the filtered signal obtained from the sensor,but the integral of the square input(ISI)index is calculated using the original signal of the process because it is the real signal being app lied to the plant.

Table 21 Performance indexes for controllers tuned by RL method(experimental results).

The results of the experimental tests for each PI controller tuned by RL method are show n in Fig.18.From this figure it is possible to see that PI controllers have partial capacity to control the level and at some stages it is not possible to take the level to the desired reference,i.e controllers do not m eet control specifications.This is somehow understandable because this tuning strategy is designed for linear plants,which is not the case of study.This situation is quite similar to that was obtained via simulations and presented at[10].

Better results are obtainedwhen using the PSO procedure to tune the controller’s parameters for both PI/PID as well as FOPI/FOPID controllers.In this case the controller parameters are shown in Table 15 together with the fitness function(IAE).The performance of these controllers is presented in Fig.20.

Fig.20 Controlled variable for experimental results using PI/PID/FOPI/FOPID controllers tuned with PSO.

From Fig.20 it is observed that all the controllers successfully follow the reference in contrast with the RL tuned controllers.The corresponding control actions for each controller are illustrated in Fig.21.Asummary of the performance of these controllers is show n in Table 22where IAE,ISI and transient response indexes are presented for each case.From Table 22 it is observed that IAE and ISI indexes for IO controllers are rather similar as com pared with that obtained for their corresponding FO counterparts,i.e.indexes of PI and FOPI,as well as PID and FOPID,are of the same magnitude.On the other hand,for PI/FOPI controllers the energy spent by the control signal is larger than the case of PID/FOPID controllers.

Fig.21 Manipulated variable for experimental results using PI/PID/FOPI/FOPID controllers tuned with PSO.

Table 22 Performance indexes for controllers tuned by PSO method(experimental results).

7 Conclusions

In this work,simulated and experimental results for level control on a conical tank have been obtained using PI/PD controllers in its IO and FO versions.Based on an accurate nonlinear model of the system the parameters of PI/PID and FOPI/FOPID controllers w ere tuned using PSO usingas fitness function the IAEper for mance index,which are summarized in Table 20 and Table 22,for experimental and simulation results,respectively.The experimental results obtained from these controllers are contained in Fig.20 for a sequence of step reference changes.For comparison purposes,PI controllers were tuned via RL method using linearized models around three operating points,as illustrated in Tables 9 and 10,whose behavior is shown in Fig.18 for the same sequence of step reference changes as above.

From this study it is possible to conclude that PI controllers tuned using RLmethod do nothaveagood tracking performance,since they do not follow the reference accurately(Fig.18).PI/PID controllers,both IO and FO,tuned using PSO method behave much better than the previous ones(Fig.20)as far as reference following is concerned.amongst them PI and FOPI present the lowest IAE indexes,and PID and FOPID exhibit the lowest ISI indexes.

The comparisons of FOPID controllers with PID controllers may result unfair.A comparison of the FOPID+PSO with FOPID tuned using other methods(e.g.,Luo‘s method[29]are called for and will be part of the future research to be perform ed.

Finally,there exists a trade off between IAE and ISI indexes suggesting that including in the PSO method a fitness function considering both,properly weighted,would lead to better solutions.This is a subject of future research and can be considered as natural extension of this work.

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