Variable sampling adaptive control of a distributed ... - INESC-ID

IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 11, NO. 5, SEPTEMBER 2003

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Variable Sampling Adaptive Control of a Distributed Collector Solar Field Rui N. Silva, João M. Lemos, and Luís M. Rato

Abstract—Distributed collector solar fields are spatially distributed engineering systems, which aim at collecting and storing energy from solar radiation. They are formed by mirrors which concentrate direct incident sun light in a pipe where a fluid, able to accumulate thermic energy, flows. From the control point of view, the objective considered here consists of making the outlet oil temperature to track a reference signal by manipulating the oil flow, in the possible presence of fast disturbances caused by passing clouds. Although this plant may be successfully controlled by methods assuming it to be modeled as a “black box” lumped parameter system, the point of view advocated in this paper is that explicit consideration of its distributed character leads to an increased control performance. In this respect, the contributions of this paper are twofold: First, a controller relying on variable sampling is developed. This is derived from the partial differential equation describing the oil temperature evolution in time and space on the field and has the effect of linearizing the plant model. Second, the resulting performance is illustrated by means of experiments performed in an actual solar field. The experiments reported show that it is possible to make fast temperature setpoint changes, with reduced overshoot. The ideas presented are applicable to other types of industrial processes, involving transport phenomena.

Fig. 1.

Collectors of the ACUREX field.

Index Terms—Adaptive control, distributed collector fields, process control, semigroups, solar energy, spatially distributed systems, variable sampling.

I. INTRODUCTION

D

ISTRIBUTED collector solar fields are spatially distributed engineering systems which aim at collecting and storing energy from solar radiation. They are formed by mirrors which concentrate direct incident sun light in a pipe, where an oil able to accumulate thermic energy flows. Figs. 1 and 2 show the field used in the experiments reported in this paper. This is the ACUREX field of Plataforma Solar de Almeria (PSA), located in the south of Spain. It consists of 480 parabolic mirrors arranged in 20 East–West oriented rows, forming ten parallel loops. The elevation of the mirrors is varied by a sun tracking controller, which will not be further considered here. Fig. 1 shows a detail of the collectors where the pipe is seen as a white rod in the focus of the mirrors and Fig. 2 provides Manuscript received November 1, 2001; revised October 25, 2001. Manuscript received in final form January 6, 2003. Recommended by Associate Editor G. A. Dumont. This work was supported in part by the research project AMBIDISC—Adaptive and nonlinear control of distributed parameter systems with environmental impact under Grant POSI/1999/SRI/36328. The work of J. M. Lemos and L. M. Rato was supported by POSI IIIrd EC Framework Programme. Experimental work at Plataforma Solar de Almeria was suported by EC-DGXII under the “Improving Human Research Potential” Programme “Access to Research Infrastructures” Activity. R. N. Silva is with FCT/UNL, Universidad Nova de Lisboa, 2829–526 Capariica, Portugal (e-mail: [email protected]). J. M. Lemos is with INESC-ID/IST, 1000-029 Lisboa, Portugal (e-mail: [email protected]). L. M. Rato is with INESC-ID/U.Évora, 1000-029 Lisboa, Portugal. Digital Object Identifier 10.1109/TCST.2003.816407

Fig. 2. Overall view of the ACUREX field.

an overall view of the field evidencing its spatially distributed structure. The oil flowing in the pipe is an incompressible fluid, able to support temperatures up to 300 . Since this oil is a very poor thermal conductor, heat diffusion effects in it may be neglected, a fact to be exploited in the plant model used for control design. Fig. 3 shows a schematic diagram of the oil circuit. The oil is extracted at low temperature from the bottom of a storage tank (seen at the right lower corner of Fig. 2), passed through the field where it is heated by solar radiation and returns to the tank, where it is injected at the top. Hereafter, the part of the oil pipe located on the focus of the mirrors will be called the “active part.” Inside the storage tank the oil forms layers at different temperatures, which do not mix, a fact making possible energy storage. From the top of this tank (hot zone inside the tank) the oil may be extracted for usage, e.g., in a desalination plant. After the energy has been used, the cool oil is reinjected at the bottom of the storage tank. The control objective considered in this paper consists of making the average of the loop outlet oil temperatures (hereafter simply referred as “outlet oil temperature”) to track a reference signal by manipulating the oil flow in the presence of fast acting disturbances caused by passing clouds. Other main disturbances are changes in radiation due to atmosphere scattered

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Fig. 3.


Schematic view of the ACUREX field.

water steam, in the temperature of the inlet oil coming from the bottom of the storage tank and in ambient temperature. Dust deposition and other factors such as wind, changing collectors shape, also act as disturbances because they alter mirror reflectivity. In [1] experimental data is shown which illustrates this fact. It consists of the daily evolution of the outlet temperature of two different rows, which are noticeably different. The above control problem may not adequately be solvable with a constant gain linear controller relying on a simple design. Again [1] provides an example in which a proportional integral derivative (PID) controller tuned for higher values of the flow (low temperatures) yields unacceptable oscillatory responses in setpoint changes. This motivated research on more sophisticated controllers of which [1]–[10], [12]–[14], [16]–[20], [23] are significant examples. The major role played by changes of solar radiation and plant uncertainty lead to the approach of [6] where a pole placement self-tuning controller with a series feed-forward compensator is used. An improvement of the adaptation mechanism and of the underlying control law was possible by resorting to predictive adaptive control techniques. Different forms of adaptive GPC [3], [7] and MUSMAR [10], [23] were then demonstrated with success. By making a frequency-response analysis under conditions which correspond to linear behavior, it is possible to recognize the occurrence of antiresonances [15]. This is confirmed by a simplified analysis based on the partial differential equation (PDE) describing collector loop dynamics [2] and lead to the design of controllers based on frequency methods [2], [16]. In [16], a prescheduled adaptive controller for resonance cancellation has been presented and in [2], an adaptive control algorithm using an internal model control structure together with a frequency-design approach has been introduced. While adaptive control already provides some form of accommodation of nonlinear behavior by adjusting the controller gains according to the operating point, explicit recognition of plant nonlinearities and their exploitation is much likely to lead

to performance and robust stability improvements (i.e., the ability of the plant to meet control objectives in a wider set of operating conditions). First steps in this direction were made by employing gain scheduled constant parameter GPC [4] and switched multiple model supervisory controllers [14], [24]. In [1], a nonlinear controller is developed which explicitly takes into account the distributed parameter model of the solar plant. By resorting to a simple space collocation method, a bilinear state-space model was derived and used as a basis for output feedback linearization design and for studying internal dynamics. The possibility of existence of internal oscillations and stationary wave temperature distributions along the pipe has been demonstrated. This complements, for a different situation, the results of [2], [15]. Using the methods of [21], a nonlinear adaptive controller was designed using Lyapunov methods. In this way, not only the spatial dependency of plant dynamics is taken into account but also, due to the adaptation mechanism, uncertainty and time variations on mirror reflectivity and other plant parameters are accommodated, thereby improving attained performance. Also departing from the PDE model of the plant, [13] proposes a design based on Lyapunov methods, using internal energy as Lyapunov function. Lyapunov methods, but using a quadratic function, are also the realm of [8], [9]. In [17] and [18], an optimal control point of view is adopted. The distributed parameter character of the plant is explicitly recognized and a maximum principle suitable to the type of dynamics considered is applied. Both optimal and suboptimal solutions are studied. Although the plant considered may be successfully controlled by methods assuming it to be modeled as a “black box” lumped parameter system, the point of view advocated in this paper is that explicit consideration of its distributed character (and consequent use of a PDE model) leads to an increased control performance. The key point to explore is the observation that the value of oil flow establishes a “natural” time scale for the system. Indeed, the higher the flow rate, the faster will the system response be. This is achieved in discrete time by indexing the sampling rate to the value of flow. As a consequence, the model equations become linear and it is possible to achieve good performance on step responses of big amplitude. The above idea may be applied to plants having a space dependency of their dynamics and involving transport phenomena. An example is provided by [26] where the modeling of a glass tube drawing bench is made. In this work, the sampling rate is chosen, according to heuristic arguments, depending on drawing speed. The main steps in the design of the controller proposed in this paper are the following. First, the PDE modeling the field is decomposed. From the physical point of view, this corresponds to a time discretization such that the update, over a small time increment, of temperature spatial distribution corresponds to a shift due to flow, composed with temperature increase due to radiation (and losses to the environment if considered). Second, the time scale is changed, depending on flow, such that the characteristic lines of the shift term become straight

SILVA et al.: VARIABLE SAMPLING ADAPTIVE CONTROL OF A DISTRIBUTED COLLECTOR SOLAR FIELD

lines. This new time scale is defined in practice by a variable sampling rate. The resulting sampled data model is then used to compute the manipulated variable by minimizing an extended, receding horizon, cost. The contributions of this paper are twofold: First, controllers relying on variable sampling are developed. These are derived from the PDE describing the oil temperature evolution in time and space on the field. Second, the resulting performance is illustrated by means of experiments performed in an actual field. The experiments reported show that it is possible to make fast temperature setpoint changes, with a value substantially higher than what is found in the literature, with reduced overshoot and comparable settling times. The paper is organized as follows: After a short plant description, a simplified PDE model is recalled. This is the basis of the control laws to be designed. By considering a suitable time variable transformation, a simple model is then obtained, which reveal plant dynamic character and allow controller design in a simple way. Different controllers are then derived on the basis of these transformed models and tested on the real plant. Part of the results presented in this paper are contained in a preliminary form in [25]. The interest of the approach presented in this paper transcends the application to solar plants. Indeed, there are several examples of industrial processes of interest, involving transport phenomena, to which similar techniques may be applied with advantage.

II. CONTROL PROBLEM Each of the field loops is approximately modeled [5] by the PDE

(1) is the increment with respect to the ambient temwhere perature of the oil temperature at location (measured along is the oil flow, is the corrected the pipe) and at time , solar radiation, is the area of the inner cross section of the is mainly conpipe and and are parameters. The first with cerned with collector optical efficiency and the second losses along the pipe. On the right-hand side of (1) the first term (first-order space derivative) reflects the change of temperature due to oil flow. The second term models oil heating by solar radiation and the third term models losses along the pipe. and Temperature measurements are made at (beginning and end of the active part of the loops). Solar radiwhich is ation is also measured. It is the temperature at controlled in this paper. The control objective consists in keeping the outlet oil temconstant in spite of changes in the inlet oil temperature , the solar radiation, and other unmeasurable disperature turbances, e.g., mirror reflectivity changes due to dust deposition.

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Fig. 4. Time propagation of a temperature distribution along the pipe, in the absence of solar heating and losses.

III. MODEL SAMPLING Model sampling (discretization) of (1) with the approximation

, leads to

(2) This equation may be obtained by performing an energy balance along the pipe [22]. For plants where an energy balance does not make sense, another possibility of deducing (2) is to resort to semigroup decomposition techniques [25]. Equation (2) actually defines a sampled data model which allows to compute from the temperature distribution along the pipe at time the temperature distribution at time , with (the sampling interval) small. It should be recalled that model (2) neglects losses ( is assumed to vanish for the sake of simplicity). In the actual control law used, however, the loss term has been considered. IV. CHANGING THE TIME SCALE For motivating the need for a discrete-time model using a variable sampling time period, consider the situation in which and there are no losses there is no heating by the sun . The temperature distribution along the pipe at will in this case evolve in time (Fig. 4) such that it is constant along the characteristic curves, given by (3) is any value in the interval . If the oil flow is where constant, these characteristic curves are straight lines. In the “warped” time scale defined by (4) is a constant with dimensions of flow, the characterwhere istic curves become straight lines and equally spaced samples in space correspond to equally space samples in time. In particular, as shown in Fig. 5, by using such a time scale, if the pipe is divided in segments, an element of oil in the first segment will always exit the pipe exactly after time sampling periods, no matter what the flow value is. It is thus natural to consider time sampling in the scale .

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Fig. 6. Pole/zero map and antiresonant frequency response of the warped time field model. Fig. 5.

Even sampling in space implies uneven sampling in time.

Let

V. DISCRETE MODEL WITH VARIABLE SAMPLING TIME Let the active part of the pipe, i.e., the part between and be divided in segments of equal length (Fig. 5) and let denote a generic sampling instant in which the output (which may temperature is measured. The sampling interval vary with the discrete-time epoch ) is defined as

denote the temperature, respectively, at the input and at the output of the active part of the field and take as manipulated variable the inverse of the flow multiplied by

(5)

(11)

Assuming the flow to be constant in each interval the sampling interval is chosen such that, during it, the oil in each pipe volume of length is transfered to the following volume element. This means that the sampling interval is such that

With these choices and the fact that the sampling interval verifies (6), the model (10) is written

(6)

Hereafter, since only discrete time is considered, for simplicity denotes . The constant has been of notation inserted to take into account losses (this may also be justified by and using similar arassuming from the beginning that guments as before). Equation (12) forms a sampled time model in warped time and is the basis for control design. It is interesting to remark that, assuming no losses and a constant radiation , (12) defines a finite-impulse response (FIR) model. For illustrative purposes, Fig. 6 shows the corresponding pole/zero map and the frequency response for . As seen, there are antiresonant peaks. Although for a different situation (because a “warped time” model is used here), this is in relation with the results of [1], [2], [15].

This nonuniform sampling is actually a way of implementing the time scale transform defined by (4). With the sampling dewill fined in this way, the oil in the first element at time exit the pipe after samples (Fig. 5), at the continuous time given by (7) Iterated application of (2) over these steps, with by (5), yields for the outlet oil temperature

defined

(12)

VI. CONTROL LAWS (8) Since the sampling is made such that (6) holds, it follows that (9) and, considering (7), (8) becomes (10)

The control laws to consider are designed such as to minimize the multistep receding horizon cost functional given by (13) is the desired where (integer) is the prediction horizon, is a penalty on reference for the output temperature and the control effort. Two types of controllers are considered hereafter. In the first, model (10) is directly used for minimizing (13). In the the second, the minimization is carried out on the basis of an equivalent state-space model and a state estimator.


A. Design With the

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Model

From (12) it is seen that the outlet oil temperature over the prediction horizon verifies the following predictive model:

(a)

(14) Prediction of the outlet oil temperature depends on the future values of the solar radiation and the manipulated variable. If there is no need for future values the horizon is such that . of the inlet oil temperature , under the constraint of Minimization of (13), with a constant value of the manipulated variable along the prediction horizon, and assuming the radiation to be constant over the horizon and equal to its value at the present instant , results in

(b) Fig. 7. WARTIC- i=o: (a) Oil outlet temperature and (b) oil flow.

It should be remarked that the presence of in 16 provides a feedforward action with respect to changes in the inlet oil temperature. B. Control Design Based on State Model (15)

Consider (10) and build the equivalent fully controllable and fully observable state-space model (18) (19)

where (16) The minimization of (13) is performed according to a receding horizon strategy: of the whole sequence minimizing , only the first element given by (15) is actually applied to the plant input at time , the whole procedure being repeated over the next sampling interval. (WARped TIme Controller The following WARTIC model) algorithm summarizes the control strategy based on proposed: algorithm WARTICAt the beginning of each sampling interval, recursively execute the following steps. 1) On the basis of plant measurements, and in estimate the parameters the model (12) by recursive least squares. 2) Apply to the plant a control given by (15). 3) Choose the duration of sampling into be given by terval (17)

where .. .

.. .

.. .

(20) (21)

is the state formed by the oil temperatures at positions , is the outlet oil temperature and is the inlet oil temperature. The variable is given as in (11), being computed by minimizing the quadratic cost ), assuming a constant reference function (13) (with along the optimization horizon. This results in

(22)

. where Since the state is not accessible for direct measurement, its value in (22) is replaced by its estimate produced by an asymptotic observer. This results in the WARTIC- State control algorithm, equal to WARTIC- , but with Step 2) replaced by (22). VII. EXPERIMENTAL RESULTS Experiments with both WARTICreported hereafter.

and WARTIC- State are

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(a)

(b) Fig. 8. WARTIC i=o: (a) Solar radiation and (b) oil inlet temperature.

Fig. 9. WARTIC i=o: Parameter estimates in warped time.

A. Experiments With WARTICFigs. 7–9 shows experimental results obtained in the ACUREX field of PSA on 18 June 1998, using the WARTIC and algorithm described above. The values are used. Fig. 7(a) and (b) shows the oil outlet temperature and oil flow, respectively. Note that this is an image of the sampling period being used, which varies in time according to (17). Fig. 8(a) and (b) describes the accessible disturbances of solar radiation and oil inlet temperature, respectively. The inlet oil temperature is kept approximately constant, while solar radiation changes according to day time, but without significant clouds disturbing it. Fig. 9 shows the evolution of the estimates of the coefficients and of the time warped discrete-time model. A major issue illustrated by this experiment is that it is possible to apply jumps of high value in the value of the reference.

Fig. 10. WARTIC-State: Outlet oil temperature and flow, T (experimental).

Fig. 11.

=

1

WARTIC-State: Estimated temperatures along the pipe, T = 1.

In Fig. 7, a jump of 40 is applied to the reference, the settling time being of the order of 12 min. This jump is significantly higher than the ones possible with several other methods, without overshoot and without increasing the delay of the response. When using methods which do not take into consideration the distributed dynamics of the plant, a setpoint of 20 is considered high. In [13], where the distributed dynamics is explicitly considered, experimental results of a jump of 30 are reported. B. Experiments With WARTIC-State Figs. 10–15 show experimental results obtained, with WARTIC- State, in the ACUREX field for different values of was chosen in order to get the horizon . The value of a sampling time between 9s and 45s. As seen, the performance grows. It is also interesting to look at the improves when estimated temperatures inside the tank, Figs. 11, 13 and 15.


Fig. 12. WARTIC-State: Outlet oil temperature and flow (experimental).

T

=

8

Fig. 14. WARTIC-State: Outlet oil temperature and flow (experimental).

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T

Fig. 15. WARTIC-State: Estimated temperatures along the pipe (experimental). Fig. 13. WARTIC-State: Estimated temperatures along the pipe (experimental).

=

16

T = 16

T = 8

Furthermore, for there are oscillations at the manipulated variable (lower part of Fig. 12) which are not present at the output temperature (upper part of Fig. 12), a fact due to the cancellation of antiresonant plant zeros. These oscillations tend to disappear when grows (compare Figs. 13 and 15). With WARTIC-State it is also possible to perform sudden high-amplitude changes with little overshoot. For instance, Fig. 16 shows a jump of 50 . The oscillations on the manipis being used. ulated variable are due to the fact that suppress these As previously discussed, higher values of oscillatory behavior. only considers explicitly the output, While WARTICWARTIC-State has the interesting feature of taking into account the temperature along the pipe. This may be important, e. g., for security reasons. However, WARTIC-State has the drawback of requiring an observer, which is avoided in WARTIC- .

Fig. 16. WARTIC-State: Sudden high amplitude setpoint jumps of temperature, T = 8 (experimental).

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VIII. CONCLUSION While the distributed collector solar field considered has been the subject of many research works, this paper considers a novel approach, viz. the use of a variable time scale indexed to flow, to develop a variable sample time controller. By explicitly considering the distributed parameter behavior of the plant, this controller is able to impose jumps of high value in the reference to follow, without overshoot or increased rise time. This is a consequence of using a modified time scale which linearizes the plant model. Furthermore, it was shown, by means of experiments performed on the plant, that internal oscillations are eliminated by extending the horizon of the multistep cost function. The strategy described may be applied to the control of other plants described by similar partial differential equations, e.g., drying processes, tubular reactors, or river pollution. REFERENCES [1] M. Barão, J. M. Lemos, and R. N. Silva, “Reduced complexity adaptive control of a distributed collector solar field,” J. Proc. Contr., vol. BF 12, pp. 131–141, 2002. [2] M. Berenguel and E. Camacho, “Frequency based adaptive control of systems with antiressonance modes,” in Prep. 5th IFAC Symp. Adaptive Systems Control Signal Processing, Budapest, Hungary, 1995, pp. 195–200. [3] E. F. Camacho, M. Berenguel, and C. Bordóns, “Adaptive generalized predictive control of a distributed collector field,” IEEE Trans. Contr. Syst. Technol., vol. 2, pp. 462–467, July 1994. [4] E. F. Camacho, M. Berenguel, and F. Rubio, “Application of a gain scheduling generalized predictive controller to a solar power plant,” Contr. Eng. Practice, vol. 2, no. 2, pp. 227–238, 1994. [5] E. Camacho, M. Berenguel, and F. Rubio, Advanced Control of Solar Plants. New York: Springer-Verlag, 1997. [6] E. F. Camacho, F. R. Rubio, and F. M. Hughes, “Self-tuning control of a solar power plant with a distributed collector field,” IEEE Control Syst. Mag., pp. 72–78, Jan. 1992. [7] E. F. Camacho and M. Berenguel, “Robust adaptive model predictive control of a solar plant with bounded uncertainties,” Int. J. Adaptive Contr. Signal Processing, vol. 11, no. 4, pp. 311–325, 1997. [8] L. Carotenuto, M. La Cava, and G. Raiconi, “Regular design for the bilinear distributed parameter of a solar power plant,” Int. J. Syst. Sci., vol. 16, pp. 885–900, 1985.

[9] L. Carotenuto, M. La Cava, P. Muraca, and G. Raiconi, “Feedforward control for the distributed parameter model of a solar power plant,” Large Scale Syst., vol. 11, pp. 233–241, 1986. [10] F. Coito, J. M. Lemos, R. N. Silva, and E. Mosca, “Adaptive control of a solar energy plant: Exploiting accessible disturbances,” Int. J. Adapt. Contr. Signal Processing, vol. 11, pp. 327–342, 1997. [11] R. Dresnack and W. E. Dobbins, “Numerical analysis of BOD and DC profiles,” J. Sanitary Eng. Division, Proc. Amer. Soc. Civil Eng., vol. SA5, pp. 789–807, 1968. [12] T. A. Johansen, K. Hunt, and I. Petersen, “Gain scheduled control of a solar power plant,” Contr. Eng. Practice, vol. 8, pp. 1011–1022, 2000. [13] T. A. Johansen and C. Storaa, “Energy-based control of a solar collector field,” Automatica, vol. 38, pp. 1191–1199, 2002. [14] J. M. Lemos, L. M. Rato, and E. Mosca, “Integrating predictive and switching control: Basic concepts and an experimental case study,” in Nonlinear Model Predictive Control, F. Allgöwer and A. Zheng, Eds. Basel, Switzerland: BirkhäuserVerlag, 2000, pp. 181–190. [15] A. Meaburn and F. M. Hughes, “Resonance characteristics of a distributed solar collector fields,” Solar Energy, vol. 51, no. 3, pp. 215–221, 1993. , “Prescheduled adaptive control scheme for ressonance cancella[16] tion of a distributed solar collector field,” Solar Energy, vol. 52, no. 2, pp. 155–166, 1994. [17] A. Orbach, C. Rorres, and R. Fischl, “Optimal control of a solar collector loop using a distributed-lumped model,” Automatica, vol. 27, no. 3, pp. 535–539, 1981. [18] C. Rorres, A. Orbach, and R. Fischl, “Optimal and suboptimal control policies for a solar collector system,” IEEE Trans. Automat. Contr., vol. AC-25, pp. 1085–1091, June 1980. [19] F. Rubio, M. Berenguel, and E. Camacho, “Fuzzy logic control of a solar power plant,” IEEE Trans. Fuzzy Syst., vol. 3, pp. 459–468, Aug. 1995. [20] F. Rubio, F. Gordillo, and M. Berenguel, “LQG/LTR control of the distributed collector field of a solar power plant,” in Prep. IFAC Symp. Adaptive Systems Control Signal Processing, Glasgow, U.K., 1996, pp. 335–340. [21] S. Sastry and A. Isidori, “Adaptive control of linearizable systems,” IEEE Trans. Automat. Contr., vol. 34, pp. 1123–1131, Nov. 1989. [22] R. N. Silva, “Time scaled predictive controller of a solar power plant,” in Proc. European Control Conference 99, Karlsruhe, Germany, 1989. [23] Silva, R. N. Silva, L. M. Rato, J. M. Lemos, and F. Coito, “Cascade control of a distributed collector solar field,” J. Proc. Contr., vol. 7, no. 2, pp. 111–117, 1997. [24] R. N. Silva, “Plant driven design of a nonlinear PID controller,” in Proc. IFAC Workshop on Digital Control, Terrassa, Spain, 2000, pp. 236–241. [25] R. N. Silva and J. M. Lemos, “Adaptive control of transport systems with nonuniform sampling,” in Proc. ECC’01, Porto, Portugal, 2001, pp. 1774–1779. [26] V. Wertz, G. Bastin, and M. Haest, “Identification of a glass tube drawing bench,” in Prep. 10th World Congr. Automatic Control, vol. 10, Munich, Germany, 1987, pp. 334–339.