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Robust controller design for uncertain systems with variable time delay M. Garc!ıa-Sanz*, J.C. Guille! n, J.J. Ibarrola Automatic Control and Computer Science Department, Public University of Navarre, Campus de Arrosadia, 31006 Pamplona, Spain Received 26 November 1999; accepted 8 September 2000

Abstract A Smith predictor controller (SPC) is designed for an uncertain pasteurisation process exhibiting variable time delay. As the SPC may be very sensitive to model-plant mismatch, especial attention must be paid to ensure robustness of the design when the model of the plant is not precisely known. This paper introduces two criteria for such cases. The ﬁrst criterion is based on bandwidth considerations and the second one introduces some guidelines to improve the design by using the quantitative feedback theory technique. An experimental veriﬁcation of the proposed methodology on the pasteurisation plant is also presented. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Robust control; Predictive control; Time delay; Uncertain dynamic systems; Biotechnology

1. Introduction Pasteurisation is a very common process in the food industry. It consists of a heat treatment applied to some food products (milk and milk-based products, fruit juices, eggs, etc.) by exposing them to a certain minimum temperature for a certain minimum time. The aim of this process is to reduce pathogenic microorganisms so public health hazards are minimised. The temperature/time combination that will suﬃciently reduce a given bacteria population varies according to the nature and conditions (viscosity, percentage of fat, solid contents, etc.) of the product to be pasteurised. Due to the required pasteurisation time, the dynamic behaviour of this process contains an inherent time delay. Design of controllers for systems with a dominant time delay is notoriously diﬃcult. The presence of such time delay greatly increases the diﬃculty of achieving satisfactory performance of feedback controllers, in particular for systems with large time delay with respect to the plant dynamics. For these systems, closed-loop performance can be substantially improved by introducing dead-time compensation. In the last few decades many research eﬀorts have focused that particular *Corresponding author. Tel.: +34-948-169387; fax: +34-948168924. E-mail address: [email protected] (M. Garcı´ a-Sanz).

problem, especially in process control. For open-loop stable processes, the most popular scheme may be the Smith predictor controller (SPC), which was early introduced by Smith (1957). The SPC contains a model of the process with time delay in an inner loop, and can be easily implemented by using low-cost digital microcontrollers. Indeed, the SPC is available as a standard algorithm in many commercial devices. This wide availability has promoted the SPC even when its especial structure, as advised by Horowitz (1983), could become inconvenient for securing the maximum feedback. Although ﬁrst suggested for single-input–single-output (SISO) systems, the SPC has been extended to multivariable systems (Alevisakis & Seborg, 1973; Ogunnaike & Ray, 1979; Bhaya & Desoer, 1985). Two major issues have been raised concerning the eﬀectiveness of the SPC. Firstly, it turns out that disturbance attenuation is not as good as are the improvements for tracking. Many modiﬁcations and extensions have been proposed on this topic (Watanabe & Ito, 1981; Palmor & Powers, 1985; among others). It has been shown (Morari & Zaﬁriou, 1989) that this issue can be addressed by the two-degree-of-freedom structure (Horowitz, 1963), where a basic controller is tuned for disturbance rejection and a preﬁlter is then chosen for good setpoint tracking. Another issue is the sensitivity of the SPC to modelling error. Generally speaking, any mathematical model of a real process is

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based on a set of parameters qi which are usually estimated between some lower and upper bounds q i and qþ , due to the uncertainty that aﬀects the description of i the plant, þ ð1Þ P ¼ PðqÞjqi 2 q i ; qi ; i ¼ 1; 2; 3; . . . ; l : In this context, it is well known that the Smith predictor technique may be very sensitive to model-plant mismatch, either in the time delay or in the rational part of the model. Under these circumstances large stability margins of the nominal plant may not guarantee stability even with small modelling errors (Ioannides, Rogers, & Latham, 1979; Palmor, 1980; Horowitz, 1983; Yamanaka & Shimemura, 1987). As pointed out by Morari and Zaﬁriou (1989), the SPC does not really increase the sensitivity to modelling error. Instead, problems arise when the controller is to be adjusted without attention to the limitations imposed by model uncertainty. Unfortunately, the eﬀect of the level of model uncertainty on controller tuning is not clear. So, it is particularly important for the use of a design technique which addresses uncertainty up front. An increasingly used engineering technique for robust control design is the quantitative feedback theory (QFT). It is a frequency domain methodology that tries to achieve robust stability and robust performance over a speciﬁed region of model uncertainty. The QFT approach (Horowitz, 1991; Houpis & Rasmussen, 1999) consists of converting plant uncertainties (represented as plant ‘templates’) and design speciﬁcations into ‘bounds’ on the open-loop transmission function. Additionally, QFT is based on the two-degree-of-freedom structure mentioned above, enabling the designer to address the regulation/tracking dilemma. Some key aspects of SPC design are unveiled when viewed from the QFT perspective, the main of these aspects is the primary role played by the particular model selected for the SPC. Chen (1984) and Laughlin, Rivera, and Morari (1987) indicated that the nominal model with all the parameters at their mean values, the usual choice rather arbitrarily adopted in the literature, does not always lead to an optimal design. In their work, nominal model parameters and controller parameters are combined within an optimisation procedure. However, initial values for the parameters must be provided, which could bias the search. This paper presents two criteria for the design of a SPC when the plant}rational part and time delay}is not precisely known. The ﬁrst criterion is based on frequency bandwidth considerations and the second one introduces some guidelines to improve the design of the SPC (basically, the selection of its nominal model) by using the QFT technique. A two-step methodology of design is then proposed, based on the above two criteria. This work revises and expands previous work presented by Garc!ıa-Sanz & Guill!en (1998). Section 2 presents the

model of a real pasteurisation plant, which motivates subsequent development. Section 3 brieﬂy summarises the basic, well-known properties of the SPC. The templates the QFT procedure uses to start from are analysed in Section 4. In Section 5, the new methodology of design is proposed. A simulated example clariﬁes it in Section 6 and in Section 7 it will be applied to the previously modelled pasteuriser. Finally, some concluding remarks are presented in Section 8.

2. Model of the pasteurisation plant The plant PCT23, manufactured by Armﬁeld (UK), is a miniature version (1.2 m 0.6 m 0.6 m) of a real industrial high temperature short time (HTST) pasteurisation process. It consists of a bench mounted process unit, connected to a dedicated control console in order to provide access to the various signals associated with measurements and control. A diagram of the plant is shown in Fig. 1. Raw product is stored in a tank, from where it is pumped to a heat exchanger of countercurrent parallel ﬂows. The product is heated to the pasteurisation temperature using a hot water ﬂow coming from a closed circuit with a heater. The product is kept at the pasteurisation temperature for a few seconds (holding time), while it crosses through a thermally insulated tube. A complete model of the HTST pasteurisation plant can be found in Ibarrola, Guill!en, Sandoval, and Garc!ıa-Sanz (1998). The following considerations clarify the control problem faced in this paper: *

*

Raw product temperature Tr is assumed constant (about 218C). Pasteurisation temperature Tp is the control variable. Temperature requirements strongly depend upon

Fig. 1. HTST pasteurisation plant.

M. Garcı´a-Sanz et al. / Control Engineering Practice 9 (2001) 961–972

*

*

*

product conditions. On the other hand, the pasteuriser parameters were shown to be nearly temperature independent (Ibarrola et al., 1998). With this in mind, pasteurisation temperatures about 408C will be demanded herein. While this may seem rather low for pasteurisation, it allows experiments to be speeded up without compromising reliability. Hot water temperature Tw is held constant at 498C. This is achieved by means of an autonomous DMC controller which manipulates power supply to the heater (Ibarrola, 1998). As disturbance input, product-ﬂuid ﬂow Fp is allowed to vary within 180 and 400 ml/min. This is a source of variable time delay, which can be computed dividing the volume of the holding tube (82 cm3) by the product ﬂow. Hot-ﬂuid ﬂow Fh is the manipulated variable, which is in the range 100–500 ml/min.

With a sampling time T of 4 s, the model of the plant has been identiﬁed as Tp ðz1 Þ Kzd ¼ ; Fh ðz1 Þ ð1 0:8825 z1 Þð1 0:8249 z1 Þ K 2 ½0:056; 0:118 ; d 2 ½3; 7 ;

ð2Þ

where the non-linearities of the process (from varying ﬂows) are incorporated through parameter uncertainty. The aim of this paper is to further investigate SPC design so as to ensure the temperature at the output of the holding tube is maintained within a speciﬁed range around setpoint value, despite disturbances and model uncertainty. The design process will be carried out in the QFT framework.

963

eﬀect of the control variable on the output and modiﬁes the feedback signal accordingly. A simple analysis of that structure shows that, if y# ¼ y and G# ðsÞ ¼ GðsÞ, the time delay is eliminated from the characteristic equation. Thus the controller CðsÞ can be synthesised without considering delay. This allows the designer to improve the closed-loop performance with respect to that achievable by a feedback regulator directly designed for the original plant. In this manner, the time response of the closed-loop system with a SPC will thus have the same shape as the response of the closed-loop system without the time delay; the only diﬀerence is that the output will be delayed by y s. ( stro. m (1977) that the Indeed, it has been shown by A Smith predictor structure provides a signiﬁcant phase lead, which justiﬁes the above statement. Many other simulation and experimental studies (see, for example, Palmor & Powers, 1985, and references therein) have demonstrated the potential improvement of the SPC over conventional controllers. The actual plant is however never known exactly, nor is it ﬁxed. In this context, Palmor (1980) and Marlin (1995) introduce some rules to achieve practical and asymptotical stability and zero steady state oﬀset when model-plant mismatch is present. An alternative approach makes use of QFT. Let the plant under consideration be described by an uncertain frequency response function: P ¼ Pðs; qÞq 2 O Rl ; ð3Þ where Pðs; qÞ ¼ GðsÞesy ¼

am sm þ am1 sm1 þ þ a0 sy e ð4Þ sn þ bn1 sn1 þ þ b0

3. Structure and basic properties of the Smith predictor

and where the coeﬃcients belong to intervals þ am 2 a y 2 y ; yþ ; m ; am ; etc:

The basic control structure of the two-degree-offreedom SPC is the model-based approach shown in Fig. 2, where CðsÞ is a conventional controller, FðsÞ is the preﬁlter and GðsÞ exp ðsyÞ represents the actual plant. A model of the open-loop process G# ðsÞ exp ðsy# Þ is used in an additional feedback loop in order to obtain an open-loop feedback signal that carries current and not delayed information. This minor loop predicts the

A grid on the parameter space can be deﬁned, leading to the approximation P ¼ Pj j j ¼ 1; . . . ; N . Thus, a discrete set of plants is obtained, where Pj can be thought of as a possible plant (and, therefore, can be selected as the nominal model of the actual plant). A simple rearrangement of the SPC diagram in Fig. 2 leaves the equivalent structure shown in Fig. 3. This structure is suitable for applying the QFT technique to the equivalent plant Geq ðsÞ. The expressions of the

Fig. 2. Two-degree-of-freedom Smith predictor controller structure.

Fig. 3. Equivalent diagram of the Smith predictor controller.

ð5Þ

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blocks are shown in Eqs. (6)–(8): G# ðsÞ sy# þ esy ; HðsÞ ¼ 1 e GðsÞ QðsÞ ¼

esy ; HðsÞ

# Geq ðsÞ ¼ GðsÞ HðsÞ ¼ 1 esy G# ðsÞ þ GðsÞ esy :

As shown in Fig. 4, there exists a complex value Dj ðs0 Þ such that ð6Þ

Gj ðs0 Þ es0 yj ¼ G# ðs0 Þ es0 y þ Dj ðs0 Þ:

ð7Þ

The equivalent plant for that speciﬁc Pj , from (8) and (11), is given by # Geq j ðs0 Þ ¼ 1 es0 y G# ðs0 Þ þ Gj ðs0 Þ es0 yj

ð8Þ

In QFT, the actual plant P will be represented by templates at certain frequencies of interest. Since the equivalent plant (8) is a function of the actual plant, it will also exhibit parameter uncertainty, leading to equivalent-plant templates. It is worth comparing them with the actual-plant templates. This is done in the next section.

4. Template analysis QFT templates are ‘geometric’ representations of the magnitude and phase uncertainty of the open loop transmission function LðsÞ in the Nichols chart. Since the compensator CðsÞ is accurate, the signiﬁcant uncertainty in LðsÞ is caused by the process. The process to be controlled is Geq ðsÞ in Eq. (8). The open loop transmission function is given by LðsÞ ¼ CðsÞGeq ðsÞ:

ð9Þ

In order to compare a actual-plant template with the corresponding equivalent-plant template at a given complex frequency, s0 , consider one out of the possible plants Pj within the uncertain plant set. To simplify the notation in (4), let Pj be written as Pj ðsÞ ¼ Gj ðsÞ esyj 2 P:

ð10Þ

#

¼ G# ðs0 Þ þ Dj ðs0 Þ:

ð11Þ

ð12Þ

Eq. (12) holds for any Pj in P, so the equivalent-plant template at s0 can be obtained by applying (12) to the discrete set of plants within the original template. Fig. 4 illustrates this procedure. Some remarks are in order here. Remark 1. While the original templates are located over the G# ðsÞ exp ðsy# Þ curve, the corresponding equivalentplant templates are over the G# ðsÞ curve. This can be seen as the nominal delay being removed from the plant. Of course, this was the primary motivation of the SPC. Remark 2. For a speciﬁc frequency, the original template translates without rotation to a new position on the complex plane to give the equivalent-plant template. In other words, the same set of vectors Dj j j ¼ 1; . . . ; N generates both templates from two diﬀerent origins. This translation leaves the shape of the template unaltered when plotted on the complex plane (Fig. 4), but signiﬁcantly impacts the shape on the Nichols Chart, because # jG# ðsÞ esy þ Dj ðsÞj 6¼ jG# ðsÞ þ Dj ðsÞj:

ð13Þ

Notice also that diﬀerent nominal models will give rise to diﬀerent equivalent-plant templates through (13). Therefore, the selection of the nominal model is expected to greatly aﬀect the bounds computation, and hence the loop-shaping stage of the QFT procedure. (The term ‘nominal model’ here is not to be confused with the concept of nominal plant for QFT bounds computation. That term will be always used in the SPC sense, unless otherwise stated.) The following example illustrates the template analysis. Example 4.1. Consider the plant K esy ; Plant: GðsÞ esy ¼ ð14Þ ts þ 1 which captures the essential dynamics of many chemical, biological and industrial processes. In addition, assume appropriate ranges for the uncertain parameters are identiﬁed as K 2 ½1; 2 ;

Fig. 4. Relation between actual-plant and equivalent-plant templates in the complex plane.

t 2 ½1; 2 ;

y 2 ½1; 2 :

ð15Þ

The nominal model of the plant is represented as K# # # Model of the plant: G# ðsÞ esy ¼ esy : ð16Þ t# s þ 1

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Now, three speciﬁc models within the space of uncertainty are selected (the selection of the ﬁrst two will be made clear in Section 6.2; the third is the ‘mean’ model): Case A: ðK# ¼ 1:0; t# ¼ 2:0; y# ¼ 1:9284Þ; Case B: ðK# ¼ 1; t# ¼ 1:7284; y# ¼ 1:0Þ; Case C: ðK# ¼ 1:5; t# ¼ 1:5; y# ¼ 1:5Þ:

ð17Þ

Fig. 5 shows the contour of the template of the original ﬁrst order plus time delay plant for a frequency of o=1 rad/s, and the three points from (17) above. Fig. 6 shows the shape variation of the templates of Geq ð j1Þ when the models A–C are used as nominals in the Smith predictor structure. Notice that the shape of the templates strongly depends on the selected nominal model of the SPC.

965

5. Methodology of design A previous analysis of the equivalent SPC structure showed that, if there is no uncertainty in the model, that is to say HðsÞ ¼ 1, the time delay is eliminated from y * ðsÞ. However, if there is any model-plant mismatch, then HðsÞ will be diﬀerent from one. As a consequence, the control system behaviour will be aﬀected by the uncertainty in HðsÞ, through the blocks Geq ðsÞ and QðsÞ in Fig. 3. In this case, the selection of the nominal model, G# ðsÞ exp ðsy# Þ, has an impact on the set of values that HðsÞ adopts when the parameters of the model vary over the space of uncertainty. For this reason, it is necessary to analyse how to select the nominal model of the SPC, and how this choice can aﬀect the control system behaviour. The methodology of design proposed in this paper divides the study in two complementary steps. The QJ;ﬁrst one analyses the inﬂuence of the nominal model selection on the block QðsÞ. The second step studies how that selection aﬀects the design of the controller CðsÞ for the system Geq ðsÞ, by using the QFT technique. 5.1. First step

Fig. 5. Actual-plant template (o=1 rad/s) for a ﬁrst order plus time delay model.

The HðsÞ term that appears in the QðsÞ block, out of the control loop (see Fig. 3), may be responsible for the deterioration of system performance if resonant peaks appear at frequencies below the system bandwidth. In that case, y * ðsÞ will be distorted when passing through the block. Some years ago, a brief paper written by Santacesaria and Scattolini (1993) introduced this problem and proposed a graphical solution for the simple case of model-plant mismatch in the time delay, y# 6¼ y;

G# ðsÞ ¼ GðsÞ:

ð18Þ

Following these ideas, an extension is proposed for the case of model-plant mismatch in both time delay and rational part, y# 6¼ y;

Fig. 6. Equivalent-plant templates (o=1 rad/s) when the nominal model of the Smith predictor is A, B or C.

G# ðsÞ 6¼ GðsÞ:

ð19Þ

The study is based on the analysis of the magnitude of QðjoÞ over the frequency range of interest and for the whole space of parameter uncertainty. Any increase in the model-plant mismatch moves the resonant peaks to lower frequencies, thus limiting achievable closed-loop bandwidth. For a speciﬁed bandwidth BW, this can be stated as follows. Criterion 1: A nominal model must be selected such that the resulting QðjoÞ does not distort y * ðsÞ for frequencies up to BW and for every possible plant in P. For example, the nominal plant must satisfy 20 log jQðjoÞj43 dB 04o4BW 8Pj 2 P: ð20Þ 10

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Table 1 Outline of the First Step computer program

Table 2 Outline of the Second Step computer program

Item

Operation

Item

Operation

1

Deﬁne a grid over P (Houpis & Rasmussen, 1999). This allows the algorithms to treat with a ﬁnite set of possible plants. Fix a desired closed-loop bandwidth BW. Select a plant Pj 2 P as the nominal model for the SPC. Compute, for the selected model, the magnitude of QðjoÞ for every possible plant in P. If any |QðjoÞj from Item 4 exhibits some ampliﬁcation or attenuation about 3 dB at frequencies lower than the desired bandwidth BW, then a deterioration of the system characteristics will be expected. Therefore, the model selected in Item 3 must be rejected as nominal model. Otherwise, the selected model could be adopted. Repeat from Items 3–5 for every possible model in P. Finally, a set of admissible models of the plant is obtained.

1

Select a plant Pj as the nominal plant of the SPC from those that have successfully passed First Step criterion. Compute the equivalent-plant templates over the frequency range of interest. Calculate the area of the templates, and then the cost function (22). Repeat from Items 1 to 3 for every possible model from those that have successfully passed First Step criterion. Select the model that results in the minimum cost.

2 3 4 5

6 7

From (7) and taking into account that log10 j1=Hj= log10 jHj, (20) is equivalent to 20 log jHðjoÞj43 dB 04o4BW 8Pj 2 P: ð21Þ 10 The computer program that implements the First Step out is outlined in Table 1. It is a general procedure that can be used for any model of the plant, with any sort of rational part and any time delay, and for any kind of model-plant mismatch. The procedure ﬁnds the subset of plant models that could be used by the SPC without the QðsÞ block causing too much distortion of the output. 5.2. Second step If there is a collection of possible models of the plant satisfying Eq. (20), then an additional degree of freedom is still available in the selection of one for the SPC. The idea is to use the amount of change suﬀered by the equivalent-plant templates as a second criterion to guide this selection. The aim is to improve the design of the controller, CðsÞ, by using the QFT methodology. As seen above, two equivalent-plant templates calculated for the same frequency and corresponding to two diﬀerent nominal models could diﬀer in shape, producing diﬀerent loop-shaping stages and hence diﬀerent controllers. This fact introduces the question about which is the best model of the plant that has to be chosen, from among those satisfying First Step, in order to obtain the least demanding templates and ease the loop-shaping of CðsÞ. Initial investigations and many experimental simulations suggest that the smaller the area of the templates, the easier the controller design becomes. Let TðoÞ represent the actual-plant templates; Tj ðoÞ the equivalent-plant templates when plant Pj has been selected as nominal in the SPC; Að Þ the area of a

2 3 4 5

template on the Nichols chart and O the (discrete) set of frequencies of interest, with no frequencies. The following cost function is proposed as a measure of suitability of a speciﬁc nominal model: 1 X wðoÞAðTj ðoÞÞ ; ð22Þ IðPj Þ ¼ no o2O AðTðoÞÞ where wðoÞ are weights that can be used to emphasise critical frequencies. This cost function is a weighted sum of normalised areas. A nominal plant model leaving invariant every actual-plant template area would be assigned a cost IðPj Þ=1 if unity weights were used. Criterion 2: From the possible nominal plants obtained by Criterion 1, select the one that presents the minimum cost (22). The computer program that implements the Second Step out is outlined in Table 2. Again, it is a general procedure that can be used for any model of the plant, with any sort of rational part and any time delay, and for any kind of model-plant mismatch. The model of the plant selected for the SPC structure with the proposed methodology will avoid distortion within the operating bandwidth (First Step) and will present the least restrictive templates to the controller design stage (Second Step). Finally, though the development has been made for continuous systems, the obtained results remain valid for sampled-data systems incorporating digital SPCs.

6. A synthesis example To clarify the above ideas consider again the plant (14)–(16) in Example 4.1. 6.1. Applying ﬁrst step As it was mentioned above, the First Step of the methodology is based on the analysis of the magnitude of QðjoÞ over the frequency range of interest and for the complete set of parameter uncertainty. For the proposed plant and model, the magnitude of QðjoÞ can be

M. Garcı´a-Sanz et al. / Control Engineering Practice 9 (2001) 961–972

written as 1 jQðjoÞj ¼ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; ð1 þ MxÞ2 þðMyÞ2 where # K 1 M¼ ; K 1 þ o2 t# 2 h i x ¼ ð1 þ o2 t#tÞ cos oy cos oðy# yÞ h i o ðt t# Þ sin oðy# yÞ þ sin oy ; h i y ¼ ð1 þ o2 t#tÞ sin oðy# yÞ þ sin oy h i þ o ðt t# Þ cos oy cos oðy# yÞ :

ð23Þ

ð24Þ

ð25Þ

967

Second Step (Table 2) for these admissible models, and using wðoÞ ¼ 1 8o, a best-cost value of 0.9437 is found for case A in Eq. (17). The worst case, given here for comparison purposes, corresponds to case B in (17) with a cost of 1.0740. The ‘mean’ model (case C) presents a cost of 0.9964. In order to illustrate the second criterion, both A and B models are selected for two alternative designs using the QFT technique. For simplicity, no preﬁlter will be used here. Eqs. (27)–(30) state the desired speciﬁcations (robust stability, bandwidth and tracking) for the control system: Cð joÞ½Gð joÞHð joÞ 1 þ Cð joÞ½Gð joÞHð joÞ 41:15

ð26Þ

Now, ﬁxing the desired closed-loop bandwidth to BW=0.5 rad/s (Item 2 of Table 1), the set of admissible models which fulﬁl the bandwidth speciﬁcation can be easily found following Items 3–7 of the First Step procedure. The result for this example can be plotted as a 3D object, where each of the three axes represents the uncertainty in a model parameter (Fig. 7). The models the SPC could adopt are those located inside the 3D ﬁgure. To obtain Fig. 7, a grid of 20 values for each parameter was used. Figs. 8 and 9 show the 3D object obtained for bandwidth speciﬁcations of 0.55 and 0.60 rad/s, respectively. As might have been expected, the larger the bandwidth speciﬁcation the smaller the set of possible models. As illustrated by Figs. 7–9, the usual selection of a ‘mean’ nominal model does not seem to be appropriate when using Criterion 1.

for o ¼ f0:01; 0:1; 0:3; 0:5; 1:0g rad=s;

ð27Þ

Cð joÞ½Gð joÞHð joÞ 1 þ Cð joÞ½Gð joÞHð joÞ 40:707 for o ¼ 0:5 rad=s;

ð28Þ

6.2. Applying second step Given the speciﬁcation BW=0.5 rad/s, there is an additional degree of freedom still available to select a model within the 3D object shown in Fig. 7. Applying

Fig. 8. Admissible models after First Step, for BW=0.55 rad/s.

Fig. 7. Admissible models after First Step, for BW=0.50 rad/s.

Fig. 9. Admissible models after First Step, for BW=0.60 rad/s.

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Cð joÞ½Gð joÞHð joÞ 4bð joÞ að joÞ4 1 þ Cð joÞ½Gð joÞHð joÞ for o ¼ f0:01; 0:1; 0:3; 0:5g rad=s;

ð29Þ

where

0:08 ; að joÞ ¼ 3 2 ð joÞ þ 2:4ð joÞ þ 0:84ð joÞ þ 0:08 0:0040ð joÞ2 þ 22ð joÞ þ 40 bð joÞ ¼ : ð joÞ2 þ 0:36ð joÞ þ 0:16

Chait, & Yaniv, 1995). In order to make the comparison easier, a very simple controller structure, k=s (a gain with an integrator), is selected for both cases. Fig. 10 shows the loop shaping of LðsÞ when the best model is selected for the SPC structure. The controller for that model is given by CðsÞ ¼

ð30Þ

ð31Þ

In addition, Fig. 11 shows the loop shaping of LðsÞ when the worst-case model is selected. The controller for that model is given by CðsÞ ¼

Following the QFT procedure, after determining bounds on the loop transmission, LðsÞ ¼ CðsÞGeq ðsÞ, for each case, it is shaped on the Nichols chart using a CAD package (Garc!ıa-Sanz & Vital, 1999; Borghesani,

0:1247 : s

0:1225 : s

ð32Þ

Looking at the loop-shape obtained by the controllers (31) and (32), it can be concluded that the ﬁrst one satisﬁes the bound constraints, while the second one

Fig. 10. Loop-shaping for case A.

Fig. 11. Loop-shaping for case B.

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does not (note the location of Lðj5Þ). This shows that the selected nominal model of the plant used in the SPC not only modiﬁes the templates and the loop shaping stage, but can also aﬀect the controller structure.

7. Application to the HTST pasteurisation plant In order to validate the proposed methodology in a non-linear process with variable time delay, the new algorithms are applied to the HTST pasteurisation plant of Section 2. As usual in QFT, a preﬁlter FðsÞ and the basic controller CðsÞ within the SPC are to be designed. Fig. 12 shows the nominal plants satisfying Criterion 1 for three diﬀerent bandwidths: 0.030, 0.037 and 0.040 rad/s. Integer values for the delay and 50 values for K were used to obtain Fig. 12. Now, specifying a desired bandwidth of BW=0.037 rad/s (Fig. 12b), the best and worst models according to Criterion 2 are Case a : best model ðK# ¼ 0:0642; d# ¼ 6Þ; Case b : worst model

ðK# ¼ 0:0704; d# ¼ 3Þ;

ð33Þ

with cost values of 1.0263 and 2.3619, respectively. Fig. 13 shows the equivalent-plant templates corresponding to both cases for a frequency of 0.04 rad/s. The actual-plant template for this frequency is also depicted. Although only integer delays were considered when computing actual-plant templates, they were proved to adequately describe partial delays, following the results in de Paor and O’Malley (1995), for the frequency range of interest. It should be mentioned, however, that template computation for the general case of sampled-data systems with partial delays does not seem to be addressed yet in the literature. Closed-loop speciﬁcations of robust stability and tracking are imposed as follows: CðzÞ½GðzÞHðzÞ 1 þ CðzÞ½GðzÞHðzÞ 41:15 ð34Þ for z ¼ e joT ; o ¼ f0:001; 0:005; 0:01; 0:03; 0:05; 0:07g rad=s; FðzÞCðzÞ½GðzÞHðzÞ 4bð joÞ að joÞ4 1 þ CðzÞ½GðzÞHðzÞ ð35Þ for z ¼ e joT ;

Fig. 12. Models that satisfy requirements in First Step criterion. (a) BW=0.03 rad/s; (b) BW=0.037 rad/s; (c) BW=0.04 rad/s.

o ¼ f0:001; 0:005; 0:01; 0:03; 0:05; 0:07g rad=s; where

0:00000125 ; að joÞ ¼ 3 2 ð joÞ þ 0:06ð joÞ þ 0:000525ð joÞþ 0:00000125 0:025ð joÞ2 þ 0:55ð joÞ þ 0:025 bð joÞ ¼ : ð joÞ2 þ 0:025ð joÞ þ 0:000625

ð36Þ

A controller structure with ﬁve zeroes, ﬁve poles and an integrator is also imposed. In order to avoid human biases in the loop-shaping stage, an automatic loopshaping is carried out for both cases a and b, starting from equal initial conditions. This algorithm looks for a controller with minimum high-frequency gain, the original notion of optimality suggested by Horowitz

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Fig. 15. Loop-shaping of the pasteurisation plant. Case b.

Fig. 13. Actual-plant template and equivalent-plant templates for the cases a and b (o=0.04 rad/s).

Figs. 14 and 15 show the Nichols plots of both designs. Though all performance speciﬁcations are met for both cases, high-frequency gain is lower for the best-nominal-plant case. This is a result of the less demanding templates for this case, since the automatic loop-shaping algorithm still retains some freedom to further reduce high-frequency gain once the constraints are satisﬁed. Fig. 16a (pasteurisation temperatures) and b and c (hot-ﬂuid ﬂows) compare the result of both designs when applied to the real pasteurisation plant. The experiment was scheduled as follows. Initial setpoint was 368C, and initial product ﬂow was 6.7 103 l/s. At t=1200 s, the setpoint was changed to 388C. At t=2400 s, the product ﬂow was set to 3 103 l/s. Finally, at t=3600 s, the setpoint was raised to 408C. As expected, the ﬁrst design clearly outperforms the second one, obtaining better setpoint tracking and disturbance rejection, and featuring a less oscillatory control signal.

Fig. 14. Loop-shaping of the pasteurisation plant. Case a.

8. Conclusions (1963). The preﬁlter, instead, is manually tuned using the QFT Toolbox of MATLAB (Borghesani et al., 1995). The results are shown in Eqs. (37)–(38) and (39)– (40) for best and worst cases above, respectively:

It is well known that the (SPC) may be very sensitive to model-plant mismatch, resulting in poor performance when model uncertainty is present. This paper has introduced two criteria for the design of a SPC when the

Ca ðz1 Þ ¼

1:75ð1 0:87 z1 Þð1 0:84 z1 Þð1 0:83 z1 Þð1 0:79 z1 Þð1 0:02 z1 Þ ; ð1 z1 Þð1 0:40 z1 Þð1 0:40 z1 Þð1 0:35 z1 Þð1 0:23 z1 Þð1 þ 0:37 z1 Þ

ð37Þ

Fa ðz1 Þ ¼

0:2565 ð1 0:9625 z1 Þ ; ð1 0:9131 z1 Þð1 0:8893 z1 Þ

ð38Þ

Cb ðz1 Þ ¼

4:70ð1 0:83 z1 Þð1 0:82 z1 Þð1 0:82 z1 Þð1 0:80 z1 Þð1 0:76 z1 Þ ; ð1 z1 Þð1 0:41 z1 Þð1 0:30 z1 Þð1 0:29 z1 Þð1 0:27 z1 Þð1 þ 0:19 z1 Þ

ð39Þ

Fb ðz1 Þ ¼

0:3419ð1 0:9622 z1 Þ : ð1 0:8914 z1 Þð1 0:8812 z1 Þ

ð40Þ

M. Garcı´a-Sanz et al. / Control Engineering Practice 9 (2001) 961–972

971

Fig. 16. Response of the pasteurisation plant: (a) Tp obtained for case a (solid line) and case b (dashed line); (b) Control signal for case a; (c) Control signal for case b.

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plant-rational part and time delay}is not precisely known. The ﬁrst criterion, based on frequency bandwidth considerations, ﬁnds those models that, if used as the nominal in the SPC design, will not cause the output to be signiﬁcantly distorted by the eﬀect of model uncertainty. The second criterion introduces some guidelines to improve the design of the SPC by using the QFT technique, minimising the resulting templates in order to present the least restrictive conditions to the controller design stage. An experimental veriﬁcation of the proposed methodology in a real HTST pasteurisation plant with variable time delay has also been presented. Acknowledgements The authors gratefully appreciate the support given by the Spanish ‘Comisio! n Interministerial de Ciencia y Tecnolog!ıa’ (CICYT) under grant TAP’97-0471.

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