Constrained Temperature Control of a Solar Furnace - IEEE Xplore

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 20, NO. 5, SEPTEMBER 2012

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Constrained Temperature Control of a Solar Furnace Manuel Beschi, Antonio Visioli, Senior Member, IEEE, Manuel Berenguel, Member, IEEE, and Luis José Yebra

Abstract—A new constrained control strategy for a solar furnace is proposed in this paper with the goal of attaining a minimum-time transition between two values of the temperature subject to constraints on both the saturation and slew rate level of the process input. In the case of set-point step-shape signals, the strategy basically consists in implementing a (model-based) feedforward control law with maximum positive and negative velocity phases in order to obtain a minimum-time transition with no overshoot. In the case of ramp-shape set-point signals (addressing the output slew-rate constraint), the suitable feedforward control law is obtained by also inverting the dynamics of the system. Implementation issues are discussed. Simulation and real experimental results demonstrate the effectiveness of the methodology. Index Terms—Constrained control, feedforward control, optimization, set-point regulation, solar furnace.

I. INTRODUCTION

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ENEWABLE energies are becoming one of the most important topics in society, industry and research. In fact, the damages caused by the pollution and the prospect of exhaustion of the fossil fuels increase the society interest towards the “green” energy. The solar energy (defined as the thermal or electrical energy produced directly from the Sun radiation energy) is one of the most interesting renewable power sources in the scientific community and in the industrial world. There are many technologies to take advantage of this renewable source, like photovoltaic panels or solar water heaters. Among these technologies, the concentrated solar power has interesting industrial perspectives because it can be integrated with the conventional energy sources and it allows economies of scale to reduce the facility cost. Concentrated solar power systems use an optical system to focus a large area of sunlight into a small area. The optical system is normally constituted by one or more mirrors with a motorized system to align them to the solar radiation direction. The concentrated solar radiation can be usefully exploited in a solar furnace for thermal and chemical treatments or resistance Manuscript received January 18, 2011; revised July 01, 2011; accepted July 19, 2011. Manuscript received in final form August 08, 2011. Date of publication September 12, 2011; date of current version June 28, 2012. Recommended by Associate Editor M. Mattei. This work was supported by the National Plan Project DPI2010-21589-C05-04 of the Spanish Ministry of Science and Innovation. M. Beschi and A. Visioli are with the Dipartimento di Ingegneria dell’Informazione, University of Brescia, Brescia 25121, Italy (e-mail: [email protected]; [email protected]). M. Berenguel is with the Departamento de Lenguajes y Computación, University of Almería, Almería 04120, Spain (e-mail: [email protected]). L. J. Yebra is with the CIEMAT-PSA, abernas, Almería 04120, Spain (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2011.2164795

tests for high temperature or thermal shocks [1]–[8], by means of heating the samples following many different temperature patterns. Solar furnaces have received a great attention since the fifties [9]–[13]. An excellent description and overview of different solar furnaces can be found in [14], where open loop computer-based control systems are used (e.g., [15]). The samples are placed in the centre of the optical system focus. A shutter regulates the quantity of solar radiation which exits from the optical system. The largest European centre for research, development and testing of concentrating solar technologies is the Plataforma Solar de Almería (PSA) located in Tabernas (Almería, Spain), which belongs to the Centro de Investigaciones Energéticas Medioambientales y Tecnológicas (CIEMAT) a research agency for the energy and the environmental technologies of the Spanish Ministry of Science and Innovation. There are several solar furnaces operating in Europe. The largest solar furnace is Odeillo furnace managed by CNRS/Promes (France), built in 1968. The furnace is capable of reaching temperatures of up to 3500 C. The high flux solar furnace of the Paul Scherrer Institute in Switzerland achieves 2200 C and is comparable to the CIEMAT-PSA solar furnace addressed in this paper to control the temperature of the samples used for materials testing purposes. Due to the complexity and diversity of sample materials and temperature trends such research plants are usually manually controlled by expert operators. Obviously, the efficiency of the operations depends on the operators’ skill and therefore the presence of a properly designed automatic control system would have the advantage of providing adequate results for different operating conditions. For this reason, different control strategies have been published in literature, ranging from adaptive control [16]–[20], fuzzy logic control [21], [22] and predictive control [23]. Many of these techniques are based on a physical model developed in [16]. From the control viewpoint, the solar furnace is a system which presents the following various interesting characteristics which make the control problem a difficult task: • the characteristics of the samples are quite different depending on their nature (steel, alumina, copper, …); • the dynamic characteristics of each sample greatly depend on the temperature and introduce a high nonlinearity in the control system, which makes the behavior of the controlled system change with the operating conditions; • the control specifications are quite severe (rate of temperature increase, rate of temperature decrease, variable step changes, etc.) and have to be achieved with small errors; • the system suffers from disturbances caused by solar radiation variations (slow variations due to the daily cycle or fast

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Fig. 1. CIEMAT-PSA Solar furnace (courtesy of PSA). Fig. 2. Sample used in the tests made of packed wires (courtesy of PSA).

and strong variations due to passing clouds), which make the exact reproduction of the conditions of a determined test impossible; • limitations exist in the maximum temperature achievable by the materials and different constraints (nonlinearities) in the actuator (amplitude, slew rate, etc.). However, in this kind of process the saturation and slew-rate limits of the actuator as well as the output constraints can be very significant (see Section II) and should be therefore taken into account explicitly when the control system is designed. Thus, this paper presents a new control strategy that provides a minimum-time process output transition subject to constraints on the process input [24]–[26]. The method is based on selecting a feedforward (model-based) control law with maximum positive and negative velocity phases. Then, in case an additional output (slew-rate) constraint is considered by selecting a ramp set-point signal, the suitable feedforward control law is obtained by employing an input-output inversion of the system. The problem of designing an optimal feedforward control law for a thermal process with similar issues has been addressed also in [27] and [28], but therein a different cost function and a different design procedure has been proposed. This paper is organized as follows. In Section II the plant is described and the control specifications are outlined. The modelling and parameter identification part is addressed in Section III. The design methodology is explained in Section IV, where simulation results are also presented. Experimental results are shown in Section V. Conclusions are drawn in the last section.

II. PLANT DESCRIPTION The CIEMAT-PSA solar furnace (see Fig. 1) essentially consists of a continuously solar-tracking, flat heliostat, a parabolic concentrator mirror, an attenuator or shutter and the test zone located in the concentrator focus [29]. The flat collector mirror, or heliostat, reflects the parallel horizontal solar beams on the parabolic dish, which in turn reflects them on its focus (the test area). The amount of incident light is regulated by the attenuator located between the concentrator and the heliostat. Under the ) focus there is a test table movable in three directions ( that places the test samples in the focus with high precision

(Fig. 2 shows a copper sample made of packed wires that has been used in the tests shown in this paper). The CIEMAT-PSA solar furnace heliostat is made up of 28 3-mm-thick flat silvered facets and has a total surface of 120 m . The facets are made of second-surface mirrors which are silvered on the back, which have a nominal 90% reflectivity and are jointed together so that they form a continuous flat surface reflecting the sunbeams but not concentrating them. The concentrator disk is the main component of the solar furnace. It concentrates the incident light from the heliostat, multiplying the radiant energy in the focal zone. Its optical properties especially affect the flux distribution at the focus. It is composed of 89 spherical facets with a total surface of 98.5 m and a nominal 92% reflectivity. Its focal distance is 7.45 m. The parabolic surface is achieved with spherically curved facets, distributed along five radii with different curvatures depending on their distance from the focus. The concentrator and distribution of the flux density in the focus is the element that characterizes a solar furnace. This distribution usually has a Gaussian geometry and is characterized by a CCD camera hooked up to an image processor and a lambertian target. The characteristics of the focus with 100% aperture and solar radiation of 1000 W m are: peak flux of 3000 kW m , total power of 58 kW and focal diameter of 0.23 m. The attenuator, or shutter, consists of a set of horizontal louvers that rotate on their axes, regulating the amount of entering sunlight incident on the concentrator. The total energy on the focus is proportional to the radiation that passes through the attenuator. It is composed of 30 louvers arranged in two columns of 15. In closed position the louvers form a 55 C angle with the horizontal and 0 C when open. Finally, the test table is a mobile support that is located under the focus of the concentrator. It moves on three axes perpendicular to each other and positions the test sample with great precision in the focal area. In the test table it is possible to place an atmosphere-controllable reaction chamber, called also Minivac. This equipment allows the user to control the atmosphere constituents, the pressure and the air flow. It is worth stressing that, for control purposes, the dynamics of the shutter can be neglected because it is much faster than the dynamics of the temperature, but its constraints have to be taken in account. In particular, the shutter angular aperture has

BESCHI et al.: CONSTRAINED TEMPERATURE CONTROL OF A SOLAR FURNACE

the maximum value normalized to 100% and the minimum to 0%. In addition, the maximum percentage angular velocity is s . In any case, the presence of the Minivac limited to limits the maximum admissible velocity. In fact, a fast shutter aperture can generate a thermal shock in this equipment. The s slew rates must be therefore set equal to a maximum of for the positive velocity and s as the negative value. The negative value is greater than the positive one because if there is a great radiation increment it is necessary to close rapidly the shutter to prevent thermal shocks. The control specifications depends obviously on the application and on the sample material. If a set-point step signal is applied, a fast settling time is desirable in addition to a small steady-state error. However, a ramp temperature profile is often required and in this case the steady-state error for a ramp should be as small as possible. In any case, a minimal overshoot is desirable. This requirement comes from the fact that it is sometimes necessary to work at temperatures close to melting point, and therefore overshoots or other types of oscillatory responses would destroy the sample. III. MODELLING The basic sample’s temperature model is described in [16]. The model consists of the following first-order nonlinear equation:

(1) where and are the reflectivity coefficients of the heliostat and the concentrator (collector) [-]; , , and are the surfaces of the concentrator, of the sample and the focus area where the 90% of the solar input energy is concentrated [m ]; is the input direct solar radiation, which is measured by using a pirheliometer [Wm ]; is the absorption capacity of the is the emissivity of the sample [Wm K ]; sample [-]; is the capacity of the sample to exchange heat with the air [-]; is the Stephan-Boltzmann constant [ Wm K ]; is the specific heat [Jkg K ]; is the sample mass [kg]; is the maximum angular aperture of the shutter [rad]; is the sample temperature [K] and supposed uniform into the body; is the environmental temperature [K] and is the angular percentage aperture [%]. In this work all the physical properties are supposed to be constant, as it has been observed in control tests that taking into account their dependence on temperature does not considerably improve the obtained results. In this way the parameters can be grouped together and (1) becomes

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Fig. 3. Comparison between simulations (dashed line) and experimental data (solid line).

(3) and can be measured beNote that, in practical cases, fore a test (typical values are 75% and 95%, which are less than their nominal values because of the unavoidable dirtiness , , and can be estipresent on the surface). Then, mated with a standard least squares method. Resulting values for a 50 mm 50 mm 20 mm copper sample made of packed wires with a diameter of 2 mm, where the temperature is measured by means of a thermocouple placed in the middle of the Km W s ), sample, are s , and s K , although these values can change in a wide range depending on the operation conditions and layout of the experiment (see Section V). Fig. 3 shows a comparison between experimental data and a simulation done with the estimated parameters. The model error is never greater than 7%. IV. CONSTRAINED FEEDFORWARD CONTROL DESIGN A. Generalities Consider the single-state system represented by the equation (4) increases monotonously when and assume that increases. The control action is limited in the interval and its derivative is limited in the interval . The goal of the control law is to move the state from the initial and to the final values values and in a minimum time subject to the following constraints:

(5) (2) where

where, for the sake of simplicity, the initial time is assumed equal to zero without loss of generality. The problem can be written in the following form: (6)

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such that

(7) The minimization problem can be rewritten as a minimum time problem by introducing a new state variable which is equal to the control action and by considering the derivative of the control of the new system. Therefore, denoting action as the input , , and , we obtain the following second-order system:

Fig. 4. Structure of the optimal input function.

Fig. 4 shows the structure of the feedforward control signal . The final solution can be expressed by the following relations: if if if

or, in a more compact version (8)

(10)

where • is the initial slope, namely

The constraints of the new system (8) are

if if •

is the final slope, namely, if if

•

is the saturation value. It is equal to the maximum , otherwise it is the minimum limit limit of if

(9)

if if

The Hamiltonian for the optimal controller is •

where is the associate multiplier. By applying the Pontryagin’s Minimum Principle (see [30]) is not saturated, the new control variit appears that, when able assumes only the values ; otherwise, if is saturated, is set equal to zero. of the system (8) can assume Thus, the control variable . Hence, we obtain that only three values, namely the optimal control law is composed by the following three time intervals. • In the first interval the control action follows a ramp with the maximum permitted slope. If the final value of is greater than the initial one, the ramp has a positive slope otherwise the slope is negative. • In second interval the control action is equal to the maximum permitted value. This interval can be null if there is not enough time to reach the saturation value. • In third interval the control action follows a ramp with inverse slope with respect to the first interval.

is the first time interval. It is equal to the smallest time interval between the time necessary to reach the saturation value, denoted as , and the time when the first interval ramp crosses the third interval ramp, called . Thus, it can be calculated as

where and

•

is the third time interval. It is equal to the difference between the final time and the largest time between the time instant when the second ramp crosses the saturation . Indeed, can value, denoted as and the time instant be calculated as:

where


•

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is the second time interval

The case when is called triangular case, the other one is the trapezoidal case. The described times depend on the value of . The correct is found by imposing the condition . If value of an analytical solution of (4) is unknown, as in the solar furnace model, a numerical approach is necessary to find and the time intervals , and . This is not a complicated problem bevaries monotonously with and therefore biseccause tion can be applied. A simple algorithm to solve the problem can be therefore posed as follows ( is a precision parameter): 1) determine suitable lower and upper bounds for (denoted and , respectively); as then go to 6 else; 2) if 3) set ; ; 4) calculate set , else set 5) if ; go to 2; 6) end. To obtain a fast convergence of the algorithm it is necessary and . Indeed, to choose initial suitable values of the initial lower bound can be set as while the initial upper bound can be set by first searching the for which and then setting value of (namely, an upper bound of the first two time intervals is added to an upper bound of the third time interval). B. Minimum-Time Set-Point Step Response The method described in the previous subsection can be applied straightforwardly to the solar furnace by considering the and . The value of is found by model (2) with and it results: imposing

(11) Notice that for the determination of the optimal feedforward input function, the radiation and the environmental tempermay be assumed equal to the latest measured value ature by the data acquisition system (thus accounting for real distur900 Wm and bances) or either to have nominal values of 300 K (typical of clear days that can be used for simulation purposes). The errors induced by this assumption are very small (as will be shown in the results), but to cope with them the constraints are set smaller than their maximum values (see Section II) because it is convenient to leave a margin for adjustments to compensate for disturbances or model mismatches. Then, the shutter constraints are assumed equal to s

s

Fig. 5. Sketch of the devised feedforward control signal. Thick solid line: with slew-rate constraints. Thin solid line: without slew-rate constraints.

It is clear that if the feedforward algorithm uses less actuator’s capacity, the feedback control can improve the performance (see Section IV-D). C. Tracking of a Ramp Signal The method described in the previous subsections can be extended to the case of a set-point ramp signal, that is, when constraints on the slew-rate of the process output are also taken into account. Assume that the system is at an equilibrium point with as output at the initial time (which is assumed to be zero) and it is desired to obtain an output transition with slope to a new equias output. The task of the control system librium point with is therefore to track the following signal: if if if

(12)

The control strategy, in the same way of Section IV-A, is to use all the capacity of the actuator to reach the condition , in a time interval as small as possible. Then, the by always actuator is used to keep the constant value . ensuring that constraints (7) are satisfied for every For this purpose the control action is determined by inverting . It is worth noting that, the equation at the end of the transient, the actuator has to decrease its action in order to reach the final point without overshoot at . To better understand the solution, the situation is represented in Fig. 5, where the unconstrained input (i.e., with no slew-rate constraints) that causes exactly the required output is plotted as a thin solid line (note that it is obtained by inverting the system dynamics), while the devised constrained control law is plotted as a thick solid line. The solution to the problem is therefore to use a (feedforward) control signal which is defined in the following five time intervals. • In the first time interval the control action follows a ramp then the with the maximum permitted slope. If ramp has a positive slope, otherwise the slope is negative. • In the second time interval the control action is equal to the maximum permitted value. This interval can be null if there is not enough time to reach the saturation value (as happens in Fig. 5). • In the third time interval the control action consists of a ramp with inverse slope with respect to the first interval.

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• In the fourth time interval the control action is determined by inverting the equation . • In a fifth interval the shutter follows a ramp with the same slope of the third interval and finally it attains the final value . Formally, the control signal is expressed as if if if if if if (13) and where (by taking into account that ): is equal to the smallest time between the time necessary • to reach the saturation value, called , and the time when the first interval ramp crosses the third interval ramp, called (case of Fig. 5). These time instants can be calculated by using the following expressions:

is underestimated then is less than . If similar considerations can be done. The algorithm can be therefore posed as follows ( is again a precision parameter): 1) determine suitable lower and upper bounds for (denoted and , respectively); as then go to 6 else; 2) if ; 3) set 4) calculate ; set , else if 5) if set ; go to 2; 6) end. To obtain a fast convergence of the algorithm it is necessary and . Indeed, can to choose adequate values of be set equal to the time interval necessary to reach from with a slope equal to while can be set equal to . . Also In order to find it is necessary to impose in this case a bisection method can be applied. The maximum value can be set equal to

while the minimum value of

•

is the third time interval. It is equal to the difference between the time and the largest time between the time when the second ramp crosses the saturation value, called , and the time . Indeed, and can be calculated by using the following expressions:

In order to verify the feasibility of the solution, namely, in order to check that the ramp can be tracked by considering the actuator slew-rate constraints, it is possible to find a relation between and as

Thus, the value of •

is the second time interval. It is trivially equal to

The case when is called triangular case, the other is the trapezoidal case. is found by solving the following equation: •

•

can be set as

should be such as with

(14)

The devised strategy can be applied straightforwardly to the solar furnace model by considering model (2) for which the inversion of the dynamics yields [see (11)]

is trivially equal to

From the above expressions, it appears that the time intervals , , , , and can be determined easily once the correct values of and are found. If and can not be determined analytically, a numerical algorithm can be employed. In particular, as in the previous case, a bisection method can be conveniently used for both and . In fact, by considering , if is overestimated then is greater than the desired value and, on the contrary, if

Remark 1: It is worth stressing that if in (13), (that is, only the positive slope ramp of the control signal is apis satisfied) then the referplied until the condition ence ramp is tracked with a constant error, namely the process output is delayed with respect to the reference signal. Indeed, if there is a (small) time interval in which, in order to set (see Section IV-E). This is the control error to zero, not a problem from a practical point of view for the solar furnace plant, however, in general, if the slope of the temperature is is a viable solution (note a hard constraint, setting again that this implies a constant tracking error).


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Fig. 7. Alternative implementation of the control scheme. Fig. 6. Implemented control scheme.

D. Implementation The devised control strategy has been implemented according to the control scheme shown in Fig. 6 where the feedforward control action is applied directly to the plant and a feedback PI controller is employed to cope with unavoidable modelling uncertainties and disturbances (the derivative action has not been used because the process is of first order without dead time and in order to avoid the excessive excitation of the actuator because of the measurement noise). Its transfer function is written as

Fig. 8. Command signal generator block scheme.

(15) where is the proportional action and is the integral time constant. The nonlinearity inherent to this system is implicit in the algorithm in the use of the nonlinear model of the system to provide the feedforward signal action, that reduces the effects of the nonlinearities significantly and almost the same performance is obtained for the different operating points, as will be seen in the following sections. The nonlinear model at the input of the closed-loop system has been applied so that the PI controller acts on the difference between the estimated output and the actual one, that is, the PI controller acts on a deviation signal so that a constant gains version of the controller can be used here also for the purpose of comparison, as it is the standard methodology. Then, the feedforward signal generator has been designed so that the feedforward signal (10) or (13) is obtained as output when the step or ramp signal parameters are employed as input. By taking into account that the deviations due to the modelling errors between the desired and the actual output can be treated as the effect of a load disturbance [26], the PI parameters have been tuned by means of an iterative numerical algorithm where the nonlinear system has been simulated (by taking account the constraints) three times using three different set-points (200 C, 500 C, and 800 C), an initial error of 60 C, a solar radiation of a cloudy day are used and the slew-rate and saturation constraints of the actuator have been considered. For each temperature set-point the mean absolute error has been calculated and the parameters set with the smallest sum of these three values has been eventually chosen. The resulting proportional and the resulting integral time constant is gain is in the case of the application shown in this paper where the copper sample made of packed wires is used, as explained in the following section. A back-calculation anti-windup mechanism [31] has also been implemented. An alternative scheme (which resembles the typical two-degree-of-freedom control scheme) that can be implemented is that shown in Fig. 7. The command signal generator is obtained simply as shown in Fig. 8.

Fig. 9. Simulation results obtained by applying a sequence of set-point step signals. Dash-dot line: Reference signal. Solid line: Control with the feedforward action. Dashed line: Control without the feedforward action.

Although it is a different approach, the idea is similar to that used in [32], [33], where a nonlinear model of a solar plant was used to obtain the free response of the plant and a simplified linear model was used to obtain the forced response (providing a predictive controller with constant gains). E. Simulation Results This section presents representative simulation results. The first one consists of a sequence of steps of amplitude equal to 200 C is applied every 10 min until the temperature of 1000 C is attained (with the exception of the first step where the initial temperature is the environmental temperature of 27 C). Then, after 50 min, the set-point is set to 400 C. A constant solar W m has been considered. The radiation of W m slew-rate and saturation constraints of the actuator have been imposed at the input of the process. The results obtained with and without the feedforward action are shown in Fig. 9. It appears that the feedforward action improves the set-point tracking performance allowing to decrease the rise time and a reduced overshoot. In any case, the use of the feedforward action reduces the effects of the nonlinearities significantly (note that almost the same performance is obtained for the different operating points). Obviously, the performance obtained with the

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Fig. 12. Control variable obtained by applying a set-point ramp signal. Dasheddotted line: Reference signal. Solid line: Control with the feedforward action. Dashed line: Control without the feedforward action. Fig. 10. Simulation results for a set-point step from 400 C to 600 C. Dashdotted line: Reference signal. Solid line: Control with the feedforward action. Dashed line: Control without the feedforward action.

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Fig. 13. Performance obtained with different values of with a set-point step signal. Solid line: control with the feedforward action. Dashed-dotted line: Control without the feedforward action. Dashed line: Control with feedforward action and tight actuator constraints.

F. Robustness to Parametric Uncertainties Fig. 11. Process variable obtained by applying a set-point ramp signal. Dasheddotted line: Reference signal. Solid line: Control with the feedforward action. Dashed line: Control without the feedforward action.

two different control strategies in the first part of the transient when the negative set-point step is applied are almost the same because the required decrease of the temperature implies that the shutter has to be fully closed in both cases (thus the system is in open-loop). For a better evaluation of the result, a zoomed view of the step response from 400 C to 600 C is shown in 145.2 s, (namely, the triangular Fig. 10. In this case 29.7 s. case) and In the second simulation, a ramp set-point signal with a slope of 20 C min is applied starting from the environmental temperature until 1000 C are attained. The output temperature obtained with and without the feedforward action are shown in Fig. 11, where a zoom of the first and last part of the transient is also provided. Note that the feedforward action allows the ramp to be tracked without a steady-state error and the absence of overshoot when the reference signal becomes constant (notice that this simulation is the nominal case and the effect of parametric uncertainties is treated in the next subsection). The while control signals are plotted in Fig. 12. Note that 48.4 s, 5.6 s, 47.7 min and 4 s, that is, the inversion of the system is applied for a large part of the control interval.

In this subsection the effect of model mismatches is evaluated. In particular, a reference step from 200 C to 700 C, has been considered and the following three performance indexes have been evaluated: obtained by varying the nom• the 1% settling time inal system parameters, normalized to the value for the nominal case for each control strategy, in such a way that its value is always one when there are no model mismatches. (the same considered in the previous section with W m and C); • the maximum overshoot obtained by varying the nominal system parameters (note that there is no overshoot in the nominal case): obtained by varying the • the integrated absolute error nominal system parameters, normalized to the value for the nominal case for each control strategy. Results are shown in Figs. 13–15 for both the case with and without (that is, with PI control only) feedforward action. It is the most important parameter and thereturns out that fore a good estimation of it is required. In any case the presence of the feedforward action improves in general the performance of the control system (note that for the overshoot, small , the most important values are obtained in any case). Indeed, parameter, should have an error greater than 20% of its nominal value to decrease significantly the performance. Obviously, the performance obtained with the feedforward action (which is


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Fig. 14. Performance obtained with different values of with a set-point step signal. Solid line: Control with the feedforward action. Dash-dot line: Control without the feedforward action. Dashed line: Control with feedforward action and tight actuator constraints.

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Fig. 15. Performance obtained with different values of with a set-point step signal. Solid line: Control with the feedforward action. Dashed-dotted line: Control without the feedforward action. Dashed line: Control with feedforward action and tight actuator constraints.

based on the system model) decreases for large modelling errors which are in any case not reasonable for the solar furnace (see Section III). It is worth stressing that the robustness can be improved, at the expense of the speed of the response, by tightening the actuator constraints when the feedforward action is determined. For example, if the saturation and slew-rate constraints are set at the 90% of their true value, the results shown as a dashed line in Figs. 13–15 are obtained, where it appears that the range of estimation errors for which there is no significant decrease of performance is enlarged. It can be deduced that the constraints employed to determine the feedforward action are an effective mean to handle the trade-off between aggressiveness and robustness, which is always a desirable feature from the operator point of view. The same analysis has been performed with the ramp setpoint signal. In this case the three performance indexes considered are the maximum and the mean absolute errors between the actual and nominal output and the tracking time, defined as the time when

where is the initial temperature. Obviously, the tracking time can be determined only for the control scheme with the feedforward action because, as already mentioned, without the feedforward action there is always a constant steady-state error. The reference to follow is a ramp with a slope of 20 C min starting from the environmental temperature and ending at 1000 C. Figs. 16–18 show the results. It can be seen that, as is most important parameter and for for the step response, this reason it requires a more accurate estimation. Obviously, in

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Fig. 16. Performance obtained with different values of with a set-point ramp signal. Solid line: Control with the feedforward action. Dashed-dotted line: Control without the feedforward action.

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the presence of large estimation errors, the absence of a tracking steady-state error can not be guaranteed. In any case, as for the step response, the presence of the feedforward action improves the performance of the control system. Note that for the ramp signal, tightening the actuator constraints does not yield a significant increment of the performance because the control input determined by the inversion of the system, where the actuator limits are not attained, plays a key role in a large part of the transient. Finally, it is worth stressing that if an estimation error is applied to all the parameters at the same time [for example, because of a variation of , see (3)] the performance decrement is similar to the case of Figs. 13–16, where the variation of (which is the most representative parameter) is addressed. V. EXPERIMENTAL RESULTS The constrained control strategy has been applied in the CIEMAT-PSA solar furnace, where the copper sample made of

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TABLE I VALUES OF THE SYSTEM PARAMETERS EMPLOYED FOR THE GENERATION OF THE FEEDFORWARD ACTION AND ESTIMATED AFTER THE EXPERIMENT

Fig. 19. Experimental results. Top: Process variable (solid line) and set-point signal (dashed line). Middle: Control variable. Bottom: Solar radiation.

packed wires with a diameter of 2 mm has been used (Fig. 2). For safety reasons indicated by the plant responsible the shutter aperture has been limited to 70% and the slew rate constraints s . A sequence of step-shape have been fixed to and ramp-shape set-point signals has been applied and the results obtained are shown in Fig. 19, where the temperature, the control signal, and the solar radiation which acts as the main disturbance are plotted. It can be seen that a satisfactory performance with a minimum-time transition with almost no overshoot in the different operating points is obtained and the posed constraints are satisfied. The mean overshoot is around 1.2% and always less than 3%, that it is obtained for negative steps as the dynamics when heating the sample are faster than the dynamics of cooling against the environmental temperature (actually, the process is asymmetric), thus slightly affecting the performance of the feedback controller. It is worth noting that the feedforward action has been determined by considering the new values of the parameters shown in the first raw of Table I, which have been obtained in a previous identification experiment using a least squares method. Notice that these values are different that the nominal ones used for simulations as they depend on the specific layout of each experiment and the operating conditions so that, those included in the first row of Table I reflect the conditions of the shown test. After doing the test, the obtained data have been used to perform an a-posteriori identification of the model parameters (second row in Table I). As can be seen, there is a large difference between the values of the parameters used by the model to generate the feedforward action and the real ones, mainly in the case of parameter, probably because the test does not provide the enough dynamical information for identification purposes and parameter to the global dynamics the contribution of the is the less representative one. Anyway, it can be seen that even under significant modelling uncertainty, the performance of the

closed-loop system is quite acceptable. Other tests performed in the installation have shown similar results (the number of tests was limited because the installation is used by many researchers), mainly because the radiation profile was that of a clear day (this installation is mainly used in clear days). The disturbance rejection problem is treated in [23], where this approach has demonstrated to adequately handle disturbances. If the results of the technique are compared with other techniques tested in the same installation, some conclusions can be drawn, although the type of samples and operating conditions were not the same. In [16] an adaptive PI plus feedforward control algorithm was implemented using the same model and taking into account the slew-rate constraints of the actuator. As in the case treated in this paper, the adaptive control law tried to achieve the same performance indexes in the whole range of operating conditions, using an on-line least squares based identification algorithm. Although the results were quite acceptable, oscillations appeared in the response, mainly due to the fact that coupling between the system dynamics and the identification dynamics was impossible to avoid. The case treated in this paper required an offline estimation of the model parameters (as it allows modelling uncertainty) but minimizes the temperature transition time avoiding overshoots, what is more desirable from the operation point of view (although the technique is slightly more complex than that in [16]). So, the main difference is that overshoots are avoided in the algorithm presented in this paper. Notice that also the use of (conservative) constraints as tuning knobs is something that positively valued operators. In [17] the same model developed by [16] was used to develop a simple algorithm based on a linearized model of the system and including an estimator of disturbances. Only simulation results are shown using a linearized model that doesn’t reflect the real behavior. In [22] fuzzy logic was used to mimic the experts way of operating on samples similar to those used in this paper. Fast responses without overshoot were achieved without using any physical model of the system, but obviously a large set of data and operating hours was required to develop such algorithm (being necessary to repeat the procedure each time a sample is changed). The responses was similar to those attained by an expert operator, and also to those obtained in this work, where the use of a physical model allows to achieve similar results reducing the need of a priori information required to build the control system. In [18]–[20], the same physical model is used (in this case taking into account the variation of the samples characteristics with temperature) and both adaptive PI and an adaptive control law based on the exact feedback linearization and Lyapunov


adaptation of the process dynamics was applied (having a structure that is comparable to an adaptive PI conroller with feedforward, rendering the commissioning easier), although the results were obtained in a different solar furnace and thus difficult to compare. The underlying idea in both cases is that of using a nonlinear model embedded in the control algorithm (family of feedback linearization controllers), which has shown to provide adequate results. VI. CONCLUSION A constrained control strategy for a solar furnace has been proposed in this paper. In particular, a suitable feedforward action, which takes into account the process input constraints is determined in order to minimize the temperature transition time. The method can be extended to the case when process output constraints are also considered, namely, when a ramp set-point signal is considered. Simulation and experimental results have shown that the technique provides the required performance and it is robust to modelling uncertainties. ACKNOWLEDGMENT The authors would like to thank the personnel of the PSA, especially I. Cañadas and L. Roca. REFERENCES [1] D. Suresh and P. K. Rohatgi, “Melting and casting of alloys in a solar furnace,” Solar Energy, vol. 23, pp. 553–555, 1979. [2] D. P. Rodríguez, V. López, J. J. Damborenea, and A. J. Vázquez, “Surface transformation hardening on steels treated with solar energy in central tower and heliostats field,” Sol. Energ. Mater. Sol. Cells, vol. 37, pp. 1–12, 1995. [3] Y. Yang, A. A. Torrance, and J. Rodríguez, “The solar hardening of steels: Experiments and predictions,” Sol. Energ. Mater. Sol. Cells, vol. 40, pp. 103–121, 1996. [4] F. Almeida, N. Shohoji, J. Cruz, and L. Guerra, “Solar sintering of cordierite-based ceramics at low temperatures,” Solar Energy, vol. 78, pp. 351–361, 2005. [5] R. Román, I. Cañadas, J. Rodríguez, M. T. Hernández, and M. González, “Solar sintering of alumina ceramics: Microstructural development,” Solar Energy, vol. 82, pp. 893–902, 2008. [6] Y. Tsuo, J. R. Pitts, M. D. Landry, C. E. Bingham, A. Lewandowski, and T. F. Ciszek, “High-flux solar furnace processing of silicon solar cells,” in Proc. 24th IEEE Photovoltaic Specialists Conf. Photovoltaic Energy Conv., 1994, pp. 1307–1310. [7] J. Fernández-Reche, I. Cañadas, M. Sánchez, J. Ballestrín, L. Yebra, R. Monterreal, J. Rodríguez, G. García, M. Alonso, and F. Chenlo, “PSA Solar furnace: A facility for testing PV cells under concentrated solar radiation,” Sol. Energ. Mater. Sol. Cells, vol. 90, pp. 2480–2488, 2006. [8] J. Petrasch, P. Osch, and A. Steinfeld, “Dynamics and control of solar thermochemical reactors,” Chem. Eng. J., vol. 145, pp. 362–370, 2009. [9] F. Trombe, “Solar furnaces and their applications,” Solar Energy, vol. 1, no. 2–3, pp. 9–15, 1957. [10] P. Glaser, “A solar furnace for use in applied research,” Solar Energy, vol. 1, no. 2–3, pp. 63–67, 1957. [11] A. Neumann and U. Groer, “Experimenting with concentrated sunlight using the DLR solar furnace,” Solar Energy, vol. 58, no. 4–6, pp. 181–190, 1996. [12] G. Flamant, A. Ferriere, D. Laplaze, and C. Monty, “Solar processing of materials: Opportunities and new frontiers,” Solar Energy, vol. 66, no. 2, pp. 117–132, 1999. [13] R. Pitz-Paal, “High temperature solar concentrators,” in Solar Energy Conv. Photoenergy Syst., J. B. Gálvez and S. M. Rodríguez, Eds. Oxford, U.K.: EOLSS Publishers, 1999. [14] D. Martínez, “Solar furnace technologies,” in Solar Thermal Test Facilities. Madrid, Spain: CIEMAT, 1996. [15] J. Giral, B. Rivoire, and J. F. Robert, “A new advanced control and operating system for the heliostats of the French CNRS’1000 KW Solar Furnace,” in Proc. 8th Int. Symp. Solar Thermal Concentrat. Technol., 1996, pp. 1592–1608.

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[16] M. Berenguel, E. F. Camacho, F. García-Martín, and F. R. Rubio, “Temperature control of a solar furnace,” IEEE Control Syst. Mag., vol. 19, no. 1, pp. 8–24, 1999. [17] A. Paradkar, A. Davari, and A. Feliachi, “Temperature control of a solar furnace with disturbance accommodating controller,” in Proc. 34th Southeastern Symp. Syst. Theory, 2002, pp. 431–434. [18] B. A. Costa, J. M. Lemos, E. Guillot, G. Olalde, L. G. Rosa, and J. C. Fernandes, “Temperature control of a solar furnace for material testing,” presented at the 8th Portuguese Conf. Autom. Control (CONTROLO), Vila Real, Portugal, 2008. [19] B. A. Costa, J. M. Lemos, E. Guillot, G. Olalde, L. G. Rosa, and J. C. Fernandes, “An adaptive temperature control law for a solar furnace,” presented at the 16th Med. Conf. Control Autom. (MED), Ajaccio, France, 2008. [20] B. A. Costa and J. M. Lemos, “An adaptive temperature control law for a solar furnace,” Control Eng. Pract., vol. 17, pp. 1157–1173, 2009. [21] D. Lacasa, M. Berenguel, I. Cañadas, and L. Yebra, “Modelling the thermal process of copper sintering in a solar furnace,” in Proc. 13th Solarpaces Int. Symp., 2006, pp. A9–P2. [22] D. Lacasa, M. Berenguel, L. Yebra, and D. Martínez, “Copper sintering in a solar furnace through fuzzy control,” in Proc. IEEE Int. Conf. Control Appl., 2006, pp. 2144–2149. [23] M. Beschi, M. Berenguel, A. Visioli, and L. Yebra, “Control strategies for disturbance rejection in a solar furnace,” presented at the IFAC World Congr., Milan, Italy, 2011. [24] A. Visioli, “A new design for a PID plus feedforward controller,” J. Process Control, vol. 14, pp. 457–463, 2004. [25] A. Wallen and K. J. Åström, “Pulse-step control,” presented at the 15th IFAC World Congr., Barcelona, Spain, 2002. [26] A. Wallen, “Tools for autonomous process control,” Ph.D. dissertation, Lund Inst. of Tech., Dept. Autom. Control, Lund, Sweden, 2000. [27] M. Mattei and F. Amato, “Robust control of a plasma wind tunnel,” Eur. J. Control, vol. 7, pp. 494–510, 2001. [28] M. Mattei, “An LMI approach to the robust temperature trajectory following for space vehicle testing,” Automatica, vol. 37, pp. 1979–1987, 2001. [29] CIEMAT, Madrid, Spain, “PSA annual report,” , 2009. [Online]. Available: http://www.psa.es/webeng/techrep/index.php [30] F. L. Lewis, “Optimal control,” in The Control Handbook, W. S. Levine, Ed. Boca Raton, FL: CRC Press, 1996. [31] K. J. Åström and T. Hägglund, PID Controllers: Theory, Design and Tuning. Research Triangle Park: ISA Press, 1995. [32] E. F. Camacho and M. Berenguel, “Application of generalized predictive control to a solar power plant,” in Proc. 3rd IEEE Conf. Control Appl., 1994, pp. 1657–1662. [33] M. Berenguel, M. R. Arahal, and E. F. Camacho, “Modeling free response of a solar plant for predictive control,” Control Eng. Pract., vol. 6, pp. 1257–1266, 1998. Manuel Beschi was born in Castiglione delle Stiviere, Italy, in 1986. He received the industrial automation engineering degree (cum laude) from the University of Brescia, Brescia, Italy, in 2010, where he is currently pursuing the Ph.D. degree in automatic control and computer science. His research interests include control applications for solar energy systems and event-based industrial control.

Antonio Visioli (S’97–M’00–SM’07) was born in Parma, Italy, in 1970. He received the Laurea degree in electronic engineering from the University of Parma, Parma, Italy, and the Ph.D. degree in applied mechanics from the University of Brescia, Brescia, Italy, in 1995 and 1999, respectively. His Ph.D. dissertation was on control strategies for industrial robot manipulators. He is currently an Associate Professor of automatic control with the Department of Information Engineering, University of Brescia. His research interests include industrial robot control and trajectory planning, dynamic-inversion-based control and process control. He has authored or co-authored two monographs, one textbook, and over 150 papers in international journals and conferences. Dr. Visioli is a member of IFAC and a member of the board of Anipla (Italian Association for the Automation).

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Manuel Berenguel (M’00) was born in Almería, Spain, in 1968. He received the industrial engineering and Ph.D. (extraordinary doctorate award) degrees from the University of Seville, Seville, Spain. He is Full Professor of automatic control and systems engineering with the University of Almería, Almería, Spain. He is the head of the research group Automatic Control, Electronics, and Robotics, University of Almería. His research interests include control education and in predictive and hierarchical control, with applications to solar energy systems, agriculture, and biotechnology. He has authored and coauthored over 200 technical papers in international journals and conferences. Prof. Berenuel has been member of the board of Governors of the Spanish Association in Automatic Control from 2003 to 2008, member of IEEE Control System Society from 2000, and a member of the IFAC Technical Committee TC 8.01 Control in Agriculture.

Luis José Yebra was born in Almería, Spain, in 1971. He received the telecommunications technical engineering degree from Universidad de Alcalá de Henares, Madrid, Spain, in 1993, and the physics degree and the Ph.D. degree in computer science engineering from the Spanish National Distance Education (UNED), Madrid, Spain, in 1997 and 2006, respectively. He has authored 3 books, 15 scientific articles published in international journals, over 25 conference proceedings papers, and participated as a researcher in more than 15 research projects.