Gap Metric Concept and Implications for Multilinear Model-Based

Ind. Eng. Chem. Res. 2003, 42, 2189-2197

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Gap Metric Concept and Implications for Multilinear Model-Based Controller Design Omar Gala´ n,†,‡ Jose A. Romagnoli,† Ahmet Palazogˇ lu,*,§ and Yaman Arkun| Department of Chemical Engineering, The University of Sydney, Sydney, NSW, 2006 Australia, Department of Chemical Engineering and Materials Science, University of California, Davis, California 95616, and College of Engineering, Koç University, Rumelifeneri, Sarıyer, I˙ stanbul, 80910 Turkey

The gap metric concept is used within the context of multilinear model-based control. The concept of distance between dynamic systems is used as a criterion for selecting a set of models that can explain the nonlinear plant behavior in a given operating range. The case studies presented include a CSTR and a pH neutralization reactor. The gap metric is used to analyze the relationships among candidate models, resulting in a reduced model set that provides enough information to design multilinear controllers. The simulation and experimental results indicate good performance and stability features. Introduction Classical linear design tools have matured to a point where one can incorporate robustness and performance requirements in a natural fashion. However, for nonlinear processes, strictly linear designs might not provide satisfactory performance unless they are suitably modified. One approach that tries to retain the features of linear design and at the same time account for nonlinearities is the multimodel approach for controller design.1 The key concept is to represent the nonlinear system as a combination of linear systems to which classical control design techniques can be applied. Controller design based on the multimodel approach requires either simultaneous plant stabilization using a single controller, subject to performance and stability constraints,2,3 or interpolation using model validity functions, where local controllers are selected as a function of the current state of the process.4,5 However, in all of these approaches, the question of how many and which models are required remains largely unanswered. Although it is common to use a large number of local models to improve the piecewise linear approximation of the nonlinear system,6 the optimization problem of solving the design problem becomes formidable when the number of local models is large. We shall formulate the multimodel control problem by assuming a set of local plants and controllers that stabilize these plants and by asking the question: Is there a reduced set of controllers that are based on models that are “close” in some sense? Determining when two systems are close to one other is a nontrivial task, and furthermore, what is meant by close is not entirely obvious. Because systems can be visualized as input-output operators, a natural distance concept would be the induced operator norm. * To whom correspondence should be addressed. Tel.: (530) 752-8774. Fax: (530) 752-1031. E-mail: anpalazoglu@ ucdavis.edu. † The University of Sydney. ‡ Current address: ABB Australia Limited Pty, VPP9 Project, 436 Gadara Road, Tumut NSW 2720 Australia. § University of California, Davis. | Koç University.

Yet, the norm cannot be generalized as a distance measure.7 The aim of this paper is to discuss the application of a distance measure between systems, the so-called gap metric, for the selection of a reduced set of models that contain nonredundant process information for robust stabilization of feedback systems based on multimodel controller design. The next section briefly reviews the theory of the gap metric to establish the background. This is followed by two examples. The first is a continuous stirred tank reactor (CSTR), and the second is a case study that includes an experimental pH neutralization reactor. The reduced set of models obtained after the gap metric analysis provides enough information to design a multilinear controller that exhibits excellent performance and stability features. Review of the Gap Metric Theory The concept of the gap between the graphs of two linear systems goes back to Hausdorf.8 Later, the gap and other metrics were used to study how close different operators are.9,10 Zames and El-Sakkary11 used the gap metric to establish a topology for quantifying the tolerable uncertainties that preserve closed-loop stability. ElSakkary12 showed that the gap metric was better suited to measure the distance between two linear systems than a metric based on norms. In the following, we review the definition and computation of the gap metric. Let K be any linear operator in a Hilbert space, K: H f H. H × H is the set of all pairs {u, v} where u and v are in H. H × H is also a Hilbert space with the inner product derived from H. The domain of K, D(K), is the set of all u in H with the property that Ku is in H. The graph G(K) of K is the set of all pairs {u, Ku} with u in D(K). As K is linear, G(K) is a subspace of the product Hilbert space H × H. K is said to be closed if its graph G(K) is a closed subspace of H × H. Definition 1.12 The gap δ between two closed operators K1 and K2 in a Hilbert space H is defined as the gap between their graphs viewed as closed subspaces of the Hilbert space H × H, i.e.

δ(K1,K2) ) δ(G(K1),G(K2))

10.1021/ie020783s CCC: $25.00 © 2003 American Chemical Society Published on Web 04/22/2003

(1)

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Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003

Let ΠG be the orthogonal projection operator that maps any element {x, y} in H × H to {u, Pu} that is given by

ΠG ) GG*

(7)

The calculation of the gap metric begins with two finitedimensional linear systems with the same number of inputs and outputs whose normalized coprime factorizations are given by

ˆ i(s) D ˆ i-1(s) P ˆ i(s) ) N

Figure 1. Geometric interpretation of the gap metric.

The gap is also the maximum of two directed graphs

δ(G(K1),G(K2)) ) h (G(K2),G(K1))} (2) max{δ h (G(K1),G(K2)), δ where

and whose respective “graph” operators Gi are defined as above for i ) 1 and 2. It can be shown that the directed gap in eq 3 can be computed using the projection operators or the coprime factorizations13

B δ (P1,P2) ) ||(I - ΠG2)ΠG1|| ) inf

Q∈H∞

||( ) ( ) || D1 D2 N1 - N2 Q

∞

(9)

Definition 2.13 The gap between two systems P1 and P2 is given by

δ h (G(K1),G(K2)) ) sup

(8)

inf

u∈D(K1),u*0v∈D(K2)

||u - v||2 + ||K1u - K2v||2

x||u||2 + ||K1u||2

(3)

according to eq 2, and using eq 9, one obtains

and δ h (G(K2),G(K1)) is defined similarly. The geometrical interpretation of the gap metric δ(K1,K2) is sketched in Figure 1. Consider H × H as the Euclidean plane E1 × E2; the gap metric is then given by

δ(K1,K2) ) sin(θ1 - θ2) Equation 3 is a formal representation of the gap metric. However, we require a representation of the gap metric that allows for its computation. Such a representation can be achieved by showing that directed graphs can be computed using techniques from interpolation theory.13 Let P be a finite-dimensional linear system. Its transfer function will be denoted by P ˆ (s). The transfer function P ˆ (s) takes on a normalized right coprime factorization

P ˆ (s) ) N ˆ (s) D-1(s)

(4)

where N ˆ and D ˆ belong to the subspace of real rational functions in H∞, D ˆ has a proper inverse, and

D ˆ *D ˆ +N ˆ *N ˆ )I

(5)

with D ˆ *(s) ) D ˆ (-s)T. These factorizations can be computed using existing techniques.14 The graph of the operator P is the subspace of H2 × H2 (Hardy space of functions) that consists of all pairs {u, y} such that y ) Pu. This is expressed as

D H ≡ GH graph(P) ) N 2 2

( )

D is denoted by G. where the operator N

( )

δ(P1,P2) ) max{δ B(P1,P2),δ B(P2,P1)}

(6)

δ(P1,P2) )

{ ||( ) ( ) ||

max inf

Q∈H∞

D1 D2 N1 - N2 Q

, inf

∞ Q∈H∞

||( ) ( ) ||}

D2 D1 N2 - N1 Q (10)

The properties of the gap described by eq 10 are as follows: 1. The gap defines a metric on the space of (possibly unstable) linear systems. 2. 0 e δ(P1,P2) e 1 The metric defines a notion of distance in the space of (possibly unstable) linear systems, which do not assume that plants have the same number of poles in the right half-plane (RHP). The computation of the gap involves solving two-block H∞ problems (eq 10). In our computations, we used the gap command in the µ-Synthesis Toolbox of MATLAB. If the gap metric is close to 0, this indicates that the distance between the two systems is small (i.e., the two systems are close). If, on the other hand, the gap is closer to 1, then the two systems’ dynamic behaviors are “apart”. The impetus behind the derivation of this metric was to quantify the perturbations of an (possibly unstable) open-loop system that would still maintain closed-loop stability (see references cited earlier). In other words, if two systems are close (under gap) in the open loop, they would be expected to be also close in the closed loop. In the following examples, we use the gap metric to distinguish between models in a given set. We use the gap analysis to select appropriate linear models for the control of a nonlinear system by a multilinear controller. Closed loop simulations are presented. Continuous Stirred Tank Reactor We consider here the benchmark CSTR (continuous stirred tank reactor) where an irreversible, first-order

Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2191

Figure 2. Steady-state input-output map for the CSTR.

Figure 3. Gap metrics between 100 local linear model pairs.

Table 1. Distances (δi,j) between Linear Models Using Gap Metric j i

1

2

3

1 2 3

0.0000 0.9999 0.3674

0.9999 0.0000 0.9999

0.3674 0.9999 0.0000

reaction takes place. The mathematical model was taken from the paper by Uppal et al.,15 where x1, x2, and u are the dimensionless reagent conversion, the temperature (output), and the coolant temperature (input), respectively. The system exhibits output multiplicity, as can be seen from Figure 2. The following set of nonlinear differential equations describes the system under study

x˘ 1 ) -x1 + Da‚r(x1,x2)

(11)

x˘ 2 ) -x2 + B‚Da‚r(x1,x2) + β(u - x2)

(12)

y ) x2

(13)

The reaction term is expressed as

(

r(x1,x2) ) (1 - x1) exp

x2 1 + x2/γ

)

The nominal values for the constants in eqs 11-13 are Da ) 0.072, γ ) 20, B ) 8, and β ) 0.3. One can identify the different regions with differing stability properties around three steady-state points corresponding to u ) 0 (Figure 2). The following models are obtained from linearizations around the three steady-state points for the CSTR:

gˆ 1(s) ) gˆ 2(s) )

0.3s + 0.35 s2 + 1.4s + 0.46

0.3s + 0.53 s + 0.36s - 0.41

gˆ 3(s) )

2

0.3s + 1.29 s2 + 1.6s + 1.6

We note that the models 1 and 3 are stable, whereas model 2 is unstable. The numerical values of the gap metrics between pairs of these models are listed in Table 1. As mentioned above, the gap metric values δij ) 0

Figure 4. Projection of the gap metric onto the plane.

and δij ) 1 stand for “identical” and “different” systems, respectively. The diagonal entries are the comparisons of each model with itself and are therefore 0. The gap metric between a stable and an unstable model is essentially 1, indicating extreme dissimilarity. One can also observe that the gap metric between the two stable models is relatively low, implying that they can “explain” one another in a reasonable fashion. We next compute the gap metrics between 100 models evenly distributed along the input-output map in Figure 2. For instance, model 50 would correspond to the middle unstable point (2). The results are illustrated in Figures 3 and 4. Each point on the gap surface in Figure 3 represents the gap metric between two particular models in the set of 100 models. Each peak corresponds to the gap between an unstable model and a stable model. The gap values close to 1 suggest that the corresponding models are different and should both be included in the minimal set of models for control. Similarly, valleys with gap values close to 0 suggest that models are close. These valleys happen to occur either when both models are stable or when both models are unstable. The symmetry in these figures should also be noted. Figure 4 is just a contour plot of the image in Figure 3 intended to illustrate how the values of the gap metric change, for instance, across the line that corresponds to model 50. The contours indicate that the

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Figure 5. Block diagram of the local (p) feedback control system.

gap value is close to 1 at first and then decreases as model 50 is approached, after which it starts to increase again. Figure 6. Closed-loop profile using models 1-3.

Controller Design: Membership-Based H∞ Control Consider a set of N local linear models that, in combination, describe the behavior of a nonlinear system in a predefined operating range. The key issue is how one implements the multimodel control scheme with a set of local controllers that are derived using the local models. For this example, the controllers for different local operating regions are combined to form a complete control system, using membership functions to create a transition region in the measured variable y16

(14)

N

∑

φp(y)

p)1

The subscript p represents the pth member of a set of N controllers. Given the output variable y, the membership function returns a number between 0 and 1 that indicates the level of contribution of the local controller at that value of the output. We define the distribution function for a local controller as

[ ( )]

φp(y) ) exp -

1 y - yjp 2 σp

2

(15)

N

∑ φˆ pup(t)

(16)

p)1

The block diagram for the local controllers is given in Figure 5. Given a SISO plant gˆ p(s), a controller kˆ p(s) is designed such that the basic requirements of stability, performance, and robustness are satisfied.17 This can be done by finding a robust controller that minimizes the mixed-sensitivity criterion18

|

||

s2 + 0.042s + 0.002 s2 + 0.004s + 1.56 × 10-5

w ˆ e(s) S ˆ p(s) w ˆ e(s) S ˆ p(s) ˆ u(s) kˆ p(s) S ˆ u(s) kˆ p(s) S ˆ p(s) ) w ˆ p(s) J ) sup w ω ˆ p(s) ˆ p(s) w ˆ y(s) T w ˆ y(s) T

w ˆ y(s) ) 1 These weight functions imply that the control efforts are centered on performance without any constraint either on the manipulated variable or on the output, with the exception of imparting integral action. The resulting local controllers are given by

kˆ 1(s) ) 1.77 × 104s3 + 2.56 × 104s2 + 9099.942s + 329.329 s4 + 1.42 × 105s3 + 1.66 × 105s2 + 719.808s + 2.576 B k 2(s) )

where yjp and σp are the mean and the standard deviation, respectively, assigned to the model p. It should be noted that these parameters are chosen by the designer. The desired contribution of combined controllers on the control signal can be represented as a function of the membership functions

u(t) )

w ˆ e(s) )

w ˆ u(s) ) 1

φp(y)

φˆ p(y) )

Note that S ˆ p(s) and T ˆ p(s) are the local sensitivity and complementary sensitivity functions, respectively. Accordingly, w ˆ e, w ˆ u, and w ˆ y are the corresponding weight (penalty) functions chosen to shape the closed-loop performance and robustness behavior. We consider now the control design for three linear CSTR models. The weight functions are chosen as below

|

∞

(17)

1.22 × 106s3 + 1.06 × 106s2 + 3.49 × 104s + 1173.186 s4 + 1.86 × 105s3 + 3.33 × 105s2 + 1441.038s + 5.159

kˆ 3(s) ) 2.31 × 103s3 + 1.49 × 103s2 + 3753.786s + 150.170 s4 + 1.8 × 104s3 + 7.95 × 104s2 + 337.445s + 1.216 We use a set of simulations for set-point tracking to demonstrate the implications of model selection for control of this nonlinear process. Figure 6 shows the case where all three models are used as part of the control design. As one can see, the performance is quite good. If only model 1 is used to control the system (Figure 7), we also get an expected result. The performance is satisfactory where the model explains the dynamic behavior but arbitrarily poor where the model fails to capture the underlying dynamics. Figure 8 demonstrates that, by using only two models (stable model 1 and unstable model 2) instead of three, one can achieve satisfactory performance as suggested by the gap metric analysis. Hence, it becomes possible


Figure 7. Closed-loop profile using model 1 only.

Figure 9. Titration curve for the simulation and experimental runs.

Figure 8. Closed-loop profile using models 1 and 2.

to effectively identify a reduced set of models for the application of multilinear control strategies. Control of a pH Neutralization Reactor The experimental study was conducted at UC Davis using a bench-scale pH neutralization reactor.3,19 An acid stream (HCl solution) and an alkaline stream (NaOH and NaHCO3 solution) were fed into a 2.5-L constant-volume, well-mixed tank, where the pH was measured through a sensor located directly in the tank. The control objective was to drive the system to different pH conditions (tracking control) by manipulating the alkaline stream flow rate. The control strategy was cascaded on linear PI flow controllers. The experimental titration curve is depicted in Figure 9. The model-plant mismatch is notable at high pH values, a result that we attribute to the presence of unidentified ions in the water used to prepare the acid and base solutions, resulting in unknown buffering characteristics. The model of the process for pH neutralization was taken from the paper by Galań et al.3 Using that model, the following transfer function models were derived for five distinct operating regions in the steady-state map (Figure 9)

G ˆ p(s) )

Kp τ ps + 1

(18)

Table 2 shows the gap metric between the pairs of five linear models representing the whole operating range for the nonlinear system (see Figure 9). The values in the table suggest that model subsets ΩHS )

Figure 10. Models used in the gap computation. Table 2. Distances (δi,j) between Linear Models Using Gap Metric j i

1

2

3

4

5

1 2 3 4 5

0.0000 0.9331 0.2561 0.8050 0.2619

0.9331 0.0000 0.8885 0.5137 0.9515

0.2561 0.8885 0.0000 0.6892 0.4183

0.8050 0.5137 0.6892 0.0000 0.8567

0.2619 0.9515 0.4183 0.8567 0.0000

{2, 4} and ΩLS ) {1, 3, 5} are different (higher gap values), whereas the models in each subset are “similar” (lower gap values). Model 2 is considered different from models 1, 3, and 5, as the gap metrics between model 2 and these models are around 1, but close to model 4. This similarity between models 1, 3, and 5 can be explained physically by the fact that these models represent low-sensitivity regions (ΩLS), that is, regions with low slope (Figure 9). In contrast, models 2 and 4 represent the high-sensitivity (steep-slope) regions (ΩHS). As in the CSTR example, the gap computation again was carried out for 100 models evenly distributed along the input-output map (see Figure 10). For instance, model 40 corresponds to pH ) 7 (model 3 in Table 2). The gap surface is shown in Figure 11. As before, the peaks indicate models that show distinctly different

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Figure 11. Gap metric computation between local model pairs.

Figure 13. Closed-loop profile and manipulated variable moves using models 1, 3, and 5.

Figure 12. Closed-loop profile and manipulated variable moves using models 1-5. Process output (continuous line), reference (dashed line). Table 3. Parameters for the Membership Functions region p

1

2

3

4

5

yjp σp

3.0 0.25

5.0 0.25

7.0 0.25

9.0 0.25

10.5 0.25

behavior and, hence, should be included in the reduced set of models for control. Now, we study the use of the gap metric for model selection using the membership-based H∞ control discussed previously. The controllers for different operating regions are combined to form a complete control system. The parameters of the membership functions (eqs 14 and 15) are given in Table 3. The desired contribution of combined controllers can be represented as a function of the membership functions, resulting in the control

signal in eq 16. As discussed before, this is a single overall control action generated by the “weighted” contributions of local controllers. It is important to emphasize that different controllers were used for different operating conditions depending on the weighting. For instance, (4, 2, 4, 2, and 4) means that controller 4 is used for region 1, controller 2 for region 2, controller 4 for region 3, and so on. In this fashion, it is possible to assess the individual effect of each controller on a specific operating region. The closed-loop profile for the complete set of models (1-5) shows satisfactory performance, as depicted in Figure 12. If we only use the subset ΩLS (1, 1, 3, 3, and 5), Figure 13 shows degradation in the performance as a result of the lack of information from the highsensitivity regions in the selected models. Here, we use model 1 to explain region 2 and model 3 to explain region 4. As indicated in Table 2, these models show distinctly different behavior, hence the lack of success of the controller in these regions, especially in region 2. More simulation runs using at least one model belonging to low- and high-sensitivity regions exhibited satisfactory performance (Figures 14 and 15). Experimental Verification To assess the closed-loop performance of these controllers based on a subset of models, real-time experiments are also performed under similar conditions. Figure 16 shows the tracking performance when all five models are included. We can compare this performance with the tracking behavior when only models from the low-sensitivity region (ΩLS) are included. Figure 17 shows that there is clearly a loss of performance,


Figure 14. Closed-loop profile and manipulated variable moves using models 1 and 2.


Figure 16. Closed-loop profile and manipulated variable moves using models 1-5. Process output (continuous line), reference (dashed line).

Figure 17. Closed-loop profile and manipulated variable moves using models 1, 3, and 5.

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especially around region 2. This validates what we have seen before in the gap metric analysis and also in the simulation results (Figure 13), as these models are not sufficiently descriptive of the whole operating region. When only two models, specifically models 1 and 2 or 2 and 3, are used in combination (one model from ΩLS and one model from ΩHS), the results (Figures 18 and 19, respectively) indicate that the first combination exhibits some degradation in performance, whereas the second offers a more satisfactory tracking behavior (especially in control action). The difference from the previous simulations (Figures 14 and 15) can be attributed to the model-plant mismatch. Because of the unmodeled buffering characteristics in the high pH region, the responses are not as “symmetric” as they were in the simulations. Conclusions The gap metric is used as a measure of the distance between two linear models, and its application to chemical process control is explored. The results of the examples show that the gap metric is a useful and rigorous measure for evaluating the number of models used in multilinear model-based control setting. Acknowledgment A grant from the Australian Research Council (ARCA89906571) is gratefully acknowledged. Figure 18. Closed-loop profile and manipulated variable moves using models 1 and 2.

Literature Cited


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Ind. Eng. Chem. Res., Vol. 42, No. 10, 2003 2197 (15) Uppal, A.; Ray, W. H.; Poore, A. B. The Classification of the Dynamic Behavior of Continuous Stirred Tank Reactorss Influence of Reactor Residence Time. Chem. Eng. Sci. 1976, 31, 205. (16) de Silva, C. W.; MacFarlane, A. G. J. Knowledge-Based Control with Application to Robots; Springer-Verlag: New York, 1989. (17) Doyle, J. C.; Francis, B. A.; Tannenbaum, A. R. Feedback Control Theory; Macmillan: New York, 1992. (18) Skogestad, S.; Postlethwaite, I. Multivariable Feedback Control; Wiley: New York, 1996.

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Received for review October 2, 2002 Revised manuscript received March 7, 2003 Accepted March 26, 2003 IE020783S

Gap Metric Concept and Implications for Multilinear Model-Based

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