17th IEEE International Conference on Control Applications Part of 2008 IEEE Multi-conference on Systems and Control San Antonio, Texas, USA, September 3-5, 2008
FrA03.5
Nonlinear predictive control of processes with variable time delay. A temperature control case study M. Sbarciog, R. De Keyser, S. Cristea and C. De Prada Abstract— Material or fluid transportation is a commonly encountered phenomenon in industrial applications, generating variable time delay that makes the design of feedback control loops more difficult. This paper investigates the applicability of MPC (Model Predictive Control) strategies to this type of processes. The experimental setup consists of a heated tank, of which the outlet temperature (measured at a certain distance from the tank) is controlled by manipulating the outlet flow. The nonlinear EPSAC (Extended Prediction Self-Adaptive Control) approach is used, which reduces the complexity of nonlinear optimization to iterative quadratic programming. It is shown that developing a process model in which dynamics are decoupled from the variable time delay leads to a Smith predictorlike control structure, that allows the proper operation of the control loop with fixed control parameters. The performance of the predictive controller is compared on the pilot plant to the performance of classic control approaches for systems with time delay.
I. INTRODUCTION Many processes include time delay phenomena in their inner dynamics, representative examples being found in biology, chemistry, mechanics, physics, population dynamics, as well as in engineering sciences. In addition, actuators, sensors, field networks that are involved in the feedback loops usually also introduce delays, which are possibly timevarying. Thus there is an increasing interest in studying time delay systems in all scientific areas, especially in control engineering [1], [2]. The presence of time delay (dead time) in the control loop is always a serious obstacle to good performance. Hence time delay compensators have received a considerable amount of attention in the literature. The oldest, but also the most widely encountered method in industry is the Smith predictor [3]. Since then, many of its modifications have been studied in the control literature [4] and successfully implemented in practice. In [5] the authors introduce a methodology based again on the Smith predictor that improves the robustness of PI-controllers, while Ingimundarson and H¨agglund [6] describe how to robustly tune dead time compensating controllers. The shortcoming of mostly all available methodologies is that they are developed assuming a stable or integrating process with constant time delay. M. Sbarciog and R. De Keyser are with EeSA - Department of Electrical Energy, Systems and Automation, Ghent University, Technologiepark-Zwijnaarde 913, B-9052 Ghent, Belgium
[email protected]
[email protected] S. Cristea and C. De Prada are with Department of Systems Engineering and Automatic Control, Faculty of Sciences, c/ Real de Burgos, s/n, University of Valladolid, Spain
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978-1-4244-2223-4/08/$25.00 ©2008 IEEE.
Nevertheless, there are industrial processes involving transportation of material which is directly depending on the manipulated variable. Consequently, these processes possess an inherent variable time delay. This increases the complexity of the control problem. Although there exist only few results concerning the control of variable time delay processes reported in literature, a reasonable assumption is that the control loop can benefit from a good estimation of the variable time delay. In [7] a technique for compensating variable time delay is proposed. The method relies on a Smith predictor structure, in which the delay parameter in the process model is continuously tuned by minimizing a performance index function. An application of variable timedelay compensation is the solar collector plant described in [8], [9]. Another interesting application has been presented in [10]. MPC (Model based Predictive Control) techniques are nowadays pretty popular in academic as well as in industrial control engineering. Many successful implementations have been reported (e.g. [11], [12], [13]). In this paper the focus is on controlling a variable time delay process using the nonlinear EPSAC technique [14], which does not impose any structure on the process model and offers a simple and effective way of calculating the optimal control input. The paper is organized as follows. Section II describes the process to be controlled. Physical and experimental modelling techniques are used to develop an accurate model of the process in Section III. Next a brief review of the basic ideas of nonlinear predictive control is presented, followed by the nonlinear EPSAC algorithm. Section V illustrates the effectiveness of the proposed control strategy by comparing the experimental results to the ones resulting from two classic control strategies: genuine PI-control and Smith predictor with PI-controller. In the end conclusions are drawn. II. P ROCESS DESCRIPTION The process considered in this paper consists of a heated tank of which the level is controlled by a mechanical float switch, thus resulting in a constant water volume (Fig. 1). A submerged electrical heater delivers a constant heat flow, which causes the liquid to warm up. Temperature control of the outlet water is achieved by changing the outflow of hot tank water, which allows an equal amount of cold tap water to flow in. A variable transport delay, which depends on the outflow (the manipulated variable), is introduced in the system by measuring the temperature of the effluent stream at a distance L from the tank.
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III. P ROCESS MODELLING V
For predictive control a process model must be available. In this contribution the process is modelled as the series connection of the tank model and tube model, which consists of dynamics and time delay. Physical modelling is used for determining the tank model, while the tube dynamics are approximated experimentally.
Q Process Tank
Sensor: T tank (t)
q(t)
Outlet tube
q(t)
A. Tank model
L
Sensor: T in (t)
Sensor: Tout (t)
Fig. 1.
The mathematical model that describes the relationship between the outflow q(t) and the tank temperature Ttank (t) follows from an energy balance:
Schematic of the process.
dTtank (t) = Q + ρ c p q(t) (Tin (t) − Ttank (t)) (2) dt where ρ and c p are respectively the density and the specific heat of water, and Q is the amount of supplied heat (which is constant). Small heat losses to the surroundings are present but they are neglected. Notice that the model is nonlinear, as q(t) is the control input (manipulated variable). The model of the tank has been validated against experimental data. In Fig. 3 the continuous line shows the real tank temperature while the dashed line represents the model output. There exists a difference in the steady state value, a kind of error which can be easily overcome by a controller with integral action. The dynamics of the estimation matches the real measurement quite well, which makes model (2) suited for being used in a predictive controller.
ρ cp V
42 Measurement Model Output
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[°C]
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tank
where S denotes the cross-section of the outlet tube. Hence an extra difficulty is introduced in the control problem: we are facing a variable time delay but also a variable time constant, in which the degree of change in the time constant is of the same magnitude as the change in the time delay. The nominal value for τ is 67 sec for q = 0.0167 l/sec. Accurate flow control is achieved using a peristaltic pump driven by a 24 V DC-motor, that provides linearity between control voltage and flow. Finally, three Pt100 sensors register the temperature: (1) of the hot effluent water Tout (t), (2) of the cold incoming tap water Tin (t) and (3) of the tank Ttank (t). Only Tout and Tin are used for control purpose; the other measurement is used for the analysis of the results. Fig. 2 shows an overview of the experimental setup.
T
The experimental setup – based on ideas presented in [18] and on the operation of a solar collector field [9] – has been designed and implemented at Ghent University [19]. It has a tank volume V = 1.13 l, a heat input Q = 1100 W and an outlet tube length L = 10 m (tube volume equal to 1.02 l). The flow range is 0.005 ≤ q ≤ 0.03 l/sec. For a constant outflow q, the tank time constant τ is approximately equal to the time delay d: V L =τ ≈d= (1) q q/S
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tank
outlet tube
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Fig. 3.
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Validation of the tank model.
pump
B. Tube model Experimentally, the tube dynamics have been fitted to a first order model in Laplace variable s: Kt T (s) = Ttank (s) τt s + 1
PC interface
Fig. 2.
Experimental setup overview.
(3)
where T (t) is a virtual signal which is related to the outlet temperature by Tout (t) = T (t − d(t)) (4) and d(t) represents the variable time delay. From identification experiments, reasonable results have been obtained with a constant gain Kt = 0.99 (small heat loss) and time constant τt = 29 sec.
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As the outlet tube includes also time delay, the validation of this model is discussed in the next section. C. Variable time delay model The temperature sensor is located in the outlet tube at a distance L from the tank. Due to the variable flow, the time the water needs to travel from the tank to the sensor varies as well and is given by t
q(τ ) dτ = LS
(5)
t−d(t)
with L the length and S the cross-section of the tube. From a physical point of view, this is equivalent to filling the tube at varying flow rates. Discretizing (5) allows the calculation of the discrete-time variable delay Nd :
the nonlinear EPSAC approach [14] to NMPC is presented and a slightly modified control loop layout is introduced to better deal with the variable time delay. A. NMPC formulation In a discrete time formulation, the objective of a model predictive controller is to find the future process input sequence that optimizes a cost function over a certain prediction horizon (N1 . . . N2 ). Thus, at each sampling instant, the process model expressing the nonlinear dynamic relationship between the process output y and the manipulated process input u (i.e. y(t) = f [y(t − 1), . . . , u(t − 1), . . .]) is used to produce output predictions. The future control sequence is the solution of an on-line optimization problem, which typically consists of minimizing the summed squares of the predicted output deviations from the setpoint r while penalizing also the control effort:
Nd
Ts
∑ q(t − i) = LS
min
(6)
u(t|t),...,u(t+Nu −1|t)
J
i=1
where t now denotes the discrete time index and Ts the sampling period. Consequently, at each sampling instant Nd is calculated as the number of flow samples to be summed LS before the total sum exceeds . This means that the variable Ts time delay depends on flows that have been applied in the past [q(t − 1), q(t − 2), . . . , q(t − Nd )]. The entire model (tank, tube and time delay) is validated against experimental data in Fig. 4. The match is acceptable, although less good than that of the tank model alone. The varying time delay is accurately estimated. The remaining model errors will have to be dealt with by the robustness of the predictive controller. 42 Measurement Model Output
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T
out
[°C]
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N2
J=
∑
k=N1
[y(t + k|t) − r(t + k|t)]2 + α
Nu −1
∑
[∆u(t + j|t)]2 (7)
j=0
y(t + k|t) denotes the prediction of the process output at discrete time instant t + k based on information available up to the discrete time instant t, r(t + k|t) is the setpoint, ∆u(t + j|t) = u(t + j|t) − u(t + j − 1|t) and α is a weighting parameter. Commonly, only Nu components of the future control sequence are allowed to vary, inputs beyond the control horizon Nu are set to the last computed value: u(t + j|t) = u(t + Nu − 1|t), j = Nu . . . N2 − 1. Feedback is introduced in the control loop by considering a receding horizon mechanism: only the first component out of the Nu optimal control moves is applied to the process. The rest of the control sequence is discarded and the entire procedure is repeated at the next sampling instant. The control law is the solution of a nonlinear programming problem, which can be formulated generically as a real time minimization of a nonlinear cost function with constraints. B. Nonlinear EPSAC formulation (NEPSAC)
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where
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Fig. 4.
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Validation of the entire model.
IV. N ONLINEAR P REDICTIVE C ONTROL Nonlinear model predictive control (NMPC) is a natural extension of the linear MPC technique. Most of the algorithms are based on the use of an internal plant model, which captures the main process characteristics and allows a proper prediction of the process output, followed by a dynamic optimization that provides the optimal manipulated variables. In this section, a brief introduction to NMPC is given. Then
EPSAC is one of the original model predictive control strategies developed in the early 1980’s [14]. Since then, it has been continuously refined and applied to many technical and non-technical systems [15], [16], [17]. Compared to the current standard MPC strategies, EPSAC considers the process output predictions as being the sum of 2 parts: a term which is independent of the future control actions and a term which depends linearly on the future control actions. This allows to obtain an analytical solution in the case of unconstrained control, or a well-known quadratic programming solution in the case of constrained control. In both cases, this leads to a quick solution of the MPCproblem, with low-complexity software compared to the more general optimization solvers. The core of the nonlinear EPSAC formulation is the replacement of the complex, time
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consuming nonlinear optimization with iterative quadratic optimization problems, whose solutions converge to the nonlinear optimal solution. C. Implementation Issues The complexity of the prediction procedure is of a higher order for systems with variable time delay than for those with constant time delay. For a system with time delay, changes in the controlled variable are noticeable once the time delay has passed. Therefore, in order to find the optimal control sequence only output predictions occurring after the time delay should be taken in the cost function. This means that the minimum costing horizon N1 should be equal to the time delay. For systems with constant time delay this is easy to do. Then the maximum prediction horizon N2 can be set to an appropriate value that ensures a stable and robust response and the control loop can be operated with fixed controller parameters. For a variable time delay however, the value of N1 (and thus also N2 ) varies with the dead-time index. In this paper the structure of the process model is exploited to design a predictive controller with constant design parameters (N1 , N2 ). Since the process under consideration is stable, a parallel structure is used for output estimation. (Fig. 5). The model of the process consists of the nonlinear tank dynamics and the tube dynamics, which are separated from the variable time delay model. At each sampling instant, the delay-free model output x(t), resulting from the process dynamics only, is calculated using the stored values [x(t − 1), . . . , u(t − 1), . . .]. At the same sampling instant, the variable time delay is computed from (6). Once Nd is known, x(t − Nd ) can be selected out of the stored x-values, such that z(t) = x(t − Nd ). Disturbance
Process Output
Process
+
Process Model dynamics
Delay-free Model Output
Non-linear tank & linear tube dynamics
y(t)
Variable time delay
Model Output
-
+
z(t) Estimation of Disturbance
n(t)
x(t) EPSAC Process Model dynamics Nonlinear tank & dynamics
u base (t+k|t)
Noise Model
C (q -1 ) D( q -1 )
x(t+k|t)
linear tube
n(t+k|t)
ubase (t+k|t) =u(t+k|t) no Process input
yes u(t) =u(t|t)
u(t+k|t)
+ 0
u(t+k|t)
Cost Function
+
ybase (t+k|t) r(t+k|t)
u(t+k|t)=u
Fig. 5.
base
n(t) =
C(q−1 ) e(t) D(q−1 )
(8)
where e(t) is white noise and C(q−1 ) and D(q−1 ) are monic polynomials in the backward shift operator. Designing the noise filter C(q−1 )/D(q−1 ) in a proper way ensures closed C(q−1 ) = loop robustness. A default, suboptimal choice is D(q−1 ) 1 . 1 − q−1 V. E XPERIMENTAL RESULTS Several experiments have been carried out in order to illustrate the effectiveness of the nonlinear EPSAC controller and the effect of the most important performance-oriented tuning parameters: the prediction horizon N2 and the noise C(q−1 ) model . The experiments have been performed with a D(q−1 ) sampling period of 4 sec. The pump cannot deliver less than 0.005 l/sec nor more than 0.03 l/sec. The corresponding output range is approximately 10 ◦ C. The control horizon has been set to Nu = 1, as this choice generally led to amazingly good results. The performance of the nonlinear EPSAC controller is compared to the one of classical control strategies. A. Nonlinear EPSAC control
+
u(t)
separation of the tank and tube dynamics on one hand and the varying time delay on the other hand. In such approach the minimum prediction horizon is no longer varying and obviously equal to one. Hence, the maximum prediction horizon is also constant. The variable time delay model is only used for noise computation n(t), which is defined as contributions in the process output that cannot be explained by the model. This signal is modelled as a colored noise process
(t+k|t) + u(t+k|t)
Control structure.
In this way, the prediction procedure is thoroughly simplified, resulting in a Smith predictor-like scheme, with
Generally, the value of N2 should be selected with respect to the time constant of the system. This favors both stability and performance. For systems with variable time constant as the one under consideration, it is reasonable to take the conservative choice corresponding to the largest time constant. In this case, N2 = 15 is a good compromise. Since Ts = 4 sec, the transient behavior is predicted for 60 seconds. This leads to a good performance and does not impose computational challenges on the computer. Whereas the nonlinear tank model is quite accurate, the tube model is more approximative because its parameters are dependent on the operating point, while in this study they have been considered constant. Instead of making the model parameters adaptive, it is useful to think of improving robustness via the disturbance model C(q−1 )/D(q−1 ). There are two reasons for choosing this approach. Firstly, accurate model parameters are not easily obtained. It requires a lot of effort to derive, test and implement adaptive parameters. Secondly, model errors can never be fully excluded. There will always be external factors or combinations of factors that the model has not taken into account.
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Fig. 7 shows a setpoint change within the design region of the PI-controller. The system reacts slowly. Outside the design region of the PI-controller (Fig. 8) the control loop loses a big deal of stability, and the settling time is unacceptable. 33.5 33 32.5
out
[°C]
32
31 30.5
1 C(q−1 ) = D(q−1 ) (1 − q−1 )(1 − a · q−1 )
(9)
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For a = 0, the default disturbance model is found, hence only the DC-frequency is filtered and the steady state error is eliminated. For a = 1 the model becomes a double integrator, assuming a ramp disturbance pattern. With the value a = 0.4 the best results were obtained. An experiment performed in the upper part of the temperature range, where the tube model is expected to lose accuracy, shows that the overshoot is much smaller with the improved disturbance model (Fig. 6).
Fig. 7.
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Genuine PI-control: process operation within the design range.
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T
out
[°C]
Setpoint Default noise model Improved noise model
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T
A spectral analysis of the disturbance signal n(t) shows that it is mainly concentrated at very low frequencies, close to the DC-frequency. That is why the default disturbance 1 , filtering only the DC component, gives model 1 − q−1 already good results. However, the response can be further improved if the low frequency component of the disturbance is taken into account in the filter. In order to tune the disturbance model, it is considered that the disturbance is not concentrated at one frequency in particular, but rather spread over a 0 − 0.02 Hz band. This is expressed by the parameter a, which is an additional tuning parameter of the filter [20], [21]:
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Fig. 8.
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Fig. 6.
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Influence of the improved noise model.
B. Classic control techniques As a reference for the performance of the nonlinear EPSAC controller, two classic control techniques were also implemented and tested on the plant: genuine PI-controller and a Smith Predictor with PI-controller. In [22] several tuning methods for PI-controllers used to control processes with time delay were presented. These methods are mainly used when no model of the process is on hand. As the development of the nonlinear EPSAC controller relies on the process model, we make use of the identified process model to tune the PI-controllers. The nonlinear process model has been linearized around the operating point q = 0.0167 l/sec and a worst case time delay of 140 sec has been considered. The PI-controllers are tuned using a CAD-tool based on system frequency response [23]. In both cases (PI and Smith PI) experiments have been performed within and outside the design region.
Genuine PI-control: process operation outside the design range.
In the Smith Predictor approach the PI-controller is designed for the linearized model without delay, while the time delay is estimated using (6). Fig. 9 depicts the step changes around the operating point of 0.0167 l/sec. Steady state levels are reached in a quite short time, comparable to nonlinear EPSAC controller. Outside of the design region however (Fig. 10), a clear overshoot is visible and the settling time increases significantly. The robustness of the Smith predictor PI controller could be improved using a robustness filter as described in [9]. VI. C ONCLUSIONS This paper evaluates the benefits of nonlinear predictive control on outlet temperature control of a heated tank system with variable time delay. Using physical and experimental modelling techniques, accurate and simple models of the tank, tube dynamics and variable time delay are developed. The availability of a model for the variable time delay of the system allows to change the standard predictive control loop layout to a Smith predictor-like structure. The advantage of this transformation lies in the fact that the complex tuning procedure of the variable controller parameters reduces to
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R EFERENCES
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out
[°C]
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Fig. 9. Smith predictor with PI-controller: process operation within the design range.
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Tout [°C]
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Fig. 10. Smith predictor with PI-controller: process operation outside the design range.
tuning a controller with fixed parameters, as it is the case for systems without delay. The nonlinear EPSAC strategy greatly simplifies the optimization procedure. The nonlinear optimization is replaced with iterative quadratic programming, reducing the computation time compared to a pure nonlinear MPC approach. Nonlinear predictive control ensures superior performance of the closed loop over classic control techniques. One of the reasons is the use of the nonlinear process model, while the genuine PI-controller and the PI-controller in the Smith predictor approach are designed based on a linearized model. That is why, when operated outside the design region, both classic control strategies fail in ensuring a good response of the closed loop. The second reason concerns the time delay. While the predictive control and Smith predictor approach benefit from a very accurate estimation of the variable time delay, the genuine PI-controller must be designed using an overestimation of the time delay to compensate for its variability. Systems with big time delay cannot be made to react fast because they lose stability. That is why the genuine PI-controller loop provides the worst result, even when the process is operated around the linearization equilibrium point.
[1] J.P. Richard, Time-delay systems: an overview of some recent advances and open problems, Automatica 39 (2003) 1667–1694. [2] J.Normey-Rico, E.Camacho, Control of Dead-time processes, Springer, Berlin, (2007) [3] O.J.M. Smith, Closer control of loops with dead time, Chemical Engineering Progress 53 (1957) 217–219. [4] Z.J. Palmor, Time-delay compensation - Smith predictor and its modifications, in: W.S. Levine (Eds.), The control handbook, Boca Raton, 1996, pp.224–237. [5] J.E. Normey-Rico, C. Bordons, E.F. Camacho, Improving the robustness of dead-time compensating PI controllers, Control Eng. Practice 5 (1997) 801–810. [6] A. Ingimundarson, T. H¨agglund, Robust tuning procedures of deadtime compensating controllers, Control Eng. Practice 9 (2001) 1195– 1208. [7] Y.C. Tian, F. Gao, Compensation of dominant and variable delay in process systems, Ind. Eng. Chem. Res. 37 (1998) 982–986. [8] C. Nunez-Reyes, J. Normey-Rico, C. Bordons, and E. Camacho, A Smith predictive based MPC in a solar air conditioning plant, Journal of Process Control, 15 (2005) 1–10. [9] J. Normey-Rico, C. Bordons, M. Berenguel, and E. Camacho, A robust adaptive dead-time compensator with application to a solar collector field, in: Proceedings of the IFAC-Workshop on Linear Time Delay Systems (LDTS’98), Grenoble, France (1998), 105–110. [10] N. Petit, Y. Creff, P. Rouchon, Motion planning for two classes of nonlinear systems with delays depending on the control, in: Proceedings of the 37th IEEE-Conference on Decision and Control, Tampa, FL, USA, (1998), 1007–1011. [11] S.I. Biagiola, J.A. Solsona, J.L. Figueroa, Use of state estimation for inferential nonlinear MPC: a case study, Chemical Engineering Journal 106 (2005) 13–24. [12] M. Mahfouf, D.A. Linkens, Nonlinear generalized predictive control (NLGPC) applied to muscle relaxant anaesthesia, International Journal of control 71 (1998) 239–257. [13] R. Khaledi, B.R. Young, Modeling and model predictive control of composition and conversion in an ETBE reactive distillation column, Ind. Eng. Chem. Res. 44 (2005) 3134–3145. [14] R. De Keyser, A ‘Gent’le approach to predictive control, in: UNESCO Encyclopaedia of Life Support Systems (EoLSS), Eolss Publishers Co. Ltd., Oxford, UK, (2003). [15] R. De Keyser, J. Donald III, Application of the NEPSAC Nonlinear Predictive Control Strategy to a Semiconductor Reactor, in Springer Lecture Notes in Control and Information Sciences Series, Allgower ¨ F., Findeisen R. and Biegler L. (Eds.), Springer-Verlag, (LNCIS 358), (2006) [16] M. Galvez, R. De Keyser, C. Ionescu, Application of a Smith Predictor based Nonlinear Predictive Controller to a Solar Power Plant, in Proc. of the 7th IFAC Symposium on Nonlinear Control Systems, Pretoria, South Africa, (2007), 188–193. [17] B. Claure-Torrico, R. De Keyser, C. Ionescu,J. Normey-Rico, Predictive control with robust dead-time compensation: Application to drug dosing during anesthesia, in Proc. of the 7th IFAC Symposium on Nonlinear Control Systems, Pretoria, South Africa, (2007), 176–181. [18] S. Cristea, C. de Prada, R. De Keyser, Predictive control of a process with variable dead-time, in: Proceedings of the 16th IFAC World Congress, Prague, Czeck Republic, 2005. [19] S. Himpe, V. Theunynck, Design and Advanced Control of a Process with Variable Time Delay, Master thesis Ghent University, Department of Electrical Energy, Systems and Automation, 2006. [20] R. De Keyser, C. Ionescu, The disturbance model in model based predictive control, in: Proceedings of IEEE Conf on Control Applications (CCA2003), Istanbul, Turkey, 2003, CD paper CF-001472. [21] R.B. Gopaluni, R.S. Patwardhan, S.L. Shah, MPC relevant identification – tuning the noise model, J. Proc. Control 14 (2004) 699–714. [22] J. Barraud, Y. Creff, N. Petit, PI controllers performances for a process model with varying delay, in: Proceedings of UKACC Control Conference, Bath, UK, 2004. [23] R. De Keyser, C. Ionescu, FRtool: a frequency response tool for CACSD in MatLab, in Proceedings of IEEE Conf on Computer Aided Control Systems Design (CACSD-CCA-ISIC), Munchen, Germany, 2006, 2276–2280.
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