IT 11 083

Examensarbete 30 hpNovember 2011

IMC-based Iterative Learning Control of a Solar Plant

Francisco Martucci Mejía

Institutionen för informationsteknologiDepartment of Information Technology

Teknisk- naturvetenskaplig fakultet UTH-enheten

Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0

Postadress: Box 536 751 21 Uppsala

Telefon:018 – 471 30 03

Telefax: 018 – 471 30 00

Hemsida:http://www.teknat.uu.se/student

Abstract

IMC-based Iterative Learning Control of a Solar Plant

Francisco Martucci Mejía

Solar thermal power is one of the best ways to convert solar energy into electricity.By using solar energy, the internal fluid of a solar collector increases its thermalenergy, which can generate electricity after certain processes, such as steamproduction to drive a steam turbine. Through the years, several control schemes havebeen studied to improve the performance of solar thermal plants in order to obtainthe best possible response of the system and hence the largest quantity of energy. Inthis thesis, three control strategies are applied to improve the performance of a solarthermal plant.

The main objective is to keep the outlet fluid temperature from the system (thecontrolled variable) at a given optimal value by manipulating the fluid flow rate. In thisproject, Internal Model Control (IMC) and Iterative Learning Control (ILC) areapplied to control a simulated nonlinear solar thermal plant. The final strategy is aunion between the two approaches above. The IMC is known for good robustnessincluding handling of systems with time delays. The design of IMC is based on thelinear model of the solar plant. ILC is a strategy for dealing with periodic disturbances.In the thesis, it is assumed that the solar radiation is approximated with a periodicdisturbance with a period time of 24 hours; however this signal is in fact the mainsource of energy of the plant. By using the ILC scheme, the systems controlperformance has the potential to improve every day due to the iterative learning. Byusing a combination of IMC and ILC, their individual advantages are used to increasethe robustness against modelling uncertainties and handling time varying delay systemsas well as decreasing the influence of the periodic disturbances affecting the plant. Theresults from simulations studies show that a combination of IMC and ILC can be aninteresting alternative to effectively control a solar thermal plant.

Tryckt av: Reprocentralen ITCIT 11 083Examinator: Anders JanssonÄmnesgranskare: Bengt CarlssonHandledare: Darine Zambrano

Contents

1 Introduction 21.1 Control problem in solar plants . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Control schemes previously used in solar plants . . . . . . . . . . . . . . . . . . 3

1.3 Objectives and methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 The solar plant 72.1 Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Solar energy collectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Mathematical description of the plant . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Nonlinear model of the solar plant . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Linear model of the solar plant . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.3 Dead-time systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4 Control strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 IMC-based ILC 153.1 Internal Model Control (IMC) . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.1.2 Controller structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Iterative Learning Control (ILC) . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 Controller structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.3 Learning convergence formula . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 IMC-based ILC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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3.3.2 Controller structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.3 Tracking error formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Controller design 224.1 IMC controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 ILC controller design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.2.1 Filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.3 IMC-based ILC controller design . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5 Simulation setup 285.1 Simulation tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 General conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5.3 Linear and nonlinear model in Simulink . . . . . . . . . . . . . . . . . . . . . . 29

5.4 IMC Simulink model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.5 ILC Simulink model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.6 IMC-based ILC Simulink model . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Experimental results and discussion 356.1 Linear and nonlinear model of the plant results . . . . . . . . . . . . . . . . . . 35

6.2 IMC for the solar plant results . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6.3 ILC for the solar plant results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

6.4 IMC-based ILC for the solar plant results . . . . . . . . . . . . . . . . . . . . . 46

7 Conclusions 50

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Acknowledgements

First of all, I am deeply thankful to my supervisor, Dr. Darine Zambrano, whose guidance fromthe beginning to the very end helped me to be able to understand and develop this thesis project.

I am grateful to my reviewer, Professor Bengt Carlsson, for his valuable and useful suggestionsin order to improve this project. Thank you very much to both for your commitment, support andmany valuable advises.

Moreover, I take this opportunity to thank everyone in the International Office and specially NoeliaOllvid for her amazing work as my International Officer and by giving me important advices thatcontributed to make these exchange studies a unique experience.

Finally, words are not enough to express how truly thankful I am to my parents for their totalsupport my whole life in order to help me accomplish not just this, but many other goals. Thisthesis project is for them.

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Chapter 1

Introduction

“Scientists have confirmed that enough solar energy falls on the surface of the Earth every 40minutes to meet 100% of the entire world’s energy needs for a full year. Trapping just a smallportion of this solar energy could provide all of the electricity America uses,” according to UnitedStates of America former Vice President Albert “Al” Gore in his famous speech in Washingtonabout renewable energies [20]. Nowadays, several energy supply strategies such as biomass, windenergy and solar thermal power are being used to overcome all the negative effects produced byfossil-fired power generation, e.g. carbon emissions or harmful gases released during the process.Therefore, fossil-fired power generation should be substituted within the next decades by lessharmful supply strategies. Nowadays, renewable energies supply only 19% of the global energyconsumption [10], counting traditional biomass, large hydro-power, among others.

Solar thermal power stations are among the most cost-effective renewable power technologies,they produce more than 70% of the present production of the world’s electricity generated directlyby solar radiation [19] and these stations might become competitive with fossil-fuel plants withinthe next decades [6]. First of all, because of its reduced operating cost; its main raw material,solar radiation, is free. Furthermore, if only 1% of the Earth’s deserts would be used for solarplants in order to produce clean electric energy, then more electricity would be generated thanthe currently amount being produced on the entire planet by fossil fuels [19]. If the process ofobtaining electricity from fossil fuels like crude oil is compared with the process of obtainingelectricity from solar energy, then the harmful effects to the environment are reduced when solarenergy is used because there is neither carbon emissions nor any kind of harmful gases released inthe process.

Nevertheless, to make it possible, the prices of the produced electricity have to be competitive.For this reason, it is necessary to create new ways to obtain truly clean and efficient electricityfrom renewable energies, to reduce costs and to make the transition from fossil fuels to renewableenergies something feasible within the next decades.

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1.1 Control problem in solar plants

Throughout the years several control schemes have been studied to improve the control processesof the solar thermal plants [8] in order to obtain the best possible response of the system and hencethe largest quantity of energy. One of the characteristics of the plant is that it is impossible tomanipulate the solar radiation, which is also the source of energy of the system. Moreover, thesolar radiation in some geographic locations has a periodic nature approximated with a signal witha 24 hour period because it keeps a similar behaviour everyday, i.e. in cloudless days. The solarradiation affects the response of the plant, as a result of that, this signal is treated as a disturbancesignal. This disturbance generates variations in the oil outlet temperature, which can be slow,such as those produced by variations in the solar radiation on clear days, or they can be fast andstrong, such as those caused when clouds block partially the sun. To overcome the effect of thedisturbances, there is one variable in the system that is used to manipulate the system, in order todecrease or to increase the outlet temperature. This manipulated variable is the oil fluid flow.

In several control strategies for nonlinear processes, such as model based control strategies,local linearization of the real plant (nonlinear model) is usually required, followed by controllerdesign based on the linearized plant (linear model). The linear model of the plant is an approxi-mation of the real plant and this characteristic generates differences between the responses of bothmodels when the system is running far away from its operating point. It is necessary a schemeso as to eliminate these uncertainties as well as keeping the system robustness. Thus the internalmodel control (IMC) scheme is applied. In the case of the solar plant, the response differencesbetween the real plant and the model of the plant are calculated. The IMC scheme is capable ofmaintaining the system robustness and dealing with modelling uncertainties as well as handlingsystems with time delays, e.g. dead-time.

In order to compensate the effect of periodic disturbances due to the solar radiation, anotherscheme is also used. The iterative learning control (ILC) approach tracks the set point by using theinformation of the oil fluid flow learned every day, which is the so called historical cycle data [11].This information is used the following iteration to execute the required control cycle. Thus aftera certain number of iterations (days), tracking the set point can be accomplished with a error veryclose to zero.

1.2 Control schemes previously used in solar plants

A survey of the different advanced automatic control techniques that have been applied to con-trol the outlet temperature of solar plants with distributed collectors during the last 25 years waspresented in [8]. As it was explained in previous sections the main source of energy, the solarradiation, cannot be manipulated and even when it can be approximated as a periodic signal it hasseveral changes related to many environmental factors, which according to control strategies act

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as a disturbance to the solar plants.

Control strategies

A summary of each control strategy described in [8] is outlined below:

1. Model-based predictive control (MPC)

Many different MPC strategies have been applied to control distributed collector system(DCS). The ideas appearing in greater or lesser degree in all the predictive control family arebasically [12]: explicit use of a model to predict the process output at future time instants(horizon), calculation of a control sequence minimizing a certain objective function andreceding strategy, so that at each instant the horizon is shifted towards the future, whichinvolves the application of the first control signal of the sequence calculated at each step.

2. Adaptive control (AC)

The main idea behind AC is to modify the controller when process dynamic changes.It can be said that adaptive control is a special kind of nonlinear control where the statevariables can be separated into two groups moving in two different time scales. The statevariables which change faster, correspond to the process variables (internal loop), whilethe state components which change more slowly, correspond to the estimated process (orcontroller) parameters.

3. Gain scheduling (GS)

Some controllers have the ability of adapting to changes in process dynamics but are notconsidered to be proper adaptive controllers. This is the case of GS controllers, where pro-cess dynamics can be associated to the value of some process variables that can be measuredrelated to the operating point or to environmental conditions. If the dynamic characteristicsof the process can be inferred from measurable variables, the controller parameters can becomputed from these variables.

4. Time delay compensation (TDC)

TDC schemes aim at designing controllers without taking the pure delay of the plantinto account and expecting the closed loop response after the delay to be that expectedfrom the design without considering the delay in the dynamics of the process. Several TDCschemes have been developed for controlling distributed control system (DCS). In [30],an alternative control scheme using a simplified transfer function model including the res-onance characteristics was developed. This controller adopts a parallel control structuresimilar to that of a Smith predictor.

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5. Linear-quadratic-Gaussian (LQG) optimal control

LQG optimal control was an important precursor to the development of the model pre-dictive control (MPC) techniques that are now widely used. This method is based on conven-tional hypothesis: the system is linear, the criterion quadratic and the random disturbancesare gaussian. A separated solution to the problem is given: firstly a state estimation is ob-tained by an observer in the deterministic case or by a Kalman filter in the stochastic one. Alinear control law is applied to this estimation which is fed back and with this the completeregulator is obtained. However, the method was extremely sensitive to imprecision in theparameters and to structural modifications.

6. Nonlinear MPC techniques (NMPC)

Because the majority of controlled processes have inherently nonlinear behaviour, thereare incentives to develop MPC control strategies based on nonlinear process models, bothobtained from physical principles or data, these are mainly black box models based on ar-tificial neural networks (ANN), see [22, 23] for a comprehensive review of applications ofANN in renewable energy systems. In these cases, a nonlinear programming problem mustbe solved in real time at every sampling period instead of the quadratic linear problem typi-cal of standard MPC. The main difficulties of these methods are that the theoretical analysisof properties of the closed loop such as stability and robustness are very complicated be-cause of the appearance of nonlinear models in the formulation and that if the solution ofsolving a nonlinear programming problem at each sampling period is adopted, it is difficultto guarantee the convergence of the algorithm in an adequate lapse of time.

7. Nonlinear control (NC)

As has been mentioned, explicit recognition of plant nonlinearities and their exploita-tion could lead to performance and robust stability improvements, but at the cost of increas-ing the controller complexity. Steps in this direction were made by employing traditionalnonlinear control strategies where nonlinear transformations of input or output variablestake place. In [29, 5], a fuzzy logic scheme is proposed including Lyapunov based adapta-tion and using a simplified plant model. For dealing with plant nonlinearities and externaldisturbances, a nonlinear transformation is performed on the accessible variables such thatthe transformed system behaves as an integrator, to which linear control techniques are thenapplied.

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8. Robust control (RC)

Robust control tries to apply principles and methods that allow the discrepancies be-tween the model and the real process to be explicitly considered. There are many tech-niques for designing feedback systems with a high degree of robustness, some of whichare commented in [15] in the scope of the control of solar plants. Based on the quantita-tive feedback theory (QFT), in [9] a robust controller was developed incorporating a seriesfeedforward controller to solve a simplified problem in which a nonlinear plant subjectedto disturbances is treated in the design as an uncertain linear plant with only one input (thereference temperature to the feedforward controller).

9. Fuzzy logic control (FLC)

Fuzzy logic provides a conceptual base for practical problems where the process vari-ables are represented as linguistic variables which can only present a certain limited numberof possible values and that then be processed using a series of rules. FLC seems to be appro-priate when working with a certain level of imprecision, uncertainty and partial knowledgeand also in cases where the knowledge of operating with the process can be translated intoa control strategy that improves the results reached by other classical strategies.

10. Neural network controllers (NNC)

In [4], an application of ANN identification was presented in order to obtain models ofthe free response of the solar plant to be used in the algorithm proposed in [14]. In [1], thelocal PID controllers of a switching strategy were previously tuned off-line, using an ANNapproach that combines a dynamic recurrent nonlinear ANN model with a pole placementcontrol design. [36] used recurrent ANN aimed at obtaining a pseudo-inverse of the plant toapply FLC techniques. Further improvements led to the works of [21], where a nonlinearadaptive constrained MPC scheme is presented using nonlinear state-space ANN and theironline training. The identification of the ANN is performed in two levels. First, a parameter-ization is obtained for the selected topology by training the ANN on a batch mode, followingan online estimation of weights in order to get rid of any model/plant mismatch due to thequality of the offline training set or the time variant nature of some plants parameters.

1.3 Objectives and methodology

The main purpose of this thesis project is to evaluate, through simulations, the performanceof the IMC-based iterative learning control applied to a solar plant. Initially, the simulatorof the plant is implemented. Later on, the controllers are analysed and designed. Finally, theresponses of three different control strategies are obtained after several simulation scenarios.The output error is computed, so as to compare the results of every control scheme.

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Chapter 2

The solar plant

2.1 Description

The ACUREX-field of Plataforma Solar de Almerıa (PSA) is located in the southern partof Spain, see [13] for a detailed description. The field is composed of 480 distributed solarcollectors, arranged in 10 parallel loops [17]. A collector uses the parabolic surface to focusthe solar radiation onto a receiver tube, which is placed in the focal line of the parabola. Theheat-absorbing fluid (oil) is pumped through the receiver tube, causing the fluid to collectheat, which is transferred through the tube surface. The thermal energy developed by thefield is pumped to the top of a thermal storage tank, whereupon the oil from the top ofthe storage tank can be fed to a power generating system, a desalination plant or to an oil-cooling system, if needed. The oil outlet from the storage tank to the field is at the bottomof the storage tank.

2.2 Solar energy collectors

Solar energy collectors are heat exchangers that use solar energy to increase the thermalenergy of the transport medium. It is composed by a solar collector, which is a device thatabsorbs the incoming solar radiation, converts it into heat and transfers this heat to a fluid,which is flowing through the collector, such as water or oil. The collected energy is carriedfrom the circulating fluid to a thermal energy storage tank from which can be drawn whenthe solar conditions are not optimal for the process. This occurs during nights or duringcloudy days. Solar energy collectors can generate different sources of energy after certainprocesses, for example, steam production to drive a steam turbine to generate electricity.

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There are basically two types of solar collectors, distinguished by their motion: stationaryand non stationary [24].

(a) Stationary collectors: These collectors are permanently fixed in a position and theydo not track the sun, they have the same area for intercepting and for absorbing solarradiation. There are three types of collectors in this category:

i. Flat plate collectorsii. Stationary compound parabolic collectors

iii. Evacuated tube collectors

(b) Non stationary concentrating collectors: Their main characteristic is to be able to trackthe sun during the day, they usually have concave reflecting surfaces to intercept andfocus the sun’s beam radiation to a smaller receiving area, as a result of that increasingthe radiation flux. They are categorized as follows:

i. Parabolic trough collectorii. Linear Fresnel reflector

iii. Parabolic dishiv. Central receiver

The one to be used in the current thesis is the parabolic trough collector, which is the mostadvanced alternative of the solar thermal technologies [24]. The solar thermal plants withparabolic trough collectors are composed by several of these collectors, where each one ofthem contains a reflecting parabolic surface that focuses the solar radiation into a receiverpipe, placed in the parabola focal line, in order to heat up the fluid circulating through it [17],as it can be seen in Fig. 2.1.

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Figure 2.1: Parabolic trough collector. Reprinted from [25]

The input oil (inlet oil) at low temperature is pumped trough a pipe from the bottom of thetank to the field and deposited, via a collecting pipe, at the top of the tank. In the field, theparabolic mirrors concentrate the solar radiation onto a metal pipe containing the workingfluid that is travelling alongside, which is being heated. The stored energy is then trans-formed via a conversion system that can be either a steam turbine for generating electricityor a desalination plant, as it can be seen in Fig. 2.2.

Figure 2.2: Parabolic trough collector thermal plant. Reprinted from [25]

2.3 Mathematical description of the plant

In this section the nonlinear model of the plant is described mathematically. Afterwards alinear model of the plant is obtained by using Taylor theorem.

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2.3.1 Nonlinear model of the solar plant

The mathematical description of this plant can be found in [17]. The solar thermal plantswith parabolic trough collector are composed by several collectors, thus the differentialequation is used to describe the physical characteristics of every individual collector. Thepipes can be divided in a series of control volumes of length ∆X . The model of the plantcan be either distributed (with different number of partitions) or lumped, depending on theadjustment in the length of the partition ∆X . The nonlinear model of the solar plant isdescribed as the time-dependent ordinary differential equation

ρCA∂Tn∂t

= KmodelGI −ρC

∆X (q(Tn − Tn−1))−HtD(Tn − Ta) (2.1)

where ρ is the oil density, C is the oil specific heat, A is the transversal area, Ta is theambient temperature, Kmodel is the optical efficiency, G is the collector aperture and D isthe inner diameter of the pipe. The parameter values are in Table 2.1. The variables of thesystem are: I is the solar radiation (W/m2), Tn is the outlet temperature (C) and Tn−1 isthe inlet temperature (C),Ht = 0.00249(Tn−Ta)−0.06133 is the heat transfer coefficient(W/m2C) and q is the oil flow (m3/s).

Table 2.1: Parameters’ valueParameter Valueρ 750 kg/m3

C 2600 J/kgCA 5.97410× 10−4m2

Ta 32.5 CKmodel 0.41G 1.83 mD 0.02758 m

The main source of nonlinearity in the mathematical description of this solar plant exists inthe multiplication between the oil flow (manipulated variable) and the variation of the oiltemperature (controlled variable) with respect to distance.

To begin with the calculation, the constants are replaced as follows

∂Tn∂t

= aI + bqTn + cqTn−1 +H1D(Tn − Ta) (2.2)

whereH1 = 0.00249δTm − 0.06133 (2.3)

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δTm = Tn − Ta (2.4)

then

H1D(Tn − Ta) = K1DTn2 − (2TaK1D +K2D)Tn + (K1DTa

2 +K2DTa) (2.5)

whereK1 = 0.00249 (2.6)

K2 = 0.06133 (2.7)

and the simplified equation of (2.1) can be written as

∂Tn∂t

= aI + bqTn + cqTn−1 + dTn2 + eTn + f (2.8)

wherea = KmodelG

ρCA= 6.44× 10−4

b = −1A∆X = −11.4335

c = 1A∆X = 11.4335

d = −K1D

ρCA= −5.895× 10−8

e = 2TaK1D +K2D

ρCA= 5.284× 10−6

f = −(K1DTa2 +K2DTa)ρCA

= −1.095× 10−4

In Section 2.3.2, the nonlinear model is linearized by using (2.8).

2.3.2 Linear model of the solar plant

In this section the goal is to obtain a linear model of the plant. By using Taylor theorem,equation (2.8) is linearized as follows

∂Tn∂t

= ∂(Tn + Tn0)∂t

' aI + (bTn0 + cTn−10)q+ (bq0 + 2Tnd+ e)Tn + (cq0)Tn−1 (2.9)

∂Tn∂t' aI + (bTn0 + cTn−10)q + (bq0 + 2Tnd+ e)Tn + (cq0)Tn−1 (2.10)

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whereTn = Tn − Tn0 (2.11)

Tn−1 = Tn−1 − Tn−10 (2.12)

q = q − q0 (2.13)

I = I − I0 (2.14)

equation (2.8) is set to zero in order to obtain the operating point of Tn,

∂Tn∂t

= aI + bqTn + cqTn−1 + dT 2n + eTn + f = 0 (2.15)

Table 2.2: Operating point valuesVariable ValueTn−10 155CI0 900 W/m2

q0 0.006 m3/s

After substituting the values of the operating point in Table 2.2 in equation (2.15), the equa-tion becomes

dTn02 + (bq + e)Tn0 + (aI0 + cq0Tn−10 + f) = 0 (2.16)

after finding the roots of this equation, Tn0 = 163.438C is obtained. This is the value ofTn at the operating point.

The linear model of the plant is given as,

∂Tn∂t' 6.441× 10−4I − 96.4762q − 0.06861Tn + 0.0686Tn−1 (2.17)

Linear plant transfer function

In this section a transfer function of the linear plant is obtained. equation (2.17) is usedand the Laplace transform is applied to convert the function from the time domain to thefrequency domain, x(t)→ X(s). The following equation is obtained

sTn(s) ' KaI(s) +Kbq(s) +Kc− Tn(s) +KdTn−1(s) (2.18)

whereKc− = −Kc (2.19)

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then,(s+Kc)Tn(s) = KaI(s) +Kbq(s) +KdTn−1(s) (2.20)

by clearing Tn(s), the following equation is obtained

Tn(s) = KaI(s)(s+Kc)

+ Kbq(s)(s+Kc)

+ KdTn−1(s)(s+Kc)

(2.21)

where the first and the third term of the second side of the equation (2.21) are the loaddisturbances associated to the plant. Whereas the second term is the one that contains themanipulated variable and therefore the oil fluid contributions to the system. This term ofequation (2.21) is used as the main part of the model of the plant. The other terms areomitted in this model and they are used later on as external disturbances associated to theplant.

Finally, the transfer function that describes the linear plant is given as

Gp(s) = Tn(s)q(s) = Kb

(s+Kc)= −96.4762

(s+ 0.0686) (2.22)

2.3.3 Dead-time systems

Dead-time exists in systems where the action of control inputs takes a certain time beforethe measured outputs are affected. The typical dead-time system is

P (s) = e−sτPr(s) (2.23)

where Pr(s) is some rational function and τ is a positive delay. A time delay is commonin real systems, e.g. systems with transport delay, where mass or energy transportation isinvolved [31]. This delay is present in solar plants, where the dead-time occurs when theoil fluid flows through the pipe. The time delay is a function of the flow rate, q, and it isinherent in the solar plant.

Mathematically, the dead-time of the solar plant, τd, can be represented as [17]

L =∫ τd

0v(τ)dτ → L = Ts

A

j=d∑j=1

q(j) (2.24)

where A is the constant cross-section of the pipe, Ts is the sampling time, q(j) the oil flowcalculated at sample j and d is the number of sampling periods needed for the oil to travelthrough the whole pipe. In other words, the dead-time can be approximated in discretesystems with

τd = d(q)Ts (2.25)

where τd is the dead-time of the system.

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2.4 Control strategies

Control systems for this plant have the objective of maintaining the outlet temperature (inthis case the average outlet temperature of all the parallel loops) at a desired value in spiteof disturbances such as solar radiation (clouds and atmospheric phenomena), irregularitiesin the sun tracking control system, collector reflectivity and inlet oil temperature variations.The oil flow rate is manipulated by the control system through commands to the pump. Itcannot be forgotten that the primary energy source, solar radiation, cannot be manipulated.

The objective of the control strategy is to ensure that the outlet temperature of the field tracksthe reference signal as well as rejecting the disturbances. The latter are mainly due to thefast variations in the solar radiation and in the inlet oil temperature. The control is performedby manipulating the oil flow pumped into the field. Moreover, there are constraints on themanipulated variable as well as on the controlled variable; these constraints are presented inTable 2.3

Table 2.3: Plant constraintsVariable Constraintqmin 2 m3/sqmax 10 m3/sTnmax 300C∆T 80C

where Tnmax is the maximum permitted oil outlet temperature, qmin and qmax are the min-imum and the maximum permitted oil flow, respectively, and ∆T = Tn − Tn−1 is thedifference between outlet and inlet oil temperatures.

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Chapter 3

IMC-based ILC

In this chapter, IMC-based ILC is explained. Initially, the internal model control and the itera-tive learning control are explained. Later on an explanation of the IMC-based ILC approach isgiven. Beginning with a description of their characteristics and followed by a brief backgrounddescription of every scheme.

3.1 Internal Model Control (IMC)

3.1.1 Background

Introduced for first time by Garcia and Morari in 1982 [39], the internal model control (IMC) isa powerful control design strategy for linear systems described by transfer function models [35].By then, however, other researchers had already used similar concepts. The design of the IMC isbased on the fact that the controller contains some representation of the process to be controlled.

3.1.2 Controller structure

The structure of the IMC approach is shown in Fig. 3.1 where Gp(s), Gp(s) and Gc(s) are thereal plant, the model of the plant and the controller, respectively. Moreover, r(t) (R(s) in fre-quency domain), y(t) (Y (s) in frequency domain) and u(t) (U(s) in frequency domain) denotethe reference signal, the process output and the control signal in time domain.

The output of the model is exactly the same as the plant output if there are no modellingerrors and no external disturbances, hence the difference between them is zero and the feedbackis not necessary. If the latter occurs, this system is stable if and only if the open loop system andthe controller are stable. For open-loop and stable systems, the standard IMC structure providesa very simple parameterization of all stabilizing controllers. If the control architecture has been

15

developed based on the exact model of the process, if there are no uncertainties associated with theprocess and if there is no presence of load disturbances, then no feedback is necessary and thenperfect tracking can be obtained because perfect control is mathematically possible [35].

As it is known, time delay deteriorates the closed loop performance. IMC is a scheme usedfor time delay compensation. Additionally, when the feedback signal is included, the closed-loop IMC structure is used in some cases to deal with uncertainties associated to the process anddisturbances, while it ensures the robustness of the system [16].

The IMC design procedure generally consists of two steps if the plant is nonlinear [38]: Lin-earization of the plant followed by controller design for the linear model [8].

Controller Gc(s)

R(s)

Process Model Ĝp(s)

Process Gp(s)

+ - + - +

+

D(s)

Y(s)

Figure 3.1: Internal model control block diagram

3.2 Iterative Learning Control (ILC)

3.2.1 Background

It is widely accepted that the idea of ILC was given by Uchiyama [41]. The idea was not widelyspread because the paper was written in Japanese. Afterwards when other researchers as Arimoto,Casalino, Bartolini and Craig published their papers in English in 1984; they described schemeswhere disturbances and model errors could be compensated for. The term iterative learning controlwas introduced in [40] and it has become the standard notion during the latter part of the time spanfor the research area [34]. However, the essential idea of iterative learning was captured evenbefore 1970, not in the control theory, but in a U.S. patent [42].

The beginning of ILC was in the early 70s when an application for a United States patent on“Learning control of actuators in control systems” was accepted, the idea was to store a commandsignal in a computer memory. A “learner” updates iteratively the signal by an amount related tothe error between the actual response and the desired response of the actuator [45].

16

ILC is a control theory that has been developed significantly in the last two decades. Its idea isstraightforward: to use control information of the next execution to improve the current execution;this is done through memory-based learning [44]. By using a mechanism of trial and error [2], thiscontrol strategy is designed so as to exploit repetitiveness in several control situations to enhanceperformance. ILC can be categorised as an intelligent control methodology; it is an approachfor improving the transient response and tracking performance of dynamic systems that operaterepetitively over a fixed time interval [3].

ILC was motivated by applications to industrial robots, where the basic is to learn from re-peated training, as they do the same task in a periodical way. One of the open areas is the ILCalgorithms for nonlinear systems. Recently, ILC has been applied to chemical processes [43],because there are many chemical processes that are operated periodically, or they are subjectedto periodic disturbances. Algorithms for different classes of nonlinear systems have been devel-oped and analysed, but a unifying theory of ILC for nonlinear systems is still under development.Other examples of issues pointed out to be important future research areas are formalization of thetrade-off regarding robustness versus performance, as well as connections to more general learn-ing paradigms. To derive general ILC methods is also one of the open problems when havingnon identical, but similar reference signals [7] as well as designing algorithms, which are simple,robust and provide good control performance for industrial usage [28].

3.2.2 Controller structure

Iterative learning controllers can be constructed in many different ways. In general, ILC structurescan be classified into two major categories: embedded and cascaded. The block diagram shown inFig. 3.2 represents one of the most commonly used ILC configurations in the embedded category.It is known as the previous cycle learning block [44].

Compensatorϒ

Process Gp(s)

+ - +

+

D(s)

Memory Yr

+

+

Memory U

Ui(s) Ui+1(s)

Yi(s) Yi+1(s)

Yi+1(s) Ri(s) Memory R

Ri+1(s)

Figure 3.2: Iterative learning control block diagram

17

where ri(t) (Ri(s) in frequency domain), yi(t) (Yi(s) in frequency domain) and ui(t) (Ui(s)in frequency domain) denote the reference signal, the process output and the control signal in timedomain, respectively at the iteration i. Gp(s) and γ denote the transfer function of the plant and thelearning gain, respectively. In the present case the system has as a periodic reference signal,Ri(s),and hence Ri+1(s) = Ri(s) [43]. The memory blocks can be thought as FIFO buffers (First infirst out) with a time delay equals to the period of every iteration. The memory blocks are: R, Uand Yr. They save the information of Ri(s), Ui(s) and Yi(s), respectively. The saved informationis used in the next learning cycle providing the learning characteristic of the ILC scheme.

ILC algorithms can be based on a number of properties, such as: linear or nonlinear, discretetime or continuous time as well as frequency domain or time domain. In this thesis, the ILCscheme is based on discrete-time, due to the fact that the signals are defined on a finite timeinterval. Time t is expressed as

t = nTs (3.1)

where n ∈ [0, N − 1] with N as the number of samples and Ts = 5 sec as the samplinginterval [42] if nothing else is stated.

3.2.3 Learning convergence formula

In the following lines it is proved that the ILC approach is convergent [44]. Let,

Yi(s) = Gp(s)Ui(s) (3.2)

where Yi(s) is known as the output signal. The tracking error, e(t), is defined as

ei(t) = r(t)− yi(t) (3.3)

the discrete-time update of the control signal is

ui+1(t) = ui(t) + γei(t) (3.4)

which is the ILC updating law. If a proper value of γ is chosen, a faster tracking response isobtained. Furthermore, by using block algebra and by converting with the Laplace transform fromthe time domain to the frequency domain, x(t) → X(s), the learning convergence condition forILC is derived as follows

Ei+1(s) = R(s)− yi+1(s) = R(s)−Gp(s)Ui+1(s)

Ei+1(s) = R(s)−Gp(s) (Ui(s) + γEi(s)) = (1− γGp(s))Ei(s) (3.5)

Ei+1(s)Ei(s)

= 1− γGp(s) (3.6)

18

if the tracking error signals of the first iteration is finite, then the error becomes smaller afterevery iteration if and only if the following condition holds

‖Ei+1(s)Ei(s)

‖ = ‖1−Gp(s)γ‖ < 1 (3.7)

where the norm ‖.‖ is the infinity norm for all frequencies, with ω ∈ Ω, Ω = [ωa, ωb] andωb > ωa ≥ 0. Ω denotes the frequency band of interest or the frequency band that matches thebandwidth of a controller [44].

3.3 IMC-based ILC

3.3.1 Background

It is well known that perfect tracking can be obtained by the standard IMC structure, only for thecases where there is no uncertainty or load disturbance associated to the process. IMC structureis used to maintain the closed-loop system robust stability and provide time delay compensation.On the other hand, if there are periodic disturbances associated to the process and perfect trackingis required, then the union between IMC and ILC seems like a great idea. This is because thebenefits of every control strategies are used.

The development of an IMC-based ILC control scheme was published [27], where the resultsshow the advantage of this approach by improving the closed-loop performance. It is importantto know that the validations in this paper were based on linear models. However, the ILC schemedeals with nonlinear plants. Moreover, if the system is sufficiently lineal, i.e. close enough to itsoperating point, then the IMC approach is used as well.

3.3.2 Controller structure

The control scheme proposed in [27] is shown in Fig. 3.3, where r(t) (R(s) in frequency domain)and d(t) (D(s) in frequency domain) denote the reference signal and the load disturbance in timedomain, respectively. The subscript i is the iteration number. Gp(t) is the process (real plant)and Gp(t) is the process model (model of the plant). Gc(t) is the IMC controller and γ is thecompensator or learning gain, which are in charge of controlling the process. ui(t) (Ui(s) infrequency domain), yi(t) (Yi(s) in frequency domain) and yi(t) (Yi(s) in frequency domain) arethe control input to the process, the process output and the output of the model in time domain,respectively.

The memory blocks are the learning section, because it provides the information that thesystem uses so as to execute the next control cycle. Finally, the IMC scheme is used when theseoutputs are compared; this difference is one of the feedback signals.

19

The controller to use in the IMC-based ILC approach, defined as Gm(s), is the one used inthe system with just the IMC controller, Gc(s), multiplied by a compensator, which is similar tothe learning gain, γ, of the ILC approach

Gm(s) = γGc(s) = γ1

(λs+ 1)nGp(s)−1 (3.8)

This controller has an adjustable tuning parameter, λ; its importance is related to the timeresponse of the controller. It is important to add that γ, the learning gain, works also as a tuningparameter, whose purpose is to track a desired trajectory; a proper value γ expedites the conver-gence to the set point.

Controller Gc(s)

R(s)

Process Model Ĝp(s)

Process Gp(s)

+ - + - + +

e-­‐θms

+-­‐ Memory

Yi(s)

Ŷi(s)

Yi-1(s)

Ŷi-1(s)

+

-

eθms

D(s) + |

+ +

Ui-1(s) Ui(s)

Yi(s)-Ŷi(s)

ϒ Vi(s)

Figure 3.3: IMC-based ILC block diagram

3.3.3 Tracking error formula

By using block algebra in the frequency domain, Vi(s), the updating law for the ILC approach isobtained. This law is used to compute the control increment for adjustment of Ui(s) and it can bederived as

Vi(s) = Yi(s)− Yi(s)− eθms(Yi(s) + Yi−1(s)− Yi−1(s)) (3.9)

while the ILC control law is given as

Ui(s) = Ui−1(s) + γGc(s)[R(s)− eθms(Yi(s) + Yi−1(s)− Yi−1(s))] (3.10)

20

then, by multiplying both sides by Gp(s) and by using the relationships below, the followingis obtained

Yi(s) = Gp(s)Ui(s) +D(s)

Gp(s)e−θms(Yi(s)− Yi−1(s)) = Gp(s)(Yi(s)− Yi−1(s))

In the current case, the system is dealing with repetitive-type load disturbance and henceD(s)can be also expressed as

D(s) = Yi(s)− Yi−1(s) = Yi(s)− Yi−1(s)

obtaining

Yi(s) =1 + γGc(s)

(Gp(s)−Gp(s)eθms

)1 + γGc(s)Gp(s)

Yi−1(s) + γGc(s)Gp(s)1 + γGc(s)Gp(s)

Yd(s) (3.11)

whereYi(s) = Yd(s)e−θms − Ei(s)

Yi−1(s) = Yd(s)e−θms − Ei−1(s)Ei(s)Ei−1(s) = 1− γGc(s)Gp(s)

1 + Gp(s)γGc(s)(3.12)

It can be concluded that the tracking error will not be enlarged from cycle to cycle, ifGm(s) =γGc(s) is designed to keep stable Ei(s)/Ei+1(s). As a consequence, the output is Yi(s) = R(s)

21

Chapter 4

Controller design

In this chapter a theoretical description is presented in order to explain the steps to follow to obtaina controller for every control strategy. With the linear model obtained in Section 2.3.2, the IMCcontroller is designed. Afterwards, by using the information of the previous sections, the ILCcontroller and the IMC-based ILC are designed.

4.1 IMC controller design

In the present section the theoretical steps to obtain an IMC controller are described. The steps toobtain the proper IMC controller are [18]:

1. If there are not disturbances or differences between plant and real model then there is noneed for feedback and the following equation is enough to control the system,

Gc(s) = Gp(s)−1 (4.1)

2. If there are any disturbances or differences between plant and real model, then the followingrules to obtain Gc(s) are required:

(a) If Gp(s) has more poles than zeros,

Gc(s) = 1(λs+ 1)nGp(s)

−1 (4.2)

i. where n has to maintainNumerator degree ≤ Denominator degree

ii. λ can be adjusted to obtain the bandwidth and the stability required for the closed-loop system.

22

(b) If Gp(s) has unstable zeros (non-minimum phase), the closed-loop system might be-come unstable.

There are two ways for modelling when Gc(s) = Gp(s)−1 is formed:

i. Ignore this factorii. Replace it with βs+ 1

(c) If Gp(s) has a time delay (the e−sτ factor)There are two ways for modelling when Gc(s) = Gp(s)−1 is formed:

i. Ignore this factorii. Approximate this factor by

e−sτ '1− sτ

21 + sτ

2(4.3)

which is known as the first-order Pade approximation.

Designed IMC controller for the solar plant

In this section an IMC controller is obtained for the plant; this is done by using the nominal modeland according to the information retrieved in this section. In Section 4.1 it was explained thatthe controller Gc(s) can be obtained from the inverse of the linear plant. This is one of the maincharacteristics of IMC. In order to simplify, the time delays were neglected in the design of thecontroller as one of the design steps suggested. Then,

Gc(s) = 1(λs+ 1)nGp(s)

−1 = 1(λs+ 1)n

[Kb

(s+Kc)

]−1

Gc(s) = (s+Kc)Kb(λs+ 1) (4.4)

where λ is a parameter so as to adjust the controller. Its value, λ = 100, is chosen in order tohave a fast response of the controller, Gc(s), whereas n = 1 is chosen to keep numerator degree≤ denominator degree.

Finally, the controller to use for the solar plant when the IMC approach, Gc(s), is

Gc(s) = −(s+ 0.06861)42.30408(100s+ 1) (4.5)

23

4.2 ILC controller design

In this section, the method to obtain the compensator for the plant is described when an ILCscheme is used to perform control actions. An optimal value for the compensator is obtained byusing the nonlinear model of the plant (2.1) and by adjusting the gain γ of the compensator, by trialand error. γ is an adjustment parameter to be tuned to reach the best performance in the trackingof the set point. The goal is to determine the value of γ that provides best settling time for the oiloutlet temperature, Tn; this occurs when a faster tracking response is obtained.

1. The following values of γi are tested:

(a) γ1 = −0.0005(b) γ2 = −0.0006(c) γ3 = −0.0007

In the Fig. 4.1 is shown that the system accomplishes to learn from the perturbation, whichis the solar radiation I . However, every value of γ generates the following responses: in the firstday the values of the oil outlet temperature, Tn, for γ = −0.0007 are too large, whereas in thesecond day the compensator gain makes the response of the system smaller than the expected one.Thus a smaller absolute value for γ is needed. γ = −0.0005 was tried but the value Tn did notchange within the days as needed; during the fifth day the tracking was not good. Thus an in-between value was tested, γ = −0.0006, and as Fig. 4.2 illustrates the values of the settling timeare smaller, which stands for a faster tracking response.

In the zoomed Fig. 4.2 is shown that the tracking error during the fifth iteration is closer tozero for γ = −0.0006 than for the other values of the compensator,

To conclude, the value of the compensator that produces the best settling time for the oil outlettemperature is

γ = −0.0006 (4.6)

After finding the most appropriate value of γ, several tests are performed to evaluate theeffectiveness of this approach to track the set point.

4.2.1 Filter design

To improve the response of the system a first-order low-pass Butterworth filter, Gb(jω), is de-signed [42] with cutoff frequency fc = 10 Hz. The implementation of the filter is done in order toprovide a smoother form of the oil flow signal, q. This filter is designed also so as to remove theshort-term fluctuations, which might make the system unstable after several iterations.

24

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

200

400

600

800

1000

1200

Time [Days]

Sola

r R

adia

tion

(I [W

/m2])

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5155

160

165

170

175

180

Time [Days]

Oil

outlet te

mpera

ture

( T

n[°

C])

γ1=−0.0005

γ2=−0.0006

γ3=−0.0007

RefSignal

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.005

0.01

0.015

Time [Days]

Oil

fluid

flo

w q

[m

3/s

]

Figure 4.1: System with ILC scheme for Compensator Gains γ = −0.0005, γ = −0.0006 andγ = −0.0007

The description of an n-order Butterworth low-pass filter obeys the following function in thefrequency domain [32],

Gb(jω)2 = |H(jω)|2 = G02

1 +(jωωc

)2n (4.7)

where |H(jω)|2 is the square magnitude of the frequency response of the filter, n is the order ofthe filter, ω is the angular frequency, ωc = 2πfc is the cutoff frequency of the filter (approximatelythe −3dB frequency) and G0 is the gain at zero frequency.

Finally the first-order low-pass Butterworth filter obtained is

Gb(jω)2 = 1

1 +(jω10

)2 (4.8)

25

5.3 5.4 5.5 5.6 5.7 5.8 5.90

200

400

600

800

1000

1200

Time [Days]

Sola

r R

adia

tion

(I [W

/m2])

5.3 5.4 5.5 5.6 5.7 5.8 5.9155

156

157

158

159

160

161

Time [Days]

Oil

outlet te

mpera

ture

( T

n[°

C])

γ1=−0.0005

γ2=−0.0006

γ3=−0.0007

RefSignal

5.3 5.4 5.5 5.6 5.7 5.8 5.92

4

6

8

10

12

14x 10

−3

Time [Days]

Oil

fluid

flo

w q

[m

3/s

]

Figure 4.2: System with ILC scheme for Compensator Gains γ = −0.0005, γ = −0.0006 andγ = −0.0007 during the sixth iteration

4.3 IMC-based ILC controller design

In this section a controller for the IMC-based ILC is designed. As it was explained before, thecontroller to be used in this approach is the one obtained in Section 4.1, Gc(s), due to the fact thatthe approach is based on the internal model. On the other hand, the compensator γ for the learninggain is obtained in the same way as it was obtained in Section 4.2, i.e. by trial and error. The bestresponse of the plant occurs when γ = 60.

Finally, the IMC-based ILC controller Gm(s) is designed according to (3.8)

Gm(s) = γGc(s) = 60 −(s+ 0.06861)42.30408(100s+ 1) (4.9)

As it was done in the previous section, in order to improve the system response, the first-orderlow-pass Butterworth filter, Gb(s), described in equation (4.8) is used. This is used for the same

26

purposes that in the ILC scheme: to provide a smoother form of the manipulated variable, q, aswell as to remove the short-term fluctuations.

After designing a controller for each control strategy, the following step is to evaluate themso as to obtain which one provides the best performance. This is done through simulations. In thefollowing chapter all the simulation details are shown.

27

Chapter 5

Simulation setup

In this section the simulations to perform are explained, as well as the parameters taken into ac-count in the execution of these tests. It begins with a description of the values of the configurationparameters used in the simulations. Followed by a description of each Simulink control schemeand a brief explanation of how all the simulations were carried out in every simulation.

5.1 Simulation tools

The platform used for the simulations was a computer MacBook Pro 7.1 (Mac OS 10.6.8 operatingsystem) with a processor Intel Core 2 Duo 2.4 GHz. The modelling tool Simulink 7.4 and thenumerical tool Matlab 7.9 were used in order to execute the simulations.

Table 5.1: Configuration parametersParameters

Start time (sec.) = 0Stop time (sec.) = 24*3600*days

Solver type: Fixed-stepSolver name: ode5 (Dormand-Prince)Fundamental sample time (sec.) = 5

The fundamental sampling time was set to 5 seconds, as described in [42]. start time andstop time are, as their names suggest, the initial and the final time, respectively, during execution.The variable days is the number of iterations to simulate. solver name and solver type areshown in Table 5.1.

28

5.2 General conditions

In this section, all the conditions to be used in the simulations are stated. As it was previouslyconsidered, the values of solar radiation, I , are similar every day, under certain circumstances,e.g. during clear days. I is the main source of energy of the system, and it affects directly theperformance of the solar plant. For simulation purposes the reference signal, R, is created basedon I values during the day. This is done by using R as a stair-like signal; The objective is to avoidsaturation of the manipulated variable, q, which would be useless for control purposes. Therefore,the best possible response at the output of the system, Tn, is to provide the highest possible oilflow with the highest temperature. The inlet oil temperature, Tn−1, is defined as a constant valuefor simplification purposes.

As it was explained in Section 2.3.3, the solar plant has a delay factor because it is a systemwhere mass is transported. This delay exists when the actions of the control input has not affectedthe measured results. In the case of the solar plant, the delay occurs when the oil flows throughoutthe pipe. This time is a function of the flow rate variation, q. This time-varying delay is representedby a block called “Variable Time Delay” (VTD). This block calculates the sum of q until the valuereaches the maximum capacity of the pipeline. The total time in completing to fulfil the pipe is thefinal delay; this is calculated by using equation (2.24) that describes the dead-time model. Thisblock is omitted just in cases when simulations require it, if nothing else is said, time delays areused.

It is important to add that the scheme is based on discrete time, due to the fact that the signalsto use in Matlab and in Simulink are defined in a finite time interval t = nTs, n ∈ [0, N − 1],where N is the number of samples and Ts = 5 sec. is the sampling period [42], unless other isgiven.

5.3 Linear and nonlinear model in Simulink

In the following section, an explanation of the simulation set-up to compare the linear and non-linear model is given. In this simulation, the outputs of the linear, Tn (TnLin in Simulink), andthe nonlinear model, Tn (TnNLin in Simulink) are compared. The comparison model shown inFig. 5.1 is mainly composed of two individual blocks: the first block is the linear model, Gp(s),described by the linearized equation (4.5) and the second block is the nonlinear model describedby the nonlinear differential equation (2.1).

The linear model of the plant is an approximation of the real plant and this characteristicgenerates differences between the responses of both models when the system is being executedaway from its operating point. The goal of these tests is to evaluate the modelling performance atthe operating point and at the proximities. This is done by comparing the response of both modelsfor different values of the solar radiation. The variables to use in the evaluations are: I , q and

29

Figure 5.1: Models comparison diagram block for the solar plant in Simulink

Tn−1. The values of I and q change after each evaluation, while Tn−1 is fixed; so as to prove thatout of the operating point the system still has a good response.

5.4 IMC Simulink model

The IMC scheme for the solar plant shown in Fig. 5.2 was implemented in Simulink, which isshown in Fig. 5.3. Two plants compose the system, a linear one, Gp(s) (according to equa-tion (2.22)) and a nonlinear one, Gp(s) (according to equation (2.1)). The used controller, Gc(s),is described by equation (4.5).

Controller Gc(s)

R(s)

Process Model Ĝp(s)

Process Gp(s)

+ - + - +

+ q(s)

Tn (s)

Tn-1, I

VTD

Figure 5.2: IMC for the solar plant in a block diagram

As mentioned earlier, IMC is a control strategy that keeps the system robustness, handlessystems with time delays and solves modelling uncertainties. The objective of the evaluation isto validate the information above, by analysing the performance of the system when the designedcontroller, Gc(s), is used. In order to evaluate IMC applied to the solar plant for compensating

30

Figure 5.3: IMC for the solar plant in a Simulink block diagram

the flow-dependent time delay, the Simulink block called “Variable Time Delay” (VTD) is used inone of the simulations. For this reason the following tests are done in Simulink:

1. Using the solar radiation I from real data

2. The effect of clouds in the IMC scheme

3. Time delays of the system

5.5 ILC Simulink model

The ILC scheme shown in Fig. 5.4 was implemented in Simulink and it is shown in Fig. 5.5. Thismodel uses the nonlinear plant, Gp(s), see equation (2.1), and as compensator, the learning gain,γ, see equation (4.6). Additionally, the use of the low pass filter, Gb(s), see equation (4.8).

In this section, the ILC scheme is tested. The objective is to evaluate the performance ofthe system with the ILC scheme for the plant. As explained in Section 3.2, the objective ofthis scheme, as in any iterative control scheme is to improve system performance through theinformation learned after each iteration. The controlled variable of the solar plant is expected toaccomplish perfect tracking in a finite number of iterations, by handling the repeated disturbances.Furthermore, the objective is to track the set point after several iterations in the shortest possibletime and with a minimum error at the system output, while load disturbances are being rejected.

31

!"#$% &'()*+#,-'./%

0.'1*##%%2)"#$%

+ - +

+

3*('.4%%5.%

+

+

3*('.4%6%

qi(s) qi+1(s)

Tni(s) Tni+1(s)

Tni+1(s) Ri(s) 3*('.4%%

!%

Ri+1(s) 789-*.%2:"#$%

Tn-1, I

+ + +

;<=%

Figure 5.4: ILC for the solar plant in a block diagram

As it was done before, the block “Variable Time Delay” (VTD) is added to represent thedead-time of the plant in one of the simulations so as to evaluate the performance of the systemagainst time delays. This brings several problems to the system because ILC schemes cannothandle systems with time delays. This is an important fact that is evaluated in the simulations.

Figure 5.5: ILC for the solar plant in a Simulink block diagram

5.6 IMC-based ILC Simulink model

The IMC-based ILC scheme for the plant was implemented in Simulink, see Fig. 5.7. ThisSimulink model was based on the block diagram in Fig. 5.6. Two plants compose the system,a linear one, Gp(s) (according to equation (2.22)) and a nonlinear one, Gp(s) (according to equa-tion (2.1)). The controller to use is composed of the compensator gain γ = 60, as described inSection 4.3, and the IMC controller, Gc(s), described in equation (4.5)

There are two ideas, when the IMC scheme is used, that can be applied during the first iteration

32

to improve system performance:

1. To ignore the cycle as suggested in [27].

2. To use the system output as if just the IMC scheme is used.

By ignoring the information at the output of the memory block and not saving anyinformation in it for the second cycle, as well as by converting the compensator gain toγ = 1.

These ideas improve the system response because the information of the first iteration is not used,which reduces the error. In this case, the second alternative is used because the simulation resultswere better.

Controller Gc(s)

R(s)

Process Model Ĝp(s)

+ - + - +

+

VTD

+-­‐ Memory

Tni(s)

Ťi(s)

Tni-1(s)

Ťi-1(s)

+

-

eθms

Tn-1, I + |

+

+

qi-1(s) qi(s)

Tni(s)-Ťi(s)

ϒ Process Gp(s)

Filter Gb(s)

Figure 5.6: IMC-based ILC for the solar plant in a block diagram

The tests consisted of: setting the reference signal of the system,R, as it was explained in 5.2,followed by simulating with real values of the solar radiation, I , for clear and cloudy days. Theobjective is to demonstrate that the system manage to track the set point, even during cloudy days.

MSE calculation

In statistics, the mean squared error (MSE) of an estimator is one of many ways to quantify thedifference between values implied by an estimator and the true values of the quantity being esti-mated. MSE measures the average of the squares of the “errors.” The error is the amount by whichthe value implied by the estimator differs from the quantity to be estimated. The difference oc-curs because of randomness or because the estimator does not account for information that couldproduce a more accurate estimate [26].

33

Figure 5.7: IMC-based ILC for the solar plant in a Simulink block diagram

In the present case, the error is defined as the difference between the outlet temperature, Tnand the set point, which is the reference signal,R. The definition of MSE [33] for error calculationof the plant is

MSE = 1n− 1

n∑i=1

(Xi −Xi

)2(5.1)

where Xi is the outlet temperature and Xi is the set point temperature. It is expected that theerror tends to decrease after every iteration.

After the simulation parameters are set up, the simulations are ready to be executed. Theconfiguration of the simulations and the analytical methods, explained before, are needed to obtainthe scheme that provides the best performance for the solar plant. In the following chapter all thesimulation results are shown as well as the analysis of the obtained results.

34

Chapter 6

Experimental results and discussion

In the following sections of this chapter, the simulation results are shown for each control strat-egy. The system performance is analysed using these results. Afterwards, numerical results areobtained by calculating MSE for each scheme. Finally, the MSE results are compared among allthe control strategies in order to determine which scheme provides best performance.

6.1 Linear and nonlinear model of the plant results

1. Response of both models at the operating point

The evaluation consist in using q and I at the operating point; according to Table 2.2.The result is that the difference between both models is zero and hence the outputs are thesame. This occurs because the linear plant was calculated in this operating point, thus whenthese values are used an ideal response is obtained.

35

2. Response of both models when the solar radiation is a stair-like signal

The objective of this test is to evaluate the linear model performance when the systemmoves away from its operating point. The solar radiation, I , was designed as a stair-likesignal with values located away from the system operating point. The indexes A and Bmake a distinction between different values of the solar radiation. TnNLin and TnLin are theoutputs of the linear and the nonlinear model, respectively.

8 10 12 14 16 18 20850

900

950

1000

Time [Hours]

Sola

r R

adia

tion

(I [W

/m2])

8 10 12 14 16 18 20−6

−4

−2

0x 10

−7

Time [Hours]

Diffe

rence o

f o

utle

t te

mpe

ratu

res(

Tn[°

C])

8 10 12 14 16 18 20

2

4

6

8

10x 10

−3

Time [Hours]

Oil

fluid

flo

w q

[m

3/s

]

Solar RadiationA

Solar RadiationB

Figure 6.1: Difference in outlet temperatures of both models with stair-like solar radiation

In Fig. 6.1 the upper plot shows the solar radiation, I , with two different values, A and B;whereas the lower plot shows the oil flow, q0 = 0.006 m3/s. The middle plot in Fig. 6.1shows the difference between TnNLin and TnLin, forA andB. The difference between bothsignals is almost zero, even when I0 6= 900 W/m2. This means that the linearization hasa good performance, even though the values are far away from the operating point. Theresult is simple the closer that the system is executing to the operating point, the better theresponse of the system is. Nevertheless, it is a very accurate response.

36

3. Response of both models when the oil flow is a stair-like signal

The objective of this test is to evaluate the linear model performance when the systemmoves away from its operating point. The oil flow, q, was designed as a stair-like signalwith values located away from the system operating point. The indexes A and B make adistinction between different values of the oil flow. TnNLin and TnLin are the outputs of thelinear and the nonlinear model, respectively.

0 5 10 15 20899

899.5

900

900.5

901

Time [Hours]

So

lar

Rad

iation (

I [W

/m2])

0 5 10 15 200

0.2

0.4

0.6

0.8

Time [Hours]

Diffe

rence o

f ou

tlet

tem

pera

ture

s (

Tn[

C])

0 5 10 15 206

6.5

7

7.5

8x 10

−3

Time [Hours]

Oil

fluid

flo

w q

[m

3/s

]

Oil Flow

A

Oil FlowB

Figure 6.2: Difference in outlet temperatures of both models with stair-like oil flow

As explained before the main source of nonlinearity in the mathematical description of thissolar plant exists in the multiplication between q and (Tn−Tn−1). In Fig. 6.2 the lower plotshows the oil flow, q, with two different values, A and B; whereas the lower plot shows thesolar radiation, I0 = 900 W/m2. The middle plot in Fig. 6.2 shows the difference betweenTnNLin and TnLin, forA andB. The results shows that the linear models difference betweenboth signals is less than 0.8C, when q = 0.008 m3/s. This means that the nonlinearity doesnot affect the good performance of the linear model, even though the values are far awayfrom the operating point.

37

4. Response of both models when the solar radiation is real data

The purpose in this test is to evaluate the system performance when a real situationoccurs. In this case I is real data obtained from the solar radiation. The simulation resultsare shown in Fig. 6.3.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

200

400

600

800

1000

1200

Time [Days]

Sola

r R

adia

tion

(I [W

/m2])

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6

−4

−2

0x 10

−5

Time [Days]

Diffe

rence in o

utlet

tem

pera

ture

s (

Tn[°

C])

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

2

4

6

8

10x 10

−3

Time [Days]

Oil

fluid

flo

w q

[m

3/s

]

Figure 6.3: Difference in outlet temperatures of both models from real data for solar radiation

The upper plot in Fig. 6.3 shows the solar radiation, I , from real data collected during aclear day; whereas the lower plot shows the oil flow, q0 = 0.006 m3/s. The middle plot inFig. 6.3 shows the difference (error signal) between TnNLin and TnLin, when this situationoccurs. Although I0 6= 900 W/m2, the error signal, E is almost zero once again, e.g. whenI = 27.83 W/m2, |E| = 5.758e× 10−5.

The result is that the model of the plant is an accurate approximation of the real plant.

38

6.2 IMC for the solar plant results

The tests presented in this section are related to the IMC performance for the solar plant. The mainpurpose here is to observe the response of the system to a reference signal, R, which is the desiredvalue to track.

1. Using the solar radiation I from real data

The system is evaluated resulting for a complete day what it can be observed in thefollowing graphic:

0 5 10 15 200

200

400

600

800

1000

1200

Time [Hours]

Sola

r R

adia

tion

(I [W

/m2])

0 5 10 15 20155

156

157

158

159

160

161

Time [Hours]

Oil

ou

tlet

tem

pera

ture

( T

n[°

C])

Tn

IMC

RefSignal

0 5 10 15 200.005

0.01

0.015

Time [Hours]

Oil

fluid

flo

w q

[m

3/s

]

Figure 6.4: System response from real data of solar radiation in a cloudy day by using the IMCscheme

The upper plot in Fig. 6.4 shows the solar radiation, I , from real data collected during aclear day; whereas the lower plot shows the oil flow, q. The set point, R, is a stair-likesignal. The middle plot in Fig. 6.4 shows the oil outlet temperature, Tn. Where it is shownthat the system performance is affected because the linear plant is moving away from the

39

operating point around which the linearization was done. Furthermore, Fig. 6.4 illustratesthat the system compensates the modelling uncertainties during the execution. The obtainedresponse is good and the values of the output track with a small error the set point.

2. The effect of clouds in the IMC scheme

Once again, the test are done by using real data for the solar radiation, I , and the sameset point. The effect of clouds is shown in the Fig. 6.5.

10 11 12 13 14 15 16 17 18 19 200

200

400

600

800

1000

1200

Time [Hours]

So

lar

Rad

iation (

I [W

/m2])

10 11 12 13 14 15 16 17 18 19 20156

157

158

159

160

161

Time [Hours]

Oil

outlet te

mpera

ture

( T

n[°

C])

Tn

IMC

RefSignal

10 11 12 13 14 15 16 17 18 19 204

6

8

10

12x 10

−3

Time [Hours]

Oil

fluid

flo

w q

[m

3/s

]

Figure 6.5: System response from real data of solar radiation in a clear day by using the IMCscheme

The upper plot in Fig. 6.5 shows the solar radiation, I , from real data collected during acloudy day; whereas the lower plot shows the oil flow, q. As it can be observed in themiddle plot in Fig. 6.5, during cloudy days, the values at the output are even more affected.The system manages to compensate the effect of the disturbance. However, the problemof using just the IMC scheme is that every day the same occurs and hence there is noimprovement; in other words, there is no “learning” after every iteration. Nevertheless,

40

the system performance shows that IMC keeps the system robustness while it is dealingwith the modelling uncertainties.

3. Time delays of the system

The goal of this test is to show the performance of IMC applied to the solar, by takinginto account the flow-dependent time delay.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

500

1000

1500

Time [Hours]

Sola

r R

adia

tion

(I [W

/m2])

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1155

160

165

Time [Hours]

Oil

outlet te

mpera

ture

( T

n[°

C])

TnIMC−DeadTime

TnIMC

RefSignal

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 16

8

10

12x 10

−3

Time [Hours]

Oil

fluid

flo

w q

[m

3/s

]

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140

60

80

Time [Hours]

Dead−

tim

e d

ela

y [s]

Figure 6.6: System response with dead time from real data of solar radiation by using the IMCscheme

The upper plot shows the solar radiation, I , from real data collected during a clear day.The third plot in Fig. 6.6 shows the oil flow, q; whereas the lower plot shows the dead-timedelay, which is, as said before, flow dependent. Furthermore, the second plot compares thesystems output: TnIMC−DeadT ime (when the system has a time delay) and Tn (when thesystem doesn’t have a time delay). The plot illustrates that IMC scheme compensates thedelay of the plant.

41

To prove numerically the latter and that the IMC technique presents accurate results, the meansquare error (MSE) is calculated with equation (5.1).

Table 6.1: MSE of the solar plant response with the IMC schemeDay IMC

1 0.00322 0.00323 0.00324 0.00325 0.00326 0.00327 0.00328 0.00329 0.003210 0.0032

Total MSE 0.0316

In Table 6.1 is shown that the values of the MSE for the IMC response are constant everyday, which is the expected response because there is no learning, thus the system does not reducethe error within the days and the performance is not optimal for the solar plant. Without learn-ing, the solar plant would be able only to produce the same performance every day, without anyimprovement.

It can be concluded that the internal model control scheme keeps the robustness of the system,handle systems with time delays and deals with the modelling uncertainties, but its usage is not to-tally adequate for the solar plant because it does not eliminate the periodic disturbances associatedto the plant and hence there is always an error in the system. That is the reason why it is necessaryto evaluate another scheme that can deal with these disturbances.

42

6.3 ILC for the solar plant results

In this section results are shown in order to evaluate the performance in the system with the ILCscheme.

1. Load disturbance rejection

An important characteristic of the ILC scheme is to be able to perform load disturbancerejection. The test consists in using real data for solar radiation of different days so as toobserve the response of the system. Three days are partly cloudy and the other days arecloudless.

4 4.5 5 5.5 6 6.5 7 7.5 80

200

400

600

800

1000

1200

Time [Days]

Sola

r R

adia

tion (

I [W

/m2])

4 4.5 5 5.5 6 6.5 7 7.5 8154

156

158

160

162

Time [Days]

Oil

outlet te

mpera

ture

( T

n[°

C])

TnILC

RefSignal

4 4.5 5 5.5 6 6.5 7 7.5 82

4

6

8

10

12x 10

−3

Time [Days]

Oil

fluid

flo

w q

[m

3/s

]

Figure 6.7: System response to differences in the solar radiation (disturbance)

The upper plot in Fig. 6.7 shows the solar radiation, I , from real data collected during fourdays: two cloudy days and two clear days; whereas the lower plot shows the oil flow, q. Asit can be observed in the middle plot in Fig. 6.7, during the first cloudy day, the system isnot able to compensate the effect caused by the clouds. During the second day the learnedinformation of the first iteration helps to compensate correctly the system. However, when

43

I changes noticeably from one day to another, the learning algorithm adds this effect on thenext day, which affects negatively the system response.

The Fig. 6.7 illustrates the fact that the ILC control strategy is not able to reject disturbancesin the following iteration if I is not a periodic signal. That is why the optimal use of theILC algorithm is during clear days, because the changes in the solar radiation are small afterevery iteration. These tracking results are shown in the following 3D Fig. 6.8, where a verygood tracking is accomplished after ten iterations and almost perfect tracking is obtainedafter thirty iterations (days).

0

5

10

15

20

0

10

20

30

40

50

60

155

156

157

158

159

160

161

Time [Hours]Number of iterations [days]

Ou

tlet

tem

pera

ture

[°

C]

TnILC

RefSignal

Figure 6.8: Tracking results for the system with ILC scheme

In this case the test demonstrated that ILC is an excellent tool for load disturbance rejection,because the signal I is totally periodic. Finally, MSE is used once again in order to comparenumerically the errors at the output of the systems using ILC and IMC.

44

MSE calculation

The MSE is calculated with equation (5.1) to compare the last scheme with previous results. Aftercalculating the MSE of the output with the ILC approach, the following Table 6.2 is obtained

Table 6.2: MSE of the solar plant response with the ILC schemeDay ILC time Delay ILC No time Delay

1 2.3150 2.038212 0.0123 0.009713 0.0075 0.005274 0.0058 0.003465 0.0054 0.002536 0.0059 0.001977 0.0073 0.001618 0.0101 0.001369 0.0155 0.0011810 0.0246 0.00104

Total MSE 2.4093 2.06636

The values obtained show the fact that when there are no time delays in the system the ob-tained response is optimal after few iterations. For example, the MSE obtained within the thirtiethday is equal to 9.96872× 10−5, which stands for perfect tracking. On the other hand, when thereare time delays in the system, like the one characteristic of the solar plant, the response of the ILCapproach is not as effective. After several iterations the error starts increasing because the systemdoes not manage to compensate the effect produced by time delays.

It can been said that the ILC approach accomplishes perfect tracking of the set point within afinite number of iterations if the task is repeated from iteration to iteration and if there are no delaysassociated to the plant. This means that ILC handles the effects of only periodic disturbances. Thatis the reason why an individual ILC approach should not be used to control the plant.

45

6.4 IMC-based ILC for the solar plant results

In this section the goal of the tests is to prove that if ILC and IMC are used concurrently, thesystem performance improves.

0 1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200

Time [Days]

So

lar

Rad

iation (

I [W

/m2])

0 1 2 3 4 5 6 7 8 9 10154

156

158

160

162

Time [Days]

Oil

outlet te

mpera

ture

( T

n[°

C])

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

Time [Days]

Oil

fluid

flo

w q

[m

3/s

]

TnIMC&ILC

RefSignal

Figure 6.9: IMC-based ILC Simulink response

The upper plot in Fig. 6.9 shows the solar radiation, I , for ten days using real values of oneday; whereas the lower plot shows the oil flow, q. The block VTD was used to emulate the effectof the flow-dependent time delay. The oil outlet temperature, Tn, is shown in the middle plot inFig. 6.9. In the first iteration is shown that Tn has a similar behaviour to the one shown in the IMCtest Fig. 6.4, this occurs because just the IMC controlled the iteration. From the second iterationon the behaviour is characterised by the ILC response, which is in charge of solving the effects ofperiodic load disturbances. However, there is an important contribution of the IMC section; it isin charge of eliminating the modelling uncertainties and the time delays associated to the process.

46

0

5

10

15

20

8

10

12

14

16

18

20

155

156

157

158

159

160

161

Time [Hours]Number of iterations [days]

Outle

t te

mp

era

ture

[°

C]

TnIMC−Based ILC

RefSignal

Figure 6.10: IMC-based ILC Simulink response

The improved performance of the IMC-based ILC for the solar plant is shown in the 3DFig. 6.10, where almost perfect tracking is obtained after few iterations. Furthermore, this 3D plotillustrates that after several iterations the difference between both signals is almost zero, whichstands for almost null error.

47

The purpose of the following test is to evaluate the load disturbance rejection of the IMC-based ILC scheme applied to the solar plant. The test consists in using real data for solar radiationof different days so as to observe the response of the system. Two days are partly cloudy and theother days are cloudless. The time delays are included.

4 5 6 7 8 9 100

200

400

600

800

1000

1200

Time [Days]

So

lar

Rad

iation (

I [W

/m2])

4 5 6 7 8 9 10155

160

165

Time [Days]

Oil

outlet te

mpera

ture

( T

n[C

])

Tn

IMC&ILC

RefSignal

4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

Time [Days]

Oil

fluid

flo

w q

[m

3/s

]

Figure 6.11: IMC-based ILC Simulink response in cloudy day

The upper plot in Fig. 6.11 shows the solar radiation, I , from the iteration four to the iterationten. Days four and five are cloudy; whereas the other days are clear days. The lower plot showsthe oil flow, q. The block VTD was used to emulate the effect of the flow-dependent time delay.The oil outlet temperature, Tn, is shown in the middle plot in Fig. 6.11. The system effectivelysolves the effects of periodic load disturbances, while the system is compensating the time delaysassociated to the process. This is shown in the last four iterations, after the effect of the clouds.

The last step is to calculate numerically the error.

48

MSE calculation

In order to be compared with previous results, the MSE is calculated once again with equa-tion (5.1). All the MSE results are gathered in Table 6.3, in order to see the advantages of theIMC-based ILC approach over IMC and ILC.

Table 6.3: Comparison between the MSE of the solar plant response with every control techniqueDay IMC ILC IMC-based ILC

1 0.0032 2.3150 0.00322 0.0032 0.0123 0.00263 0.0032 0.0075 0.00224 0.0032 0.0058 0.00195 0.0032 0.0054 0.00176 0.0032 0.0059 0.00167 0.0032 0.0073 0.00168 0.0032 0.0101 0.00159 0.0032 0.0155 0.001410 0.0032 0.0246 0.0014

Total MSE 0.0316 2.4093 0.0290

In Table 6.3 is observed that using both schemes at the same time the system performanceimproves. The IMC-based ILC MSE is smaller if it is compared to the MSE of ILC and IMC. Itis shown that the value of IMC remains constant everyday, this is due to the lack of learning andhence the performance never improves within the iterations.

The individual characteristics of every control strategy contribute to improve the performanceof the solar plant with an IMC-based ILC approach. The IMC section of this approach keepsthe system robustness, handles the system time delays and deals with the modelling uncertainties.Whereas the ILC section shows its capacity to track the set point after few iterations while it isdealing with periodic disturbances caused by the solar radiation. The system response against thiseffect improves every day thanks to the learning characteristic of this one and hence after severaliterations the obtained error is almost zero.

49

Chapter 7

Conclusions

In this thesis, analysis and evaluation of control techniques for a solar thermal plant have beencarried out. Specifically, IMC-based ILC has been analysed, as well as the two individual controltechniques on which it is based, i.e. IMC and ILC. The choice of this control technique wasbased on the following facts: a) the solar plant has time-delays related to the mass transport, and,b) the solar radiation under certain conditions, i.e. clear days, can be considered as a periodicdisturbance. The last fact allows using information from previous days to improve the currentperformance of the plant.

The experimental evaluation illustrates that by using the IMC-based ILC for a solar thermalplant. It is possible to reduce the error at the output of the system, which stands for good perfor-mance. The results proved that its MSE at the output is smaller than when other control techniques,such as IMC and ILC, are used. According to the test results IMC-based ILC can be applied tocope with time delay uncertainties, to maintain the control system robust stability and to track witha minimum error the set point of the solar thermal plant.

An IMC scheme is an excellent tool to cope with modelling errors, to handle systems withtime delays and to maintain the control system robust stability of the plant. The first advantageappears when the IMC scheme accomplishes to cope with the modelling differences caused whenthe solar plant is running far away from the operating point. Furthermore, this control strategyeffectively compensates the negative effects of the dead-time in the solar plant. On the otherhand, when ILC is used for the solar plant, it accomplishes perfect tracking after several iterationsas well as dealing with periodic disturbances; this occurs only when time delays are omitted.The advantage of an ILC scheme in order to control the solar plant relies on the fact that itslearning characteristic reduces the error daily and hence the system response improves within theiterations. However, perfect tracking is not accomplished when the dead-time of the plant is notomitted, because this scheme does not handle systems with time-varying delays. If the disturbancesignal is not periodic, i.e. it has big variations from one iteration to another, the error increases

50

noticeably. This occurs because wrong information is used from the previous cycle to compensatea system that has a different behaviour. The use of a first-order low-pass Butterworth filter for theILC controller provides a smoother response at the output of the system by only learning fromfrequencies where the model errors are small [42]. Both schemes are composed of few tuningparameters. The compensator, γ, also known as the learning gain provides faster responses totrack the set point when ILC or IMC-based ILC are applied to control the solar plant; whereas, theIMC scheme is tuned with λ, which provides a fast controller response.

Further studies could consider the case of not having periodical disturbances. In this case, thesystem should not learn from signals that are far away of the periodical disturbance. This wouldavoid the negative effects of learning of wrong data, e.g. when a cloud blocks the solar radiation.Nevertheless, by taking advantage of the individual properties of IMC and ILC, the performanceof an IMC-based ILC scheme for the solar plant is showed in a simulation study and demonstratedto be more effective in coping with these control problems.

In summary, it can be concluded from the experimental results that by using the IMC-basedILC approach, all the individual results of IMC and ILC are improved. This stands for an en-hanced response of the solar plant. For an even more enhanced result a slightly different schemeis developed to let the IMC scheme control individually the first iteration and afterwards let thesubsequent iterations be compensated for by the whole IMC-based ILC approach.

51

Bibliography

[1] J. Henriques A. Cardoso and A. Dourado. Fuzzy supervisor and feedforward control ofa solar power plant using accessible disturbances. European Control Conference ECC99,1999.

[2] H.S. Ahn, K.L. Moore, and Y. Chen. Iterative learning control: robustness and monotonicconvergence for interval systems. Communications and control engineering. Springer, 2007.

[3] Hyo-Sung Ahn, Yangquan Chen, and K. L. Moore. Iterative learning control: Brief surveyand categorization. IEEE Transactions on Systems, Man and Cybernetics, Part C: Applica-tions and Reviews, 37(6):1099–1121, 2007.

[4] M.R. Arahal, M. Berenguel, and E.F. Camacho. Neural identification applied to predictivecontrol of a solar plant. Control Engineering Practice, 6(3):333–344, 1998.

[5] M. Barao. Dynamics and nonlinear control of a solar collector field. PhD thesis, Universi-dade Tecnica de Lisboa, Instituto Superior Tecnico, Lisboa, Portugal, 2000.

[6] M. Becker, W. Meinecke, M. Geyer, F. Trieb, M. Blanco, M. Romero, and A. Ferriere. Solarthermal power plant, 2000.

[7] D. A. Bristow, M. Tharayil, and A. G. Alleyne. A survey of iterative learning control. ControlSystems Magazine, IEEE, 26(3):96–114, 2006.

[8] E.F. Camacho, F.R. Rubio, M. Berenguel, and L. Valenzuela. A survey on control schemesfor distributed solar collector fields. Part I: Modeling and basic control approaches. SolarEnergy, 81(10):1240–1251, 2007.

[9] M. Berenguel C.M. Cirre, J.C. Moreno. Robust QFT control of a solar collectors field. D.Martınez. (Ed.), IHP Programme, 2003. Research results at PSA within the year 2002 accesscampaign, CIEMAT.

[10] REN21 Steering Committee. Renewable energy policy network for the 21st century. Renew-ables 2010 Global Status Report, 2010.

52

[11] D.A. Mellichamp D.E. Seborg, T.F. Edgar. Process Dynamic and Control. John Wiley &Sons, New York, USA, 2nd edition, 2004.

[12] C. Bordons E.F. Camacho. Model Based Predictive Control. Springer, 2004.

[13] F. Rubio E.F. Camacho, M.Berenguel. Advanced Control of Solar Power Plants. Springer-Verlag, London, UK, 1997.

[14] M. E.F. Camacho, Berenguel. Application of generalized predictive control to a solar powerplant. The Third IEEE Conference on Control Applications, 3:1657 1662, 1994.

[15] M.R. Arahal E.F. Camacho and M. Berenguel. Modeling free response of a solar plantfor predictive control. Proceedings of the 11th IFAC Symposium on Systems IdentificationSYSID’97, 3:1291 1296, 1997.

[16] I. Farkas and I. Vajk. Internal model-based controller for solar plant operation. Computersand Electronics in Agriculture, 49(3):407–418, 2005. Modelling and control in agriculturalprocesses.

[17] M. Galvez-Carrillo, Robin De Keyser, and Clara Ionescu. Nonlinear predictive control withdead-time compensator: Application to a solar power plant. Solar Energy, 83(5):743–752,2009.

[18] T. Glad and L. Ljung. Control theory-multivariable and nonlinear methods. Real-time Sys-tems, 2000.

[19] Pilkington Solar International GmbH. Experience, prospects and recommendations to over-come market barriers of parabolic trough collector power plant technology. Status Report onSolar Trough Power Plants, 1996.

[20] Albert Arnold Gore. A generational challenge to repower america.http://blog.algore.com/2008/07/a-generational-challenge-to-re.html, 2008.

[21] J. Henriques, P. Gil, P. Carvalho, H. Duarte-Ramos, and A. Dourado. Recurrent neuralnetworks and feedback linearization for a solar power plant control. Proceedings of theIEEE International Join Conference on Neural Networks, 2002.

[22] Soteris A. Kalogirou. Applications of artificial neural-networks for energy systems. AppliedEnergy, 67(1-2):17–35, 2000.

[23] Soteris A. Kalogirou. Artificial neural networks in renewable energy systems applications: areview. Renewable and Sustainable Energy Reviews, 5(4):373–401, 2001.

[24] Soteris A. Kalogirou. Solar thermal collectors and applications. Progress in Energy andCombustion Science, 30(3):231–295, 2004.

53

[25] Bruce Knight. Solar thermal for electricity. http://greenterrafirma.com/solar-thermal-for-electricity.html, 2007.

[26] E. L. Lehmann and G. Casella. Theory of Point Estimation. Springer, New York, USA, 2ndedition, 1998.

[27] Tao Liu, Furong Gao, and Youqing Wang. Imc-based iterative learning control for batchprocesses with uncertain time delay. Journal of Process Control, 20(2):173–180, 2010.

[28] Richard W. Longman. Iterative learning control and repetitive control for engineering prac-tice. International Journal of Control, 73(10):930 954, July 2010.

[29] J. M. Lemos M. Barao and R. N. Silva. Reduced complexity adaptive nonlinear control of adistributed collector solar field. Journal of Process Control, 12(1):131–141, 2002.

[30] A. Meaburn and F. M. Hughes. A simple predictive controller for use on large scale arraysof parabolic trough collectors. Solar Energy, 56(6):583–595, 1996.

[31] G. Meinsma and H. Zwart. OnH∞ control for dead-time systems. IEEE Trans. Aut. Control,45(2):272–285, 2000.

[32] Roger G.T. Mello, Liliam F. Oliveira, and Jurandir Nadal. Digital butterworth filter forsubtracting noise from low magnitude surfaceelectromyogram. Computer Methods and Pro-grams in Biomedicine, 87(1):28 – 35, 2007.

[33] A. Mood, F. Graybill, and D. Boes. Introduction to the Theory of Statistics. McGraw-Hill,3rd edition, 1974.

[34] K. L. Moore. Iterative Learning Control for Deterministic Systems. Springer-Verlag, Lon-don, England, 1993.

[35] M. Morari and E. Zafiriou. Robust Process Control. Prentice Hall, Englewood Cliffs, 1989.

[36] J. Henriques P. Gil and A. Dourado. Recurrent neural networks and feedback linearizationfor a solar power plant control. Proceedings of EUNIT01, 2001.

[37] A. Porwal. Internal model control (IMC) and IMC-based PID controller. PhD thesis, Na-tional Institute of Technology Rourkela, Orissa, India, 2009.

[38] Daniel E. Rivera and Manfred Morari. Low-order siso controller tuning methods for theH2,H∞ and µ objective functions. Automatica, 26(2):361–369, 1990.

[39] Daniel E. Rivera, Manfred Morari, and Sigurd Skogestad. Internal model control: PIDcontroller design. Industrial & Engineering Chemistry Process Design and Development,25(1):252–265, 1986.

54

[40] F. Miyazaki S. Arimoto, S. Kawamura. Iterative learning control for robot systems. In pro-ceedings of annual conference of the IEEE Industrial Electronics Society. IECON, 1984.

[41] M. Uchiyama. Formulation of high speed motion pattern of mechanical arm by trial. Trans-actions on Society of Instrumentation and Control Engineering, 14(6):706–712, 1978.

[42] Johanna Wallen. Estimation-based iterative learning control. PhD thesis, Linkoping Uni-versity, Automatic Control, The Institute of Technology, 2011.

[43] Jian-Xin Xu, Qiuping Hu, Tong Heng Lee, and Shigehibo Yamamoto. Iterative learningcontrol with Smith time delay compensator for batch processes. Journal of Process Control,11(3):321–328, 2001.

[44] J.X. Xu, S.K. Panda, and T.H. Lee. Real-Time Iterative Learning Control: Design andApplications. Advances in industrial control. Springer, 2008.

[45] K. L. Moore YangQuan Chen. Learning control of actuators in control systems-commentson united states patent 3,555,252. Proceedings of the 2000 International Conference onAutomation, Robotics and Control, December 2000.

55