CEP 07 Dryer

Control Engineering Practice 1

r

o,

Sup

acc

13

oce

istu

as

ng p

ba

f th

Keywords: Process control; Industrial processes; Rotary dryer; Controllability analysis; Robust control; H1 Mixed sensitivity problem

consists in the reduction of the outlet moisture content of a

which is not easy to achieve in manual mode.

In fact, there is not a complete knowledge of the drying

Douglas et al., 1993; Duchesne et al., 1997; Reay, 1979).Most agree that it is very difcult to carry out a ne tuning

ARTICLE IN PRESS

Corresponding author. Tel.: +349 54486037; fax: +34 9 54487540. of the model parameters. Furthermore, the tuning proce-dure is different for each specic material, and it is validonly for very limited operating conditions.

0967-0661/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.conengprac.2006.09.005

E-mail addresses: [email protected] (M.G. Ortega),

[email protected] (F. Castano), [email protected] (M. Vargas),

[email protected] (F.R. Rubio).product to a desired value. This process uses a continuousrotary drum in which the wet material is tumbled, ormechanically turned over, usually in the same direction,while extremely hot air is continuously owing.The main control objective is to regulate the outlet

moisture level, but it is also interesting to control theoutput temperature of the exhaust air, since it may beemployed as supply for other processes. These require-ments impose a strict and permanent control of the plant,

process, since too many phenomena all connected to eachother, such as heat and mass exchange, movement ofsolids, evaporation, capillarity, supercial tension, diffu-sion, and so on, are involved in it. This implies a highlycomplex behavior that is quite difcult to model.The scientic community has been interested in the

modelling problem of these kinds of processes, and so far,some mathematical models have been developed (see, forinstance, Brasil and Seckler, 1988; Deich and Stalskii, 1975;1. Introduction

Despite the importance of drying processes for theindustry, the automatic control of this kind of system is stillin a precarious state, since many installations are usuallyoperated in manual mode. This paper offers an analysisand study of a continuous rotary dryer, showing a numberof problems connected to these kinds of processes.Basically, the drying process in a rotary installation

The difculty in controlling this type of plants lies in thephysical properties of the product to be dried, as well as inthe change of these properties with the different moisturepercentages. The movement of the material throughout thedrum, the adhesion of the particles to one another or to thedrum blades, the evaporation speed, etc., are examples offactors strongly inuenced by the moisture level. Thisimplies a quite different system behavior depending on theoperating conditions.Multivariable robust control of a

M.G. Ortega, F. Castan

Depto. Ingeniera de Sistemas y Automatica, Universidad de Sevilla, Escuela

Received 27 April 2005;

Available online

Abstract

This paper describes the analysis and control of an industrial pr

Two process variables are controlled simultaneously: the outlet mo

air. To do this, the ow of wet material and the ow of fuel are used

inputs and two outputs. The system is identied at several operati

any constraints in its performance. A multivariable robust control

and experimental results are included, using different adjustments o

attained agree with the controllability analysis conclusions.

r 2006 Elsevier Ltd. All rights reserved.5 (2007) 487500

otary dryer: Analysis and design

M. Vargas, F.R. Rubio

erior de Ingenieros, Camino de los Descubrimientos sn, 41092 Sevilla, Spain

epted 7 September 2006

November 2006

ss usually controlled in manual mode: a continuous rotary dryer.

re of the dried material and the output temperature of the exhaust

control variables. Thus, this constitutes a MIMO system, with two

oints, and a controllability analysis is performed in order to nd

sed on the H1 mixed sensitivity problem is proposed. Simulatione weighting matrices based on the performed experiments. Results

www.elsevier.com/locate/conengprac

This paper describes the control of a rotary dryer thatuses sand as a working material. The output temperature ofthe exhaust air and the outlet moisture of the sand areregulated simultaneously, making a multivariable controlapproach suitable.From the point of view of control, the application of

several control techniques can be found in the literaturefor these types of systems, ranging from elementaryPID control (Arjona et al., 2005) to predictive control(Didriksen, 2002), fuzzy control (Yliniemi, 1999) or others(Courtois, 1997; Savaresi et al., 2001), though they allimplement monovariable control loops.Being aware of the complexity of the drying process, it

could be appropriate to implement a controller which cancope with the coupling and strong uncertainties thatmay exist when modelling these systems. Accordingto this, the development of a robust controller seemssuitable. This paper describes one in particular: a multi-variable H1 controller, based on the mixed sensitivityproblem.The remainder of this article is organized as follows: in

2. Plant description

The system considered corresponds to a co-current rotarydryer (see Fig. 1), which is located on the terrace roof of thelaboratories of the Dept. Ingeniera de Sistemas y Auto-matica of the University of Seville. The plant makes use of adistributed control system that allows the control andintegral monitoring of the drying process.A simplied diagram of the rotary dryer is depicted in

Fig. 2, basically showing three different areas:

The feeding area, which consists of a burner and twohoppers. The burner uses gas as fuel to heat theincoming ow of air. This current of hot air is employedto dry the wet matter. The hoppers contain the wetmaterial, which is carried into the drum by means of aworm gear and a conveyor belt. This congurationallows us to control both the ow of fuel and the inletow of wet matter.

The rotary drum area. The wet material from the feedingarea is continuously hauled by the rotation of the drum,

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500488Section 2, a description of the process is presented, andits model is obtained in Section 3. This model is validatedwith real data obtained from the process, and employedfor deriving some transfer matrices at different workingpoints. A controllability analysis of the linearized systemis carried out in Section 4, in order to show the behaviorlimitations of the plant. The synthesis of the H1 robustcontroller is described in Section 5. Simulation andexperimental results attained by means of this robustcontroller are presented in Sections 6 and 7, and,nally, the main conclusions to be drawn are given inSection 8.Fig. 1. Co-current roand dropped into the hot air stream circulatingthroughout the drum. The cylinder (4m in length and0.8m in width) has inside a number of continuousblades, so while it is turning, these blades take thematerial and propels it into the gaseous current. Thedrum usually turns at a speed between 3 and 10 rpm,moved by a 3 kW electrical motor coupled to a gearreduction unit. The air speed ranges between 1.5 and4m/s, depending on the size of the particles to be driedand on the quantity of ne powder produced during theprocess. The speed of rotation, pitch angle and airvelocity determine the material delay time.tary dryer.

Frfemimpr

3.

3.

manplco

eq

ARTICLE IN PRESS

Y D

m

e co

Wi

Fgi-1+Fsti-1

Fsi-1+Fwi-1Fsi+Fwi

Fgi+Fsti

Fig. 3. ith element of the dryer drum.

ineerated in the rst stages of the drying process. This makesit possible for high temperatures to be reached in theincoming air without reaching dangerously high tem-peratures in the material to be dried. Since thetemperatures of the air and of the solid materials tendto converge when both ows reach the drum outlet, thetemperature of the solids that leave the cylinder can becontrolled to reach their maximum value, while main-taining the advantage of having a wide range oftemperatures in the rst stages.The output area. In this area, the dried product and theexhaust air are extracted from the drum and takenoutside the process. On one end, the solid material dropsinto a small hopper, and from this one to anotherFor the sake of efciency, it is essential to set up the owof solid materials helped by the air stream (see Fig. 2),since a signicant amount of moisture must be evapo-

FEEDING AREA ROTAR

Fuel

Air

Hot Air

Wet material

wat

erFig. 2. Diagram of th

M.G. Ortega et al. / Control Engconveyor belt. On the other end, the exhaust air is guidedthrough a dust lter, which is located before the suctionfan used to create the air stream.

om this output area, the exhaust air could be used foreding another process, provided that the air currenteets the adequate conditions. Therefore, it may beportant to control not only the moisture of the driedoduct, but also the temperature of the exhaust air.

Process model

1. Thermodynamic model

In rotary dryers, water is removed from a material byeans of a hot air stream. This transfers heat to the solidd reduces its humidity content. Heat transmission takesace mainly by conduction and convection in adiabaticnditions.Thus, the system can be modelled by a set of nonlinearuations describing energy and mass balances (RubioRUM AREA OUTPUT AREAExhaust air

Dried material

wat

er

wat

er

otor

Air filter

-current rotary dryer.

ring Practice 15 (2007) 487500 489et al., 2000; Rubio et al., 2001). Due to the drumsgeometry, strong gradients of concentration and tempera-ture appear. Therefore, the application of mass andenergy balances gives rise to partial derivatives anddifferential equations, which can be avoided dividing thedryer drum into a nite number, n, of elements in series(Savaresi et al., 2001). Fig. 3 shows a generic element, towhich balance equations are applied. In this gure, Fg isthe mass ow of dry gas, Fst is the mass ow of steamin the gas, Fs is the mass ow of dry solid, Fw is themass ow of water in the solid, and W is the ow ofevaporated water.Balance equations come from the analysis of the ows

that go across the ith section. Assuming that the conditionsat the dryer input are known, the outputs of each sectionare computed and used as the input values for the nextsection. Many variables appear in each element, such asspecic heat of water and solid, specic heat at constantpressure of the dry gas and water steam, latent heatof vaporization, volumetric heat transfer coefcient,among others.

In the same way, it was necessary to develop a model ofthe combustion process in the feeding area, which is alsogoverned by differential equations. A complete model of aburner should include a combustion performance study, aheat transmission analysis, along with other information.The variables involved in the combustion process are quitesimilar to those mentioned in the previous paragraph.For the sake of completeness, a more detailed but

concise description of the equations and physical variablesinvolved in both parts of the nonlinear model is given inAppendix A.

3.2. Nonlinear model validation

This model has been validated using data obtained fromthe real plant. These data were used to estimate many of theparameters that appear in the model which are not perfectlyknown, since they depend on the operating conditions.

mass ow of wet material (Fp). The controlled variables arethe outlet moisture content of the dried product (Mo) andthe output temperature of the exhaust air (To).Several models were obtained at different operating

points. Fig. 6a shows the working space in terms of controlvariables, whilst Fig. 6b shows it in terms of outputvariables. The nominal operating point, denoted as NP, aswell as other extreme points are marked. These values wereobtained from the real plant, where the variation of the fuelmass ow is around 70% and the variation of the mass owof wet material is approximately 33%, both with respect tonominal conditions.It should be noticed that the plant does not work

properly if the ow of fuel is low and the solid ow is highor vice versa. In the rst case, a small amount of heat isprovided for too much wet material, which causes tosignicant bonding of the material. In the second extremecondition, too much heat is supplied for a small amount ofwet material and, therefore, there may be risk of a re.In order to describe the fundamental dynamic behavior of

the system, some linear models have been identied atdifferent operating points by means of classical identica-tion techniques (Camacho and Bordons, 1995; Ljung, 1986).In particular, multivariable ARX models were computed

applying pseudorandom binary sequences at the non-linear

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500490Fig. 4 shows a comparison between real and simulatedoutputs obtained from real input data. It can be seen thatthe nonlinear model is able to reproduce the fundamentaldynamics, with a relatively small steady-state error undersignicant changes on either output: the output tempera-ture of the exhaust air (To), and the outlet moisture contentof the dried product (Mo).The data used as input for these model validation

experiments come from independent PRBS-type (pseudor-andom binary sequence) variations in the inputs around theconstant DC values of the corresponding operating points.

3.3. Linear models

Linear models have been developed in order tosynthesize the robust controller. Thus, the plant has beenmodelled as a multivariable system (see Fig. 5) whosecontrol variables are the mass ow of fuel (F c) and the

0 50 100 150 200 250 30040

45

50

55

60

time (minutes)

T o: Ex

haus

t air

tem

pera

ture

(C

)

0 50 100 150 200 250 3000

0.20.40.60.8

1

time (minutes)

Mo : O

utle

tm

ois

ture

( %)

real

simulated

real

simulatedFig. 4. Measured and simulated outputs. Left: signicant chanmodel inputs. As a result, the identied nominal model was

Gs 1ds

n11s n12sn21s n22s

!, (1)

0 50 100 150 200 250 300404550556065

time (minutes)

T o: Ex

haus

t air

tem

pera

ture

(C

)

0 50 100 150 200 250 3000

0.20.40.60.8

1

time (minutes)

Mo: O

utle

tm

ois

ture

( %)

real

simulated

real

simulated

Rotary dryer

To

Mo

Fc

Fp

Fig. 5. MIMO system.ges in output To. Right: signicant changes in output Mo

where

n11s 3:7724s 3:3333 102s 5:8870 103s 5:4580 104s 4:2980 104 2:4298 104 j,

n12s 6:1057 103s 4:7402 102s 3:3333 102s 4:8541 103s 1:3053 103s 5:2376 104,

n21s 2:8079 102s 3:3333 102s 1:0052 102s 2:7027 103s 5:5607 104 3:1026 104,

n22s 1:8574 104s 3:3333 102s 2:0368 102s 5:8427 103s 2:7544 103s 5:5013 103,

ARTICLE IN PRESS

Fc (Kg/h)

F p (K

g/min)

3

2

4

2.6 40.7

NP

P1

P2

P3

P4

Control variablesTo(C)

Mo(%

)1.08

1.79

37.627.8

NP

P1P2

P3P4

5728.4

0.43

1.62

56

0.30

Output variables(a) (b)

Fig. 6. Working area of the rotary dryer: (a) control variables; (b) output variables.

0 200 400 600 800 10000

2

4

6

8

10

12

T o

Step in Fc

0 200 400 600 800 10001.5

1

0.5

0

0.5

time (minutes)

Step in Fp

0

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500 491time (minutes)

0 200 400 600 800 1001.4

1.2

1

0.8

0.6

0.4

0.2

0

time (minutes)

MoFig. 7. Comparison of the nominal (thick line) and non-nominal time res0 200 400 600 800 10000.05

0

0.05

0.1

0.15

0.2

0.25

0.3

time (minutes)

ponses under step changes in inputs F c, Fp for each operation point.

ds s 3:3049 102s 5:1752 103s 5:5221 104s 3:9957 104 2:2141 104 j.

It can be seen that the system is stable, which does notimply any strong constraint in the closed-loop behavior.However, the system has some zeros in the right half plane(RHP), which impose some limitations on the systemperformance (Astrom, 2000). The most restrictive RHP-

ARTICLE IN PRESS

3

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500492These continuous-time transfer matrices have been ob-tained for each operation point after conversion fromdiscrete-time domain using a bilinear inverse transforma-tion (sampling time equal to 60 s). In the next section, acontrollability analysis is made based on the continuousnominal model, though keeping in mind that the results areno longer valid for frequencies near the sampling rate.In order to help the reader perceive the differences of the

ve transfer functions, Fig. 7 shows the corresponding stepresponses under changes in either input.

4. Controllability analysis

In this section, a basic inputoutput controllabilityanalysis (Skogestad and Postlethwaite, 1996) on thenominal linear model is made in order to check limitationsof the closed-loop performance. This analysis was alsocarried out on the other operating points, and the attainedresults were similar to those presented next.First, the system has to be adjusted in scale, in agreement

with the maximum allowed deviation of each input andoutput. In this application, the following values have beenchosen as maximum variations:

DF c max 1:4 kg=h; DTo max 5 C,

DFp max 0:5 kg=min; DMo max 0:3.Taking these values into account, the system can be scaledby means of the following expression:

G^s D1e GsDu,where Gs is the original linear model obtained in thepreceding section and G^s is the scaled system. The twoscaling matrices De and Du are built as follows:

De DTo max 0

0 DMo max

" # 5:0 0

0 0:3

,

Du DF c max 0

0 DFp max

" # 1:4 0

0 0:5

.

The rst step of the analysis consists in studying the polesand zeros of the system (see Skogestad and Postlethwaite,1996, for instance). Table 1 shows these values in thecontinuous time domain.

Table 1

Poles and zeros of the nominal model

Poles: 3:305 102, 5:175 103,Zeros: 7:065 103, 2:287 103 2:04zeros are those with the smallest magnitudes. In ourcase, such zeros are located at zRHP 2:287 1032:041 103j. This yields an approximate upper boundfor the control bandwidth oc equal to (Skogestad andPostlethwaite, 1996):

ocojzRHPj2:8

1:09 103 rad=s

which gives the following approximate lower bound for therise time, at least for one output:

tr p2oc

41434 s 24 min .

However, since the system is a multivariable one, it is alsoimportant to consider the direction of the RHP-zeros, atleast their output directions. In our case, zRHP has thefollowing output direction:

To

Mo

!

0:145 0:250 j0:156 0:94457 j

!.

It can be seen that the rst component (its absolute value,equals to 0:289) is much smaller than the second one(whose absolute value is equal to 0:957). Therefore,although the degrading effect of a RHP-zero can be movedto a given output (Holt and Morari, 1985), its natural effectis focused on the second output variable. This way, thecontrol of the outlet moisture of the dried product is morelimited than the control of the output temperature of theexhaust air. Therefore, if a tight control on the outletmoisture is imposed, the behavior of the exhaust airtemperature would be strongly degraded.Another important item is the analysis of the singular

values of the system as a function of the frequency. Inparticular, the minimum singular value is useful as acontrollability index. Singular values of the nominal modelare depicted in Fig. 8 and Table 2 shows their directions(see Skogestad and Postlethwaite, 1996) at low frequencies.It can be observed that the minimum singular value issmaller than one for all frequencies. This implies thatindependent output changes cannot be made. In particular,as the output direction of the minimum singular value ismainly in the direction of Mo, some troubles may beexpected if tight control of the outlet moisture is requiredusing the allowed variations of the control variables.

:995 104 2:214 104j, 5:522 1041 103j , 3:333 102, 3:333 102

ARTICLE IN PRESS

1ue

s o

ineeFrom Fig. 8 it can be inferred that the condition numbercorresponding to the scaled nominal plant is slightly highat low frequencies, and it experiences a signicant increasestarting from frequencies around 5 104 rad=s, mainlydue to the drop-off in the minimum singular value at thesefrequencies. Therefore, the system can be considered ill-conditioned from these frequencies on, making it strongly

105 104

104

102

100

magn

itude

freq

Minimum singular values

Fig. 8. Singular value

Table 2

Directions of the singular values at low frequencies

Singular value Input direction F c;FpT Output direction To;MoT

Maximum 0:495 0869T 0:997 0:080TMinimum 0:869 0:495T 0:080 0:997T

M.G. Ortega et al. / Control Engsensitive to uncertainties.This fact gives an additional approximate lower bound

for the rise time. In this case, this bound is the following:

tr p

2 5 10443141 s 52 min .

Other factors have been analyzed, like the relative gainarray (RGA) along the frequency, but no other noticeablerestrictions have been found.

5. Controller synthesis

As stated in the Introduction, the main control objectiveof this process is to regulate the moisture percentage ofthe dried product, as well as to control the outputtemperature of the exhaust air. To do this, the wet solidow and the fuel ow of fuel are used as controlvariables. Therefore, as already mentioned, this constitutesa multivariable control problem, with two inputs and twooutputs.A centralized multivariable robust controller is pro-

posed, taking into account that the system must workproperly at different operating points. It is difcult to ndout the way variations in each model parameter affect thesystem outputs. Therefore, it is not reasonable tosynthesize a controller based on parametric uncertainty,as is usual in very complex systems. Thus, an H1controller, based on unstructured uncertainty, is designedin this section, using the mixed-sensitivity problem ap-proach.The feedback H1 controller design can be formulated as

an optimization problem, which can be posed under thegeneral conguration shown in Fig. 9. In this gure, Ps is

03 102 101

ncy (rad/s)

Maximum singular values

f the nominal model.

K(s)

z

vu

P(s)

Fig. 9. General formulation of the control problem.

ring Practice 15 (2007) 487500 493the generalized plant, Ks is the controller, u represents thecontrol signals, v the measured variables, o the exogenoussignals and z stands for the so-called error variables.The optimal H1 control problem with this conguration

consists in computing a controller in such a way that theratio g between the energy of the error vector z and theenergy of the exogenous signals o is minimized.This optimal problem is still open, but a solution exists

for the suboptimal case (see Doyle et al., 1989 for adescription in the continuous time domain and Iglesiasand Glover, 1991 for discrete time). Therefore, the value ofthe energy ratio g is decreased as much as possible bymeans of an iteration procedure. This is the synthesismethod which has been implemented in well-knownsoftware packages such as Balas et al. (1995) or Chiangand Safonov (1998).The conguration used for building up the generalized

plant is given by the S=KS=T mixed sensitivity problem(for example, see Zhou et al., 1996), which is shown inFig. 10. In this case, the expression of the resulting closed-loop transfer function, Tzos, is as follows:

Tzos WSsSos

WKSsKsSosWT sTos

264

375,

WTdiag s 100s 0:01s 0:0125 .

ARTICLE IN PRESSineewhere Sos is the output sensitivity transfer matrix, Tos isthe output complementary sensitivity transfer matrix, andKsSos is the so-called control sensitivity transfer matrix:

Sos I GsKs1,KsSos KsI GsKs1,

Tos GsKsI GsKs1.

The factors WSs, WKSs and WT s constitute theirrespective weighting matrices, which allow to specify therange of frequencies of relevance for the correspondingclosed-loop transfer matrix.As it is known, proper shaping of Tos is desirable for

tracking problems, for noise attenuation, and for robuststability with respect to multiplicative output uncertainties.On the other hand, a convenient shaping of Sos will allowto improve the performance of the system. In addition,

G(s)

WS(s)

WKS(s)

e

u

y

z1

z2

= r

-

+

v

P(s)

z

WT(s) z3

K(s)

Fig. 10. S=KS=T mixed sensitivity conguration.

M.G. Ortega et al. / Control Eng494matrix WKSs helps us avoid some numerical problems inthe synthesis algorithms.Therefore, since the controller is obtained from the

generalized plant, the synthesis problem with this cong-uration is reduced to the design of some appropriateweighting matrices which will impose the control specica-tions. Based on this, the generalized plant can be built up,and consequently the controller can be calculated on acomputer by using a synthesis algorithm.The selection of the weighting matricesWSs andWT s

has been accomplished following the design rules fromOrtega and Rubio (2004). Thus, once the scaled nominalmodel is available (see Eq. (1)), the multiplicative outputuncertainty can be estimated as follows:

E^oPi s G^Pis G^sG^s1,where G^s is the nominal model and G^Pis stands for eachnon-nominal identied model at the extreme operatingpoints. Maximum singular values of these estimateduncertainties are shown in Fig. 11.Next, matrix WT s is designed as a square diagonalmatrix with all its diagonal elements with the same transferfunction, that is

WT s WTdiag sI22,where the transfer function WTdiag s must be stable,minimum phase, with a high gain at high frequencies,and with magnitude greater than the maximum singularvalue of the uncertainty previously computed, for eachnon-nominal model and frequency, that is,

jWTdiag jojXsmaxE^oPi jo 8o; 8Pi.In this application, WTdiag s has been chosen as follows:

105 104 103 10210

5

0

5

10

15

20

25

30

35

Max

. S.V

. (dB)

WTdiag

max(oP2)

max(oP1)

max(oP3)max(oP4)

frequency (rad/s)Fig. 11. WT s matrix as an upper bound of the maximum singular valuesof the multiplicative output uncertainty.

ring Practice 15 (2007) 487500The magnitude of this transfer function is plotted inFig. 11. It can be seen that it satises all the requirementsimposed on it.Matrix WSs is taken as a square diagonal matrix of

transfer functions:

WSs WSTo s 0

0 WSMo s

" #,

where each diagonal element WSis is designed with thefollowing expression:

WSis ais 10ki1oTs bi10ki1oT

; i To;Mo,

where oT is the crossover frequency ofWTdiag s (see Fig. 11),and whose value is about 7:75 105 rad=s. Parameters aiand bi are the transfer function gains at high and lowfrequencies, respectively. According to the design rulesstated in Ortega and Rubio (2004), the following valueshave been chosen: aTo aMo 0:5 and bTo bMo 104.

Finally, dimensionless parameters kTo and kMo determine thecorresponding transfer function bandwidth, and their valuesmust be higher as the desired response speed of thecorresponding output increases. An initial value equal tozero is recommended, which yields slow and rarely oscillatoryresponses. The nal adjustment of these parameters must becarried out on the real plant.Then, the weighting design is completed after making

weight WKSs equal to the identity matrix (i.e.WKSs I22) in order to avoid numerical problems inthe synthesis algorithm.Once the weighting matrices have been designed, the

generalized plant (see Fig. 10) can be built up, andtherefore, the controller can be computed. At this point,it is important to remember that the controller has beensynthesized from a scaled model. To gure out thecontroller to be implemented in the real application it isnecessary to carry out a reverse scaling procedure, that is

Ks DuK^sD1e ,where matrices De and Du were introduced in Section 4 forthe scaling of the plant.

A nominal controller (labelled as Controller 1 in Fig. 12)was synthesized using values for kTo and kMo equal to 3and 2.5, respectively. It can be observed that these valuesyield an appropriate response for both outputs, with atrade-off between the time responses of To and Mo, andwith no substantial overshoot in any response.Two other controllers were synthesized from slight

variations in the nominal values of these parameters. Oneof them, marked as Controller 2 in Fig. 12, aims to obtain abetter response speed of To by increasing the value of kTofrom 3 to 3.5. It can be observed that, as expected from thecontrollability analysis (see Section 4), an improvement inthe behavior of To does not signicantly affect theperformance of Mo. The other one, noted as Controller 3in Fig. 12, attempts to improve the response speed of Moby increasing the value of kMo from 2.5 to 3. Thisimplies that the natural degrading effect of the mostrestrictive RHP-zero is moved to the output Mo. In thiscase, as expected from the controllability analysis, thereis a signicant worsening of the transient response ofvariable To.Finally, Fig. 13 shows a comparison between the

performance achieved with the previous nominal controllerand that attained by means of a nely tuned multivariable

ARTICLE IN PRESS

me

me

ont

Mo

5

Mo=

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500 4950 50 100 150 20035

36

37

38

39

40

41

42

ti

To (o

C )

0 50 100 150 2000.95

1

1.05

1.1

1.15

1.2

1.25

ti

Mo ( %

)

Controller 2: To=3.5 Mo=2.5

Controller 1: To=3 Mo=2.5

C

Controller 3: To=3

Controller 1: To=3 Mo=2.

Controller 2: To=3.5 6. Simulation results

Several tests were carried out on the nonlinear model ofthe plant in order to adjust controller parameters kTo andkMo . Fig. 12 shows the responses obtained for a set ofvalues of kTo and kMo .Fig. 12. SimulaPID controller. A similar performance can be observedin the response of Mo, while the evolution of To is faster(and almost without any overshoots) in the case of theH1 controller. However, implementing a multivariablePID had, in our case, led to an annoying 12-parametertuning procedure, while using the proposed methodology,

250 300 350 400 450 500

(minutes)

250 300 350 400 450 500(minutes)

roller 3: To=3 Mo=3

=3

2.5tion results.

ARTICLE IN PRESS

39C ) Controller 1: To=3 Mo=2.5

le P

inee36

37

38T o (

Multivariab40

41

42

M.G. Ortega et al. / Control Eng496only two parameters had to be tuned (once oTdiag is chosenfrom Fig. 11) following a few intuitive rules in a systematic way.

7. Experimental results

In this section, step responses of the output variables onthe real plant have been obtained to evaluate theperformance provided by the controllers.

0 50 100 150 20035

time

0 50 100 150 2000.95

1

1.05

1.1

1.15

1.2

1.25

time

Mo ( %

)

Controller 1: To=3 Mo=2.5

Multivariable PID controller

Fig. 13. Simula

0 20 40 60 80 100 120 140 160 180303540455055

time (minutes)

T o (C

)

0 20 40 60 80 100 120 140 160 1800.3

0.35

0.4

time (minutes)

Mo (%

)

Fig. 14. Experimental results with Controller 1. Step change iID controller

ring Practice 15 (2007) 487500Unlike the previous experiments performed on thesystem model, the experiments reported in this sectionhave been made at non-nominal operating points.The step responses of the real system obtained using the

nominal controller (Controller 1 with kTo 3 andkMo 2:5) are presented in Fig. 14. From these responses,two main issues can be pointed out. First, the achieved risetimes are similar to those attained by simulations using thenonlinear model. These times are approximately equal to70 and 100min for To and Mo, respectively. In addition,

250 300 350 400 450 500(minutes)

250 300 350 400 450 500(minutes)tion results.

0 20 40 60 80 100 120 140 160 18040

45

50

55

time (minutes)

T o (C

)

0 20 40 60 80 100 120 140 160 1800.2

0.250.3

0.350.4

0.45

time (minutes)

Mo (%

)

n Mo reference (left). Step change in To reference (right).

with this controller, changes at the reference of each outputvariable have no noticeable effect on the other output. Thecorresponding control inputs are plotted in Fig. 15.Fig. 16 shows the experimental responses obtained by

means of Controller 3 (with kTo 3 and kMo 3) understep change in the reference of Mo. It can be observed thatthe response ofMo is faster (with a rise time about 50min)than the one of Fig. 14, as expected from the increment ofthe value of kMo . This faster response is achieved by meansof an excessive control effort, which makes the system to

reach its upper saturation level for the Fp control variable(see Fig. 17). Furthermore, it must be pointed out thedegradation of the response of To (more evident than theone corresponding to the simulation experiments) due tothe natural deteriorating effect of the most restrictive RHP-zero (see Section 4).It should also be noticed that in this experiment no

steady-state data have been recorded as the transientperiod of the experiment is long enough to show the maineffects of Controller 3.

ARTICLE IN PRESS

0 20 40 60 80 100 120 140 160 1802

2.22.42.62.8

33.23.43.63.8

4

time (minutes)

F p (K

g/min)

0 20 40 60 80 100 120 140 160 1802.5

3

3.5

4

time (minutes)

F c (K

g/h)

0 20 40 60 80 100 120 140 160 1802

2.22.42.62.8

33.23.43.63.8

4

time (minutes)F p

(K

g/min)

0 20 40 60 80 100 120 140 160 1802.5

3

3.5

4

time (minutes)

F c (K

g/h)

Fig. 15. Experimental control variables with Controller 1. Step change on Mo reference (left). Step change in To reference (right).

0 10 20 30 40 50 6030354045

5055

6065

time (minutes)

T o (C

)

3time

M.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500 4970 10 200.35

0.4

0.45

0.5

Mo (%

)Fig. 16. Experimental results with Contr0 40 50 60 (minutes)

oller 3. Step changeon Mo reference.

ARTICLE IN PRESS

30

3.2

me

/min

30time

th C

ineeFinally, it is interesting to note that the rise time is

Fig. 17. Experimental control variables wi0 10 202

2.22.42.62.8

3

ti

F p ( K

g

0 10 202.5

3

3.5

4

F c ( K

g/h)3.43.63.8

4

)

M.G. Ortega et al. / Control Eng498greater than 52min for an appropriate performance,as stated in the controllability analysis performed inSection 4.

8. Conclusions

This article has presented the analysis and robust controlof a rotary dryer. A multivariable system has beenconsidered, with the outlet moisture of the dried productand the temperature of the exhaust air as output variables.The ow of the incoming wet material and the ow of fuelhave been used as control variables. Linear models havebeen obtained at different operating points from a testednonlinear simulator of the plant. A controllability analysisof the linear system has been carried out to show someconstraints in its performance. A multivariable robustcontrol, based on the H1 mixed sensitivity problem, hasbeen designed for the rotary dryer. Finally, simulation andexperimental results have been presented, which are inagreement with the conclusions provided by the controll-ability analysis.

Acknowledgements

The authors wish to acknowledge CICYT for foundingthis work under Grants DPI2004-06419 and DPI2003-00429.Appendix A. Nonlinear dryer model

40 50 60 (minutes)

40 50 60 (minutes)ontroller 3: step change on Mo reference.

ring Practice 15 (2007) 487500The set of physical parameters and variables involved inthe nonlinear model of the rotary dryer are listed below:Set of parameters and their numerical values:

Cs specic heat of solid (1.5 kJ/kg 1C )Cw specic heat of water (4.180 kJ/kg 1C)Cg specic heat at constant pressure of dry gas

(1.007 kJ/kg 1C)Cst specic heat at constant pressure of water steam

(1.890 kJ/kg 1C)R constant of perfect gas (8314 J/kmol 1C)l latent heat of vaporization at 0 C (2502 kJ/kg)U volumetric heat transfer coefcient

0:3 kW=m3 CV volume of each section 0:201m3rs density of solid 1400 kg=m3mw molecular weight of water (18 kg/kmol)mg molecular weight of gas in the output (29 kg/kmol)

The set of physical variables are given now. Most of themwill appear with the subscript i, meaning that they aredened for every drums slice: i 0 n, where i 0represents variables at drums input (input to the rst slice),while i n means the drums output (output from the lastslice).

Fsi mass ow of dry solid (kg/s)Fwi mass ow of water in the solid (kg/s)

FgFsm

m

m

T

T

X

Y

M

W

hsvvaP

Invaanco

Fp Fs0 Fw0 0:05 kg=s,

an

ARTICLE IN PRESSineeTo Tgn 37 C,

Mo Mn 0:01.The set of equations relating these variables and para-meters are given next:

Dry solid balance (msi and Fsi are unknowns):qmsiqt

Fsi1 Fsi .

Water balance in solid and gas (where mwi , Fwi , msti , Fstiand Wi are unknowns):

qmwiqt

Fwi1 Fwi Wi,

qmstiqt

Fsti1 Fsti Wi.

Dry gas balance:qmgiqt

Fgi1 Fgi .

Energy balance in solid (the overall heat transfercoefcient is known):

qCsmsi Cwmwi Tsi1 qt

UV Tgi Tsi Fs Cs Fw CwTsorder to determine typical operation points for theseriables, the relations of the system inputs (F c and Fp)d outputs (To and Mo) with some of them have to bensidered:i humidity of gas in dry basis (kg water/kg dry gas)

i moisture of solid in wet basis (kg water/kg wetsolid)

i mass ow of evaporated water (kg/s)

ti steam enthalpy (kJ)

pi vaporization speed (1/s)gas pressure (Pa)isti mass of steam in the gas (kg)

si temperature of solid (1C)gi temperature of gas (1C)

moisture of solid in dry basis (kg water/kg drysolid)mgimass ow of dry gas (kg/s)

ti mass ow of steam in the gas (kg/s)

si mass of dry solid (kg)

wi mass of water in the solid (kg)

imass of dry gas (kg)

M.G. Ortega et al. / Control Engi1 i1 i1

FsiCs FwiCwTsi Wihsti . syis, the mean time employed by the solid to cross thedrum:

msi tresFsi .

This time has been estimated through a correlationproposed in Saeman and Mitchell (1954), as a value ofapproximately 25min.

Therefore, it can be observed that there are 13 equationsd 13 unknowns.The relation involving the vaporization speed of thesolid:

Wi vvapmsi .

Finally, the equation that links mass and ow ofdry solid, using the concept of residence time, thatFgi mgi

Mi Xi

1 Xi.

The other static equations are given by the relationbetween the mass of dry gas and the mass of dry solid,which can be expressed as

mgi V msirs

mgmw

mgY i mwP

RTgi 273.There are also other static equations as the unknownfactors relate to one another. It is clear that:

Xi FwiFsi

; Xi mwimsi

,

Yi Fsti ; Yi

msti ,Cstmsti Cgmgi qTgiqt

UV Tgi Tsi Fgi1Cg Fsti1CstTgi1 Tgi WiCstTgi Tsi .and using the previous mass balance equations in thesolid:

Csmsi Cwmwi qTsiqt

UV Tgi Tsi Fsi1Cs Fwi1CwTsi1 Tsi WiCst CwTsi l.

The mass balance equation of the gas:Considering that the enthalpy of the vaporized water is

hsti CwTsi lTsi CstTsi l

ring Practice 15 (2007) 487500 499On the other hand, the dynamics of the combustionstem is quite fast compared to the drums dynamics.

TY

Fg

Q

F

Pa

signals are related. In particular, in this case it can be

m-Analysis and synthesis toolbox users guide (2nd ed.). Natick, MA:The MathWorks Inc.

Brasil, G. C. & Seckler, M. M. (1988). A model for the rotary drying of

granular fertilizers. Proceeding sixth international drying symposium,

pp. 247256.

Camacho, E. F., & Bordons, C. (1995). Model predictive control in the

process industry. London: Springer.

Chiang, R., & Safonov, M. (1998). Robust control toolbox users guide

(2nd ed.). Natick: The MathWorks Inc.

Courtois, F. (1997). Automatic control of drying processes. Computerized

Control of Systems in the Food Industry, 295316.

ARTICLE IN PRESSM.G. Ortega et al. / Control Engineering Practice 15 (2007) 487500500written as

F c F comb 0:00072 kg=s.The new set of equations, governing the combustionprocess is:

Gas balance:Fg0 Fgenv F comb.

Energy balance:FgenvCgenvT env F combHcombZcomb CgTg0Fg0 .

Equation of perfect gas:Fg0

Qg0mgPatRTg0 273

.

Humidity of gas:Y 0 Y env.

References

Arjona, R., Ollero, P., & Vidal, F. (2005). Automation of an olive waste

industrial rotary dryer. Journal of Food Engineering, 68, 239247.

Astrom, K. J. (2000). Limitations on control system performance.

European Journal of Control, 6, 220.c

t atmospheric pressure (Pa)

Again some of these variables of the input and outputcg0ow of dry gas from the combustion sectionm3=s

omb mass ow of combustible (equal to controlvariable F , but time units are seconds) (kg/s)g0 output temperature from the combustion stage(input to the rst drums slice, i 0) (1C)

0 humidity of combustion gas (kg steam/kg dry gas)

0mass ow of dry air from the combustion section(input to the rst drums slice) (kg dry gas/s)FgA number of new parameters and variables appear. The listof parameters is:

Hcomb combustible heating power (39083 kJ/kg comb.)Zcomb combustion performance (0.95)Cgenv specic heat of inlet air (1.006 kJ/kg 1C)

The variables involved in the combustion model are:

Tenv input temperature (1C)Y env input humidity (kg water/kg dry air)

envmass ow of inlet dry air (kg dry air/s)Deich, V. G., & Stalskii, V. V. (1975). Optimum control of drying

processes in a rotary drum dryer. Theoretical Foundations of Chemical

Engineering, 9(1), 8590.

Didriksen, H. (2002). Model based predictive control of a rotary dryer.

Chemical Engineering Journal, 86, 5360.

Douglas, P. L., Kwade, A., Lee, P. L., & Mallick, S. K. (1993). Simulation

of a rotary dryer for sugar crystalline. Drying Technology, 11(1),

129155.

Doyle, J. C., Glover, K., Khargonekar, P., & Francis, B. (1989). State-

space solutions to standard H2 and H1 control problems. IEEETransactions on Automatic Control, 34(8), 831847.

Duchesne, C., Thibault, L., & Bazin, C. (1997). Modelling and dynamic

simulation of an industrial rotary dryer. Developments in Chemical

Engineering & Mineral Processing, 5(3/4), 155182.

Holt, B. R., & Morari, M. (1985). Design of resilient processing plants

VIthe effect of right-half-plane zeros on dynamic resilience.

Chemical Engineering Science, 40, 5974.

Iglesias, P., & Glover, K. (1991). State-space approach to discrete-time

H1 control. International Journal of Control, 54(5), 10311073.Ljung, L. (1986). System identification toolbox. The Math Works, Inc.

Ortega, M. G., & Rubio, F. R. (2004). Systematic design of weighting

matrices forH1 mixed sensitivity problem. Journal of Process Control,14(1), 8998.

Reay, D. (1979). Theory in the design of dryers. Chemical Engineering,

501506.

Rubio, F. R., Bordons, C., Holgado, J., & Arjona, R. (2000). Modelling

and PID control of a rotary dryer. Proceedings of PID2000.

Rubio, F. R., Bordons, C., Holgado, J., Rivas, I. S. (2001). Modelling and

H1 control of a rotary dryer. Proceedings of European controlconference, Oporto, Portugal.

Saeman, W. C., & Mitchell, J. R. (1954). Analysis of rotary dryer and

cooler performance. Chemical Engineering Progress, 59(9), 467475.

Savaresi, S. M., Bitmead, R. R., & Peirce, R. (2001). On modelling and

control of a rotary sugar dryer. Control Engineering Practice, 9,

249266.

Skogestad, S., & Postlethwaite, I. (1996). Multivariable feedback control.

analysis and design. New York: Wiley.

Yliniemi, L. (1999). Advanced control of a rotary dryer. Ph.D. thesis,

University of Oulu.

Zhou, K., Doyle, J. C., & Glover, K. (1996). Robust and optimal control.

Englewood Cliffs, NJ: Prentice-Hall.Balas, G., Doyle, J., Glover, K., Packard, A., & Smith, R. (1995).

Multivariable robust control of a rotary dryer: Analysis and designIntroductionPlant descriptionProcess modelThermodynamic modelNonlinear model validationLinear models

Controllability analysisController synthesisSimulation resultsExperimental resultsConclusionsAcknowledgementsNonlinear dryer modelReferences