A Comparison of Robust Control Design Methods

223b

A Comparison of Robust Control Design Methods Applied to an Ill-Conditioned Distillation Column

3

Rasmus H. Nystr om , Jari M. B oling, Hannu T. Toivonen and Kurt E. H aggblom

Department of Chemical Engineering Abo Akademi University FIN-20500 Abo (Turku), Finland. Fax: Intl+358-2-2154479

Email: rnystrom, jboling, htoivone, [email protected] 3 On leave at: Laboratory of Process Systems Engineering, Department of Chemical Engineering, Kyoto University, Japan.

Keywords:

Distillation control, ill-conditioned and uncertain plants, multimodel techniques, optimal and robust control.

Prepared for presentation at the 1998 Annual Meeting, Miami Beach, FL, Nov. 15-20 Session 223: Process Control Applications. Copyright c Rasmus H. Nystrom, Jari M. Boling, Hannu T. Toivonen and Kurt E. Haggblom Abo Akademi University, Finland, Oct. 5, 1998. Unpublished. AIChE shall not be responsible for statements or opinions contained in papers or printed in its publications.

Abstract

A set of multiple linear models of an ill-conditioned two-product distillation column is used for controller design. A number of controller design methods are studied, each with the aim of utilizing the description of plant variations and uncertainties that the model set provides, for achieving sucient robustness and performance. The methods investigated are multimodel H2 optimal control, mixed H2 /H1 control with nominal H2 performance, mixed H2/H1 control with robust H2 performance,  synthesis based on loopshaping, IMC and decentralized PI control. The controllers are tested experimentally on the distillation column and the results discussed.

1

1

Introduction

Identi cation and control of distillation columns has been a subject of frequent study due to their illconditioned nature. The model of a high-purity distillation process has a steady-state gain matrix with a high condition number; the gain matrix is almost singular and its determinant is close to zero. The sign of the determinant may be aected by quite small model errors, and if the determinant of the gain matrix for the model and for the plant have dierent signs, there is no controller with integral action that can stabilize both the model and the plant (Grosdidier et al. 1985). Control design for an ill-conditioned plant is thus very sensitive to unstructured model uncertainty. A related feature of ill-conditioned processes is that during identi cation it is dicult to get good excitation of the column in both the high and the low gain directions, and in the model tting stage it is dicult to attain a t that represents the true plant satisfactorily in both directions. A possible approach to overcome the above problems is to perform separate excitations of the low-gain and high-gain directions and to identify a multimodel set consisting of a number of linear models, each based on a step experiment. In a recent study by Haggblom & Boling (1998), an ill-conditioned twoproduct distillation column was identi ed by determining such a set of models. The multimodel set should give a reasonable description of the plant variations and uncertainty. In this study, the set of models is used for implementing and comparing several controller design methods. The controllers are designed to track step changes in the setpoint in such a way that sucient robustness is attained for the set of identi ed models. The methods investigated are multimodel H2 optimal control, mixed H2 =H1 control, robust H2 control,  synthesis, multivariable IMC and decentralized PI control. The controllers are tested experimentally on the distillation column. In the direct multimodel H2 controller design technique (MacMartin et al. 1991, Makila 1991), a xedstructure controller is calculated by gradient-based minimization of a sum of H2 cost functions. The method guarantees closed-loop H2 performance for the discrete set of models. However, neither performance nor stability can be guaranteed for models that are intermediate with regard to the set. This drawback can be addressed by using a norm-bounded uncertainty description for controller design. For this purpose, a multiplicative uncertainty model is determined from the model set by Boling et al. (1997). A multiplicative uncertainty structure gives an advantage over an additive one, because it results in a less conservative design. This is due to the multiplicative uncertainty having directionality properties similar to the nominal model. Based on the nominal model with multiplicative uncertainty, a mixed H2 /H1 controller (Jacques et al. 1996) with robust stability and nominal performance is calculated and tested, as well as a mixed H2 /H1 controller with robust performance (Stoorvogel 1993, Toivonen & Pensar 1995, Toivonen 1996, Haddad et al. 1996). Another well-established method for H1 -based controller design is H1 loop-shaping, in which shaping of the sensitivity function is used for achieving robust performance (Skogestad & Postlethwaite 1996). This in combination with utilization of the uncertainty model gives a  design, which is also tested. For further comparison, a multivariable, multicriterion IMC design is implemented (Garcia & Morari 1982). Here, the robust performance of standard IMC design is improved by tuning the lters of the IMC controller to the whole model set. Finally, a decentralized PI-controller is optimally tuned using a multimodel H2 criterion. The experimental results show that the multivariable controllers provide signi cantly better performance than the decentralized PI-controller. The dierences between the performance achieved by the various multivariable methods are smaller, but re ect the assumptions on which the designs are based. The methods guaranteeing robust performance provide better performance than the mixed H2 =H1 method based on nominal performance and robust stability. The IMC controller performs almost as well as the 2

and the robust H2 controllers. The best overall performance is achieved with the non-conservative multimodel H2 method. Both the multimodel H2 and the robust H2 controllers are of low order (nc = 3) compared to the controllers designed using -synthesis (nc = 16) and IMC (nc = 28). This indicates that distillation control need not require high-order nor complex controllers. The study also shows that the multimodel set provides a convenient means of capturing model uncertainty arising from bad conditioning, and the relatively simple and minimally conservative multimodel H2 design provides a convenient method of utilizing this multimodel set for obtaining controllers with robust performance. 

2

Modeling of an ill-conditioned distillation column

2.1 Multimodel identi cation

The distillation column is a pilot-scale two-product column which separates a mixture of ethanol and water (Waller 1992). It consists of 15 bubble-cap trays and is 0.30 m in diameter. The column is equipped with a thermosyphon-type reboiler and a total condenser and operates under atmospheric pressure. An LV control structure has been selected for control of the column (Haggblom & Boling 1998). The controlled variables xD and xB , i.e. the top and bottom concentrations, are measured on-line by capacitive analyzers (Gustafsson 1994). The sampling rate is 2 samples/minute. Also, the control variables L and V , i.e. the re ux ow rate and the ow rate of steam to the reboiler, are manipulated at the same rate. In Haggblom & Boling (1998) models of the column are identi ed at two operating points. In both cases the feed has the ow rate F = 150 kg/h and the composition xF = 25 wt % ethanol. The low-gain and high-gain directions of the operating points are obtained by analysis of ow gains. At each operating point, a series of six step excitations is performed, three in the low-gain and three in the high-gain direction. A model is tted to the input-output data of each of these step experiments. Furthermore, a model is tted to the whole sequence of experiments, yielding a total of seven models for each of the two operating points. Here, operating point \B" (L = 160 kg/h, V = 105 kg/h) has been selected for study. The seven models each have the structure      y1 ln(xD0 0 xD ) 0 y10  =  G3DL G3DV u1 := (1) y2 ln(xB0 0 xB ) 0 y20 G3 G3 u2 where

BL



3

3

BV



1 + K 2 e0 s = T K 1s+1 T 2 s+1  K3 1 K3 2 0 s G3DV = T + s+1 T s+1 e 1 2  (2) K3 1 K3 2 0 s G3BL = T s+1 + T s+1  e 1 2  3 K3 1 2 e0 s G3BV = T 1 s+1 + T K 2s+1 The inputs and outputs are de ned as deviations from stationary values L0 , V0 , y10 and y20 . The inputs are given by [u1 u2 ]T := [L 0 L0 V 0 V0 ]T . For the outputs, logarithmic transformations have been introduced in order to capture part of the static process nonlinearities. The constants xD0 = 0:94, xB0 = 00:02, and the numerical values of the other parameters in (2) are given in H aggblom & Boling (1998). G3DL

DL;

DL;

DL;

DL

DL;

DV;

DV;

DV;

DV

DV;

BL;

BL;

BL;

BL

BL;

BV;

BV;

BV;

BV;

3

BV

In order to perform discrete-time controller synthesis, the models are discretized so that exact step response is maintained. From the fact that the time constants satisfy TDL;i = TDV;i and TBL;i = TBV;i, discrete-time state-space models of order 4 are obtained. Further, incorporating the time delays into the discrete-time models yields models of order 14 to 23, since the estimated time delays are in the range  = 1:5 : : : 6 min and exact multiples of the sampling interval Ts = 0:5 min. The discrete-time models with time delays are denoted Gn , n = 0 : : : N , N = 6. They can be used for the multimodel characterization GMM = fGn ; n = 0; : : : ; N g (3) The model G0 is based on all of the step experiments, and it is used as nominal model in the design methods that require one. The models Gn, n = 1 : : : 3 are associated with the low-gain direction and the models Gn, n = 4 : : : 6 with the high-gain direction. It is therefore a fair assumption that these models describe well the direction they portray (Haggblom & Boling 1998). Also, they should contain some information about the nonlinearity around the operating point. 2.2 Representation using output multiplicative uncertainty

Many robust controller design methods require an associated frequency-weighted, norm-bounded uncertainty description in addition to a nominal plant model. The discrete-time equivalent of a method presented in Boling et al. (1997) is used for determining such an uncertainty model. The method utilizes a multimodel description as well as knowledge about the excitation signals from the identi cation experiment and gives an uncertainty model in an output multiplicative form: 



GNB = (I + W 1)G0 ; k1k1  1

(4)

where G0 is the nominal model and 1 is an unstructured and stable LTI uncertainty. W is an uncertainty weight, which in the study at hand is set to scalar. It is given by kG0 (ei!T )un (ei!T ) 0 Gn (ei!T )un (ei!T )k jW (ei!T )j = max (5) n kG0 (ei!T )un (ei!T )k where un is the input signal corresponding to model Gn, transformed into the frequency-domain. s

s

s

s

s

s

s

The expression (5) is based on not-invalidation of the experimental data. A scalar lter Wf of suitable order can be adapted to describe the weight W by minimizing its overall magnitude subject to jWf (ei!T )j  jW (ei!T )j (6) In the study at hand a rst-order lter Wf with the the transfer function 1:90 0 1:80z01 Wf (z ) = (7) 1 0 0:80z01 is selected. Its frequency response is shown in Figure 1, together with a plot of the weight W . s

s

4

1

Uncertainty magnitude

10

0

10

−1

10

−3

10

−2

10

−1

10 Frequency (rad/min)

0

10

1

10

Figure 1: Estimated uncertainty weight W (solid line) and tted uncertainty lter Wf (dashed line). 2.3 Modeling for step disturbances: the H2 cost

The control problem studied in this paper consists of controlling the distillation column against step changes in the setpoints of the outputs yi . Discrete controllers of the form u = K (y 0 r ) (8) are assumed, where  0; t < 0 r (t) := (9) rstep ; t = 0; 1; 2; : : : describes a setpoint change. The control performance of the H2 -based controllers is de ned as the LQ-type cost for the closed-loop system, J2 (G; K; rstep ) := ky 0 rstep k22 + p2 k1uk22  1 X := [y(t) 0 rstep]T [y(t) 0 rstep ] + p2 1u(t)T 1u(t) (10) =0

t

where 1u(t) := u(t) 0 u(t 0 1). The input weight p2 is introduced in order to suppress excessively large variations of the inputs. The linear system y = Gu and the cost (10) do not de ne a standard H2 (LQ) control problem due to the presence of the constant signal r. The problem is transformed to standard form as follows. Consider the plant G, described by the discrete state-space equations xG (t + 1) = AG xG (t) + BG u(t) (11) y (t) = CG xG (t) 5

De ne 1xG(t) := xG(t) 0 xG(t 0 1) and consider the dierence equation equivalent to (11), 1xG(t + 1) = AG 1xG (t) + BG 1u(t) y (t) = CG 1xG (t) + y (t 0 1) De ning the augmented state   1xG(t) x(t) := y (t 0 1) 0 r(t 0 1) and the signals   y (t) 0 r(t) z0 (t) := p1u(t) v0 (t) := r(t) 0 r (t 0 1) we can de ne the system G such that 

z0 y







= G 1v0u

with the state-space representation x(t + 1) = Ax(t) + B0 v0 (t) + B2 1u(t) z0 (t) = C0 x(t) + D00 v0 (t) + D02 1u(t) y (t) 0 r(t) = C2 x(t) 0 v0 (t) where       AG 0 0 BG A := ; B0 := CG I 0I ; B2 := 0 ;      2 3 CG I 0 I 0 C0 := 0 0 ; C2 := CG I ; D00 := 0 ; D02 := pI

(12) (13) (14) (15) (16)

(17)



(18)

The response of the system y = Gu subject to a step change in the setpoint according to (9) can be described by the augmented system G in (17) with the input  rstep ; if t = 0 v0 (t) = (19) 0; otherwise The quadratic cost (10) is related to an H2 cost de ned for the augmented system (17) as follows. Denote the closed-loop system from v0 to z0 by Fz0v0 (G; K ). Then m

X (  ) := kFz0v0 (G; K )k22 = J2 (G; K; ei)

J2 G; K

=1

i

where m = dim(v0 ) = 2 and ei is the ith unit vector of Rm.

(20)

The fact that the system description (17) contains an integrator implies that any controller which achieves a nite cost must contain an integrator, and it is thus able to eliminate steady-state osets. This is important in practice, because the distillation column is subject to slowly drifting disturbances. As the model representation G has input 1u, we will in the sequel write the controllers in the form 1u = K (y 0 r) (21) 6

3

Robust H2 control of the distillation column

3.1 Multimodel H2 control

In the multimodel H2 (LQ) problem (MacMartin et al. 1991, Makila 1991, Pensar & Toivonen 1994), a controller is computed which stabilizes the discrete models Gn in the multimodel characterization GMM of equation (3), and makes the quadratic cost small for all the individual models simultaneously. The problem can be formulated by introducing a weighted sum of H2 costs evaluated for the individual models, JMM

(GMM ; K ) :=

N X

=0

n

(

cn J 2 G n ; K

)

(22)

Here cn are nonnegative weights which re ect the importance of the models in the quadratic cost. The multimodel H2 problem can be stated as follows. H2 1u = K (y 0 r) Gn n = 0; : : : ; N In this approach robustness is achieved solely by requiring closed-loop stability and acceptable performance for the discrete model set fGng. The multimodel controller design can, however, be generalized to the case when the individual models are equipped with norm-bounded uncertainties as well (Pensar & Toivonen 1994). The multimodel H2 problem has no closed-form solution, and it must therefore be solved by numerical optimization. Special purpose algorithms for the optimization problem are discussed in the references. In this study a multimodel H2 optimal controller is designed using the identi ed models Gn, n = 0; : : : ; 6. The input weight p = 1=50 is used in the quadratic cost (10). The value is found iteratively and is based on the trade-o between the con icting goals of achieving a fast output response and small input variations. In the multimodel cost (22), all models are given equal weights (cn = 1). Table 1 shows the minimum cost obtained when controllers of dierent orders are calculated. It is seen that no signi cant decrease of the cost can be achieved by increasing the controller order over three. Therefore the experiments reported in section 5 are performed using a controller of that order. Multimodel

problem: Find a linear controller

which stabilizes the models

,

, such that the cost (22) is minimized.

nc

0 1 2 3 4 5 6

multimodel cost 613.8 260.6 219.7 211.4 208.1 207.4 207.1

Table 1: Multimodel H2 costs achieved with dierent controller orders nc. The multimodel H2 controller is experimentally tested as described in section 5. The experimental results are shown in Figures 4 and 9. 7

3.2 Mixed H2 /H1 control with nominal H2 performance

For design of controllers which achieve good H2 performance subject to robustness against norm-bounded uncertainties, various mixed H2/H1 problems have been proposed. One approach is to minimize the nominal H2 performance subject to robust stability (Ridgely et al. 1992, Jacques et al. 1996). This is equivalent to a problem with mixed H2 and H1 objectives. In order to express the robustness of the closed-loop system in terms of an H1 norm bound, the closedloop system consisting of the plant de ned in (16), the multiplicative uncertainty (4), and the controller (21) is written in the equivalent form shown in Figure 2, where 2 2 0 G0 3 3  0  P =4 0 (23) 0 5 G W

w1 v0

1u

1 

-

z1

- z0

P K

y0r



Figure 2: Control of uncertain plant. The closed-loop system in Figure 2 is stable for all norm-bounded uncertainties 1 such that k1k1  1 if and only if the closed-loop transfer function Fz1 w1 (P; K ) from w1 to z1 of the nominal system (1 = 0) satis es the norm bound kFz1 w1 (P; K )k1 < 1 (24) The nominal H2 cost J2 (G 0; K ) de ned according to (20) can be expressed in terms of the augmented plant as  0 ; K ) = J2 (P; K ) := kFz0 v0 (P; K )k22 J2 ( G (25) The mixed H2 /H1 problem for optimal nominal performance subject to robust stability is stated as follows. Mixed

H2 /H1 problem: Find J2 P; K subject to

a stabilizing linear controller

K

for the plant

P

which minimizes the

( ) H1 Numerical methods for the computation of a xed-order controller which solves the mixed H2 /H1 problem have been described by Jacques et al. (1996) and Pensar & Toivonen (1994). In this study, the toolbox of Jacques et al. (1996) is used for the controller synthesis. A potential drawback of the H2 /H1 problem is that it ensures good H2 performance for the nominal plant model only. The method provides only robust stability with respect to the plant uncertainty, and there is no guarantee for robust performance. One approach to achieve robust performance in addition quadratic cost

the

type robust stability bound (24).

8

to robust stability is to generalize the procedure to the multimodel setting (Pensar & Toivonen 1994). In the present study, however, no tractable uncertainty description existed for the separate models in the set GMM . Mixed H2 /H1 optimal controllers are designed for the uncertain distillation column model (4). In Table 2, the nominal H2 costs for H2 /H1 controllers of dierent orders nc are shown. All have H1 norm approximately equal to one. In this case, as in the one with the multimodel H2 controller in section 3.1, it is clear that it is not feasible to increase the controller order over four. The H2 /H1 controller with the same order as the augmented plant has the nominal H2 cost J2 = 23:6. This can be compared with an H2-optimal controller designed for the nominal plant, which has the H2 -cost 23.2 and the H1 norm 1.60. For comparison, the multimodel controller designed in section 3.1 achieves the nominal H2 cost 24.2. The numbers show that robustness of standard H2 control can be increased substantially, with only a marginal degradation of nominal performance. nc

0 1 2 3 4 5 6 ...

nominal H2 cost 74.6 30.6 23.9 23.8 23.8 23.8 23.8 ...

Table 2: Nominal H2 cost achieved with mixed H2 /H1 controllers of dierent orders nc. The mixed H2/H1 controller is experimentally tested as described in section 5. The experimental results can be found in Figures 5 and 10. 3.3 Robust control with robust H2 performance

In this section a worst-case control problem which achieves robust H2 performance with respect to normbounded uncertainties is applied to the distillation column. Consider the uncertain plant in Figure 2. Let Fz0 v0 (P; K; 1) denote the transfer function from v0 to z0 and Fz1 v0 (P; K ) the transfer function from v0 to z1 for the nominal system (1 = 0). We introduce the worst-case H2 cost with respect to the norm-bounded uncertainty block 1, 

I (F (P; K )) := sup kFz0 v0 (P; K; 1)k22 : k1k1  1; 1Fz1 v0 (P; K ) strictly causal 1



(26)

The assumption that 1Fz1v0 (P; K ) is strictly causal, i.e. either Fz1v0 (P; K ) or 1 (or both) contain a time delay is made because it is compliant with a large class of standard uncertainty descriptions, including the one considered here, equation (4). By contrast with the H1 worst-case norm, the value of the worst-case H2 cost (26) depends on whether strict causality or only causality is assumed. The evaluation of robust H2 performance de ned by (26) is very hard. Therefore it is well-motivated to study more tractable robust performance measures. In particular, a useful upper bound on the cost (26) 9

can be obtained as follows. Recall that a signal representation of the worst-case H2 cost (26) is I (F (P; K )) := sup

(

1

m X 2

)

kz0k22 j v0 = ei0 : k1k1  1; 1Fz1 v0 (P; K ) strictly causal

=1

i

3

(27)

where 0 is the unit impulse. The fact that 1 is norm-bounded implies the inequality (cf. Figure 2) m X 2

=1

i

kz1 k22 0 kw1 k22 j v = ei0  0 3

(28)

An upper bound on the worst-case H2 cost (27) is therefore obtained by solving the constrained maximization problem sup T

(

m h X

i

kz0 k22 j v0 = ei0 : w1 = T z1 ;

=1

i

T

strictly causal

)

(29)

subject to the inequality (28). The constrained maximization problem can be solved by a Lagrange function approach. Introduce the Lagrange-type function associated with the cost in (29) and the constraint (28), L(F (P; K ); c) := sup

(

m X 2

3 kz0 k22 + c2 [kz1 k22 0 kw1 k22 ] j v0 = ei 

strictly causal (30) Here c is a real-valued parameter which corresponds to a Lagrangian multiplier associated with the constraint (28). It can be shown (Stoorvogel 1993) that if the nominal closed-loop system has H1 norm from w1 to z1 less than one, i.e., (24) holds, then there exists c > 0 such that L(F (P; K ); c) is bounded. Moreover, the worst-case cost (30) is an upper bound on the robust H2 performance measure (26), I (F (P; K ))  L(F (P; K ); c) (31) T

=1

i

:

w1

= T z1;

)

T

By contrast with the robust performance measure (26), the worst-case cost (30) is straightforward to evaluate. More precisely, the cost (30) is de ned in terms of an LQ-type maximization problem, and it can therefore be expressed via the solution of a Riccati equation. In view of the inequality (31), it is feasible to introduce a robust H2 performance problem as follows. H2 K P c>0 L(F (P; K ); c) Notice that the cost (30) can be written as Robust

problem: Find a stabilizing linear controller

worst-case cost

for the plant

and a

such that the

is minimized.

( "  2 m  X

z0

cz1 T l2 i=1

L(F (P; K ); c) = sup

0 kcw1 k2l2 j

#

v

= ei 0 :

w1

= T z1;

T

strictly causal

)

(32) For a xed c, the cost (30) is a mixed H2 /H1 cost of the form studied in Doyle et al. (1994), Stoorvogel (1993), Toivonen & Pensar (1995), and a dual version of the cost studied in Bernstein & Haddad (1989), Khargonekar & Rotea (1991), Kaminer et al. (1993). A controller which minimizes the cost can be found by convex optimization techniques (Kaminer et al. 1993), and the minimizing value of c can be determined by a direct search (Stoorvogel 1993). The optimal controller has the same order as the plant. Low-order controllers which minimize the worst-case cost (30) can be calculated by parametric optimization techniques (Toivonen & Pensar 1995), (Haddad et al. 1996). 10

The worst-case cost (30) is also an upper bound on robust performance with respect to the set of nonlinear and time-varying perturbations 1 (Stoorvogel 1993). When a linear time-invariant uncertainty is assumed, a less conservative upper bound can be obtained by replacing the real-valued weighting parameter c by a scaling lter d 2 H1 (Zhou et al. 1994). This results in a signi cantly more complex minimization problem (Toivonen & Pensar 1995, Haddad et al. 1996). However, numerical studies have indicated that the reduction of the upper bound which is achieved using a dynamic scaling lter instead of a real-valued weighting parameter is often small (Toivonen & Pensar 1995). The robust H2 performance problem described above is formulated for the uncertain distillation column model described in section 2.2. The input weight p = 1=250 is used in the quadratic cost (10). A smaller value is chosen than in the previous design methods because the robust performance measure by itself introduces more conservatism in the design. Table 3 gives the minimum worst-case cost (30) and the nominal H2 cost achieved for the distillation column model with optimized controllers of various orders. As in the multimodel problem, no signi cant decrease of the cost is achieved by increasing the controller order over three. Thus the experiments reported in section 5 are performed with a controller of that order. nc

0 1 2 3 4 5 6

robust H2 cost nominal H2 cost 171.5 104.1 52.7 37.4 38.6 27.2 35.7 25.1 34.2 24.7 34.1 24.5 34.0 24.4

Table 3: Minimum robust H2 costs and nominal H2 costs achieved with dierent controller orders nc. For comparison, de ning the nominal H2 cost using the input weight (p=1/50) which is used to compute the multimodel H2 and mixed H2 /H1 controllers, the optimal third-order controller achieves the cost J2 (P; K ) = 28:9. This is somewhat larger than the nominal cost of the multimodel H2 and the mixed H2 /H1 controllers. The design for robust performance results in a larger degradation of nominal performance than the other robust design methods. In return, the design ensures robust performance with respect to the norm-bounded uncertainty. The robust H2 controller is experimentally tested as described in section 5. The experimental results can be found in Figures 6 and 11. 4

Other methods for robust control

4.1 -synthesis with loop-shaping

In addition to the above methods, a continuous  design is implemented and tested. The time delays of the nominal model G0 are approximated by a rst-order lter, and the problem is formulated as a standard H1 -based loop-shaping problem, in which the sensitivity function (S ) and the input function (KS ) are shaped by appropriate weights (Skogestad & Postlethwaite 1996). The weight for shaping S is 11

obtained from the control objectives (integral action, small oscillations and maximum bandwidth). The weight for KS , which is needed for restraining the energy of the input signal, can be included in the uncertainty description. The procedure is described in more detail in (Boling et al. 1997). The augmentations and  synthesis give a controller of order nc = 16 : : : 22 , depending on how far the  synthesis is taken. However, the decrease of the cost function which can be obtained by increasing the controller order from nc = 16 is negligible. Thus, a controller of that order is used in the experiments described in section 5. For the experiments, the controller is discretized using a bilinear transform. Experimental results are found in Figures 7 and 12. 4.2 IMC design with a multiple performance criterion

For further comparison, an IMC design (Garcia & Morari 1982) is implemented. The controller consists of an internal model, an inverted internal model and an IMC lter f . For the internal model the nominal model G0 with time delays is used. The time delays are removed prior to constructing the inverted model. For the IMC lter f , a diagonal lter with rst-order diagonal elements is found to be sucient. This gives two tuning parameters; the time constants of the two elements. The standard IMC design does not address robust performance or robust stability subject to a speci ed uncertainty. Thus, the IMC controller is designed for robustness by tuning the time constants so that good step response is obtained for all the models in the set GMM . A time constant  = 10 min for both outputs is found to give satisfactory performance and robustness. Since the IMC controller contains the nominal plant model, the inverted plant model (with removed time delays) and the lter f , it has relatively high order (28). The experimental results (cf. section 5) are depicted in Figures 8 and 13. 4.3 Multimodel PI controller design

In order to demonstrate that the plant is not trivial to control, a decentralized PI controller is tuned to the model set GM M using a multimodel H2 criterion. The performance of the controller is evaluated by closed-loop simulations of setpoint changes for all models in the model set. The setpoint changes are made separately for each output. The four parameters of the decentralized PI controller   KP;1 + KI;1 1s 0 KP I (s) = (33) 0 KP;2 + KI;2 1s are obtained by simplex-based optimization of the cost function JP I

=

6 X 2 TX sim X =0 k=1 t=0

n

[yn;k (t) 0 rstep;k (t)]T [yn;k (t) 0 rstep;k (t)] + p2 1un;k (t)T 1un;k (t)

(34)

where Tsim is the simulation time horizon and rstep;k is a setpoint change. The value of the input weight is set to p = 1=50. The index k denotes the number of setpoint change simulation (one for each output) and n is the number of model Gn. The controller is experimentally tested only for a setpoint change of xD (cf. section 5). The result is shown in Figure 3. 12

5

Experimental results

The controllers designed in the previous sections are tested experimentally on the distillation column by introducing step changes in the setpoints. Two types of setpoint changes are tested. Figures 3, 4, 5, 6, 7 and 8 show the resulting top and bottom compositions xD and xB and inputs L and V of the closed-loop system obtained by introducing step changes in the setpoints of the two outputs separately. First, the setpoint of xD is changed (step 1 and step 2), then that of xB (step 3 and step 4). Figures 9, 10, 11, 12 and 13 depict xD , xB , L and V as a result of introducing steps in the low-gain direction (step 1 and step 2) and in the high-gain direction (step 3 and step 4). The controllers are allowed to reject a step for 90 minutes before a new step is applied. An exception is the PI-controller (Figure 3) which requires twice the time for settling, and for which only a setpoint change of xD is tested. To facilitate comparison of the performance of the controllers, a nite-horizon version of the quadratic cost J2 (G; K; rstep) in (10) is calculated for each step change and each controller. The input weight p = 1=50 is used. Table 4 contains the quadratic costs of the rst experiment type (separate-output steps) and Table 5 those of the second type (low- and high-gain steps). Controller multimodel H2 H2 =H1 robust H2  IMC

total 0.8313 0.8450 0.9108 0.8934 0.9204

step 1 0.2737 0.2899 0.2600 0.3320 0.3543

step 2 0.2841 0.2484 0.2955 0.3033 0.2541

step 3 0.1394 0.1339 0.1717 0.1252 0.1731

step 4 0.1338 0.1726 0.1828 0.1324 0.1374

Table 4: Quadratic costs for separate-output step experiments (xD : step 1 and 2, xB : step 3 and 4). Controller multimodel H2 H2 =H1 robust H2  IMC

total 0.2806 0.4209 0.4077 0.3550 0.3731

step 1 0.0807 0.0987 0.0855 0.1164 0.1244

step 2 0.0697 0.1003 0.1607 0.1295 0.1147

step 3 0.0615 0.1075 0.0981 0.0450 0.0640

step 4 0.0686 0.1142 0.0633 0.0630 0.0694

Table 5: Quadratic costs for low-gain (step 1 and 2) and high-gain (step 3 and 4) step experiments. The experiments show that the dierences between the multivariable controllers are small. This indicates that the nominal model is a good description of the process. However, the experimental results can be regarded in view of the assumptions on which the various design methods are based. The optimal multimodel H2 controller is optimized for precisely the identi ed set of linear models. Hence it should give the best over-all performance as long as the model set is an accurate description of the plant. The mixed H2 /H1 optimal controller is optimized for nominal performance subject to robust stability only, and there is no guarantee for robust performance. The robust H2 controller is designed to minimize the worst-case performance, and hence in general it results in the most conservative design for the nominal plant model. 13

In the experiments, the multimodel H2 optimal controller (Figures 4 and 9) gives the best over-all performance for both kinds of step changes. The result indicates that the model set GMM provides an accurate description of the process, cf. Haggblom & Boling (1998). However, in some operation areas the reduced conservatism visibly leads to somewhat decreased robustness (steps 3 and 4 in Figure 9). The comparatively poor performance of the mixed H2 /H1 optimal controller (Figures 5 and 10) indicates that optimization of nominal performance is not sucient to achieve good control performance of the distillation column. In light of the experiments the robust H2 controller (Figures 6 and 11) gives one of the most conservative designs. This is seen especially for step 2 in Figure 11, where the robust H2 controller gives a particularly sluggish response. On the other hand, it gives good performance in the same experiment for step 4. It is also expected to perform well in situations where the process behaviour changes. The  (Figures 7 and 12) and IMC (Figures 8 and 13) controllers are designed using slightly dierent performance criteria, but perform well in the tests. For the IMC-controller, this is expected since its robust performance has been increased by multimodel tuning. However, it seems to be somewhat conservative and gives the worst overall result of the separate-output step experiments (cf. Table 4.) The good performance of the  design shows that sensitivity function loop shaping can give results that are comparable to using a quadratic cost as performance criterion. The experimental results described above can only be viewed as general indicators. During the course of the experiments, spontaneous disturbances occur in the concentration measurements (eg. in Figure 4, before step 2 or in Figure 13, after step 1). The frequency and magnitude of the disturbances are unequal for the dierent controllers. It is also evident from the experiments that the column is subject to heavy drifting, partly due to startup phenomena. Thus, the experiments illustrate the need for robustness when designing controllers for an ill-conditioned distillation column.

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6

Conclusion

Methods for robust controller design proposed in the literature have been applied to an ill-conditioned distillation column with a multimodel description. The control objective is to achieve good quadratic (H2 /LQ) performance despite modeling uncertainties and process nonlinearities. The various design methods dier in the statement of the performance and robustness speci cations. Experimental tests have veri ed that all the multivariable design methods achieve good performance and robustness, and that a low-order controller is sucient for achieving near-optimal control. The experimental results to some extent re ect the dierent design speci cations which are used in the controller design. As it is often well motivated to de ne performance speci cations in terms of a quadratic cost, the robust H2 controller methods oer a promising class of techniques for the robust control of uncertain plants. The results also show that it is feasible to use multimodel identi cation and control for an ill-conditioned distillation column, and that multimodel design can be used in conjunction with a vast range of control design methods. Acknowledgements

This work was supported by the Neste Foundation, the Academy of Finland under grants 31202 and 34131, and the Japanese Ministry of Education (Monbusho). References

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