ap

B

vers

ersit

rm

presence of recycle streams, can have a signicant impacton the operability characteristics of the process (e.g., [24]). Ignoring operability considerations during process

Most of these operability indicators can be easily com-puted, especially when they refer only to the steady-stateinformation of the process. However, these operabilityindicators are generally limited to linear systems. Also, theyusually reveal the eect on operability of only some processcharacteristics, e.g. non-minimum phase zeros, delays,

* Corresponding author. Tel.: +61 2 9385 6755; fax: +61 2 9385 5966.E-mail address: j.bao@unsw.edu.au (J. Bao).

Journal of Process Control 171. Introduction

Process operability refers to the inherent property of aprocess to achieve acceptable control performance in spiteof unknown but bounded disturbances and model uncer-tainty, using the available manipulated variables and sen-sor measurements [1]. Process design decisions, such asthe number of trays in a distillation column, the energyintegration interconnection between equipment and the

design may lead to costly retro-tting and re-design if theprocess is found to be dicult to control duringcommissioning.

A large number of operability indicators have been pro-posed and developed in the literature to assist in assessingthe operability properties of the process during the designstage. These indicators include the process singular values[1,5,6], the relative gain array (RGA) [7,8], the dynamic rel-ative gain array, the closed-loop disturbance gain [9], etc.Abstract

Current process operability indicators are mostly restricted to linear approximations of the process dynamics. Other operability anal-ysis approaches that have the capability to include full nonlinear process models rely on mixed integer dynamic optimisation techniqueswhich, in general, require large amount of computations. In this paper we propose a dynamic operability analysis approach for stablenonlinear processes that can be readily applied during process design and can be solved eciently using a limited amount of computa-tions. The process nonlinear dynamics are approximated by a series interconnection of static nonlinearities and linear dynamics, repre-sented by the so-called HammersteinWiener models. These type of models can often be obtained during process design where detailedsteady-state nonlinear models are available, combined with some (usually limited) information on the process dynamics. Using anextended internal model control (IMC) framework, we investigate the interaction between the static nonlinearities and linear dynamicson the operability of the process. The framework extends the well-known equivalence between operability and invertibility of linear pro-cesses to nonlinear systems. In particular, by exploiting some results from the theory of passive systems we provide conditions that guar-antee the existence of the inverse of the static nonlinearities. We show that the inverse can be attained inside a specic input/outputregion. This region imposes a constraint on the maximum magnitude of the signals that appear in the closed-loop and represents theeect of the static nonlinearities on the operability of the overall process. Dynamic operability is then quantied using a linear matrixinequality (LMI) optimisation approach that minimises a given performance criterion subject to the constraint imposed by the staticnonlinearities. 2006 Elsevier Ltd. All rights reserved.

Keywords: Process operability; Nonlinear systems; HammersteinWiener model; Passivity; Static nonlinearity; Input constraintsA dynamic operability analysis

Osvaldo J. Rojas a, Jiea School of Chemical Sciences and Engineering, The Uni

b Chancellery, Level 2, 160 Currie Street, The Univ

Received 22 May 2006; received in revised fo0959-1524/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.jprocont.2006.09.001proach for nonlinear processes

ao a,*, Peter L. Lee b

ity of New South Wales, Sydney, NSW 2052, Australia

y of South Australia, Adelaide, SA 5000, Australia

25 August 2006; accepted 9 September 2006

www.elsevier.com/locate/jprocont

(2007) 157172

roceinput constraints, etc. Thus, they may fail to quantify thecombined eect of these factors on dynamic operability.

Dynamic operability indicators for nonlinear processesare still not widely available. This is partly due to the fun-damental complexities of nonlinear dynamics. One (indi-rect) approach to study the operability of nonlinearprocesses is to quantify the degree of nonlinearity of theprocess so as to assess whether linear control is able toachieve the desired performance. This approach has beenpursued using a variety of nonlinearity measures [1014].Even though the question whether linear control is su-cient to control the process is relevant per se, it does notfully identify the inherent operability properties of the non-linear process.

A dierent approach to process operability analysis isbased on mixed integer dynamic optimisation (MIDO).This approach has been explored by several authors, nota-bly by Perkins and co-workers in the context of their simul-taneous process and control design methodology [1517].The aim of this methodology is to determine both the bestprocess design as well as the best control structure andparameters in one single integrated framework. The mostappealing advantage of this approach is that it can incor-porate, with great exibility, a variety of process and con-trol considerations. For example, both economic indexes(e.g. expected total annualised cost) and dynamic perfor-mance indexes (e.g. integral square error) can be included.Structural process and control design decisions can beincluded via boolean variables. Actuator constraints andprocess variables constraints can be considered explicitly.Finally, in principle, the approach can make use of detailednonlinear process models, thus leading to potentially morerealistic analysis results. Unfortunately, the main disadvan-tage of the optimisation approach is that the complexity ofthe resulting optimisation problem grows very quickly evenfor moderately small dimensional processes [17]. Sakizliset al. have conceded in a recent overview of the approach[15] that one of the challenges that lies ahead in this areais the need for a rigorous and ecient solution of theunderlying optimisation problem. To make the problemtractable several simplifying assumptions are usuallyrequired, such as assuming that the process is controlledwith PI controllers, considering only a nite time horizonin the optimisation, etc. These simplications may lead toan unrealistic estimate of the process operability. In addi-tion, the results may depend heavily on the specic timedomain prole assumed for the disturbance and referencesignals.

Recently, Georgakis and co-workers have proposed ageometric approach for process operability analysis[1820]. The advantage of this approach is that it is appli-cable to both linear and nonlinear processes and requiresonly limited computation. In its simplest version the geo-metric approach considers only the steady-state nonlinearinformation of the process [20]. The user denes the avail-

158 O.J. Rojas et al. / Journal of Pable input space (AIS) for the manipulated variables andthe desired output space (DOS) for the controlled vari-ables. Using the nonlinear steady-state information theAIS can be mapped into the output space and comparedto the DOS; equivalently, the DOS can be mapped intothe input space and compared to the AIS. This procedureprovides an indication of whether the desired range ofoperation can be achieved in steady-state. However, theapproach is essentially open-loop. It does not guaranteethat the closed-loop will have acceptable operability prop-erties. An extension to the steady-state geometric approachthat considers the stability of the closed-loop has recentlybeen developed by Rojas et al. [21]. Ekawati and Bahri[22] have also used the geometric interpretation of the oper-ating spaces, such as the DOS and AIS, to simplify thedynamic operability framework [23] for regulatory cases.

In this paper, we propose a methodology to analyse thedynamic operability of stable nonlinear processes. Ourmain objective is to go beyond existing operability analysisapproaches by incorporating explicit information on theprocess nonlinearity. We also aim at developing anapproach that can be readily applied during process designand that can be solved eciently using limited amount ofcomputations.

Detailed steady-state nonlinear models of the processare routinely accessible during process design. These mod-els are well understood and have been extensively testedand validated. Indeed, several commercial process designsoftware, such as Aspen, have a comprehensive libraryof detailed nonlinear steady-state models for the most com-mon process units in industry. On the other hand, only lim-ited information on the process nonlinear dynamics isusually available during the process design stage. However,some studies have shown that the process nonlineardynamics can often be described with sucient accuracyusing static nonlinearities in conjunction with lineardynamics [2427]. Linear approximations to the processdynamics can be obtained from process owsheet datausing, for example, the approach proposed by Lewin andco-workers [28,29]. Thus, in principle, relying only on ow-sheet data, one may derive a nonlinear dynamic model ofthe process based on a series interconnection of static non-linearities and linear dynamics as follows:

y hofGpshifugg 1where hifg : Rm ! Rm and hofg : Rm ! Rm are input andoutput static nonlinearities, respectively, and Gp(s) is a lin-ear multivariable transfer function. The model in (1) is saidto have a HammersteinWiener type structure.

The operability analysis approach described in thispaper considers a process model as in (1) and is based onan extended internal model control (IMC) framework (sim-ilar to Chan et al. [30] for multivariable control design). Inthis way, one can investigate how the static nonlinearitiesand the linear dynamics Gp(s) interact with each other toaect the overall operability of the process. A key conceptrelated to the issue of process operability is that of process

ss Control 17 (2007) 157172invertibility [5,31,1]. The close connection between processoperability and process invertibility is highlighted by the

There exists a large body of literature that has addressedthe question of quantifying in detail what can be achievedwith feedback control when a linear process P admits noexact inverse [41,44,31,45,46,42]. However, only limitedstudies are available for nonlinear processes [47]. The cur-rent results in both the linear and nonlinear case point con-sistently to the conclusion that what is achievable withfeedback control depends almost entirely on the inherentproperties of the plant and is independent of the controller.Thus, operability is an inherent property of the process.

Considering the IMC framework shown in Fig. 1, we

roceIMC framework. Ideally, to achieve perfect control, theIMC controller should implement a perfect inverse of theprocess. However, both theoretical and practical issuespose limitations on the feasibility of implementing theinverse of a dynamic system (e.g. non-minimum phasezeros, time-delays, unstable poles, limited bandwidth ofphysical components, model uncertainties and actuatorconstraints). Thus limitations on process invertibility leadto limitations on closed-oop performance and operability[31].

Of particular interest to our approach is to determinewhen it is possible to calculate the inverse of the static non-linearity h{}. In many cases, especially when h{} is a mul-tivariable nonlinear map, it may be either impractical ornot possible to derive an analytical inverse h1{}. Weaddress the issue by exploiting some results from the theoryof passive systems [3234]. In particular, based on an incre-mental passivity property we derive sucient conditionsthat guarantee the existence of the inverse h1{}. It isshown that h1{} can only be attained inside a specicinputoutput region. This region imposes a constraint onthe maximum magnitude of the signals that appear in theclosed-loop. This constraint represents the eect of the sta-tic nonlinearity h{} on the operability of the overall non-linear process.

To quantify the dynamic operability of the process weconsider the worst-case integral square error (wISE) whenthe disturbances that aect the process are assumed to havebounded energy. A key issue is to determine which systemnorm is the most adequate to express the required con-straints caused by the conditions on the invertibility ofthe static nonlinearity h{}. A good candidate to achievethis objective is the generalised H2 norm introduced byWilson [35] and Rotea [36]. The generalised H2 normhas been found useful in process operability analysis forlinear processes [37] and has the advantage of being amena-ble to very ecient computation via a Linear MatrixInequality (LMI) approach [38,39].

The paper is organised as follows: Section 2 describesthe framework proposed to analyse the dynamic operabil-ity of stable nonlinear processes. Section 3 studies the con-ditions required to guarantee the existence of the inverse ofthe static nonlinearities on the basis of a passivity analysis.Section 4 presents the methodology that we propose toquantify the operability of the nonlinear process. Finally,Section 5 illustrates the application of the proposeddynamic operability approach to a nonlinear neutralisationprocess and a multi-unit reactor-separation process.

2. A framework for dynamic operability analysis

In order to study the dynamic operability of a stablenonlinear process P we will adopt an internal model con-trol (IMC) framework as shown in Fig. 1. The IMC frame-work has several advantages that makes it especially

O.J. Rojas et al. / Journal of Psuitable for operability studies. First, when there is no pro-cess-model mismatch, i.e. P0 = P, the closed-loop relationbetween the process output y(t) and the reference signalr(t) is ane with respect to the IMC controller Q:

yt PQrt 2This facilitates the design of the IMC controller Q to meetthe required control objectives. Notice that in the nonlinearcase, the same closed-loop relation (2) holds except that PQdenotes the series interconnection of the nonlinear systemsP and Q. Secondly, when the process is linear, a necessaryand sucient condition that guarantees the internal stabil-ity of the closed-loop in Fig. 1 is that the IMC controller Qbe stable. Unfortunately, this simple condition cannot beextended directly to the general nonlinear case, and addi-tional assumptions need to be considered [40]. However,the dynamics of nonlinear processes with a Hammer-steinWiener model are linear, thus the stability conditionsfor the IMC closed-loop shown in Fig. 1 are essentially thesame as those for linear systems.

The closed-loop relation in (2) highlights one of the fun-damental properties of feedback control, namely that afeedback control loop implicitly implements the inverseof the process (usually over a limited frequency range).When the process P admits a realisable and stable rightinverse P1 then, in principle, one could select Q = P1

and perfect control [31,1] is attainable, i.e. y(t) = r(t)"t.From an operability point of view, processes that admitperfect control are those with ideal operability character-istics. However, it should come as no surprise that thisideal behaviour is rarely achievable in practice due to sev-eral limitations. These limitations include unstable zeros,time-delays, actuator constraints and model uncertainty[4143,1] and, eectively, they indicate that a perfectinverse of the process P is not attainable.

Fig. 1. Internal model control scheme for stable processes.

ss Control 17 (2007) 157172 159can state the problem of assessing the dynamic operabilityof a nonlinear stable process P as follows:

Dynamic operability assessment problem (Stable nonlin-ear case)

For a given performance criterion for operability, ndthe realisable and stable IMC controller Q that achievesthe best performance.

The above denition does not specify the type of opera-bility analysis carried out on the nonlinear process P. Thisissue lies entirely on the selection of the performance crite-rion for operability. Depending on the specic applicationconsidered, one performance criterion may be more appro-

dicult to solve numerically, let alone being amenable toan analytical solution for QOPT such as (4). In this paper,we propose a methodology to study the dynamicoperability of stable nonlinear processes that can be mod-elled using a series interconnection of static nonlinearitiesand a linear dynamic block such as in (1). To simplifythe discussion, we will focus on the following Hammer-stein-type model:

y Gpshfug 6

160 O.J. Rojas et al. / Journal of Process Control 17 (2007) 157172priate than others. Observe that the optimality impliedwhen referring to the best performance achieved by theIMC controller Q in the denition ensures that the opera-bility is determined by the process inherent properties andnot by the controller. To illustrate this point consider, forexample, the integral square error (ISE) as the chosen per-formance criterion for operability. The operability assess-ment problem in this case amounts to nd Q such that:

Jopt , minQ

Z 10

krt ytk2 dt 3

When the process P is linear and the reference signal r(t) isassumed to be a unit step, the optimal Q that minimises theISE is given by [48,49]:

Qopts P1m s 4where the process P(s) is decomposed as:

P s LsPms 5and Pm(s) represents the minimum phase part of P(s) andL(s) is an all-pass factor. Observe that the optimal IMCcontroller QOPT inverts only the invertible part of the pro-cess, namely its minimum phase factor. This result conrmsthat the operability of the process is independent of thecontroller and is an inherent property of P(s). Anotherinteresting observation is that if the linear process P(s) isminimum phase and admits a right inverse, then the oper-ability analysis based on the ISE yields the resultQOPT(s) = P(s)1 (since Pm(s) = P(s)). Thus perfect con-trol is attainable and JOPT = 0.

Extending dynamic operability analysis to general non-linear processes is a formidable task. One of the main dif-culties is that nonlinear optimisation problems areFig. 2. Closed-loop with extwhere it is assumed that ho{} = I and hi{} = h{}. How-ever, the method can readily be extended to models havingboth input and output static nonlinearities.

One of the key elements of the methodology is the use ofan extended version of the IMC framework shown in Fig. 1in order to take full advantage of the specic structure ofthe Hammerstein-type process model. In particular, weconsider an IMC controller Q formed by the series inter-connection of a linear system Qp(s) and a static nonlinear-ity g{}, i.e.

Q gfQpsg 7

This structure allows one to analyse the operability of thenonlinear process P by separating the inuence of the staticnonlinearity h{} from that of the linear dynamics Gp(s).Notice that the stability of the closed-loop is guaranteedif Qp(s) is stable, regardless of the choice for g{}. However,both static nonlinearities g{} and h{} are required to gen-erate a bounded output whenever a bounded input isapplied.

Fig. 2 shows the equivalent closed-loop when Q has theform shown in (7) and the process P can be approximatedby a Hammerstein-type model. When there is no process-model mismatch, we obtain the following closed-looprelation:

y GpshfgfQpsrgg 8

In the next Section we will study in more detail how theinteraction between the static nonlinearity h{} and the lin-ear dynamics Gp(s) aect the operability of the nonlinearprocess P. We will see that the presence of the static non-linearity h{} in the process model will impose additionalended IMC architecture.

invertible in the entire domain X . We will address the issue

roceu

of nding the invertible region in Xu by exploiting someresults from the theory of passive systems [3234]. Forlimitations on the operability of P compared to the case ofP being a purely linear process P = Gp(s).

3. The eect of the static nonlinearity on dynamic operability

In order to assess the dynamic operability of the processP, we need to determine the best static nonlinearity g{}and best linear transfer function Qp(s) with respect to thechosen performance criterion. In most practical cases, theperformance criterion for operability will include (directlyor indirectly) a term related to the size of the process out-put y(t). This is the case, for example, if we choose the ISEas the performance criterion for operability. Thus, for mostmeaningful performance criteria the best static nonlin-earity g{} is one that implements a right inverse of the pro-cess static nonlinearity h{}, i.e. g{} = h1{}. However,similar to the dynamic case discussed earlier, the inverseof a multivariable nonlinear static map h{} may not existor may be dened only over a limited range.

If the inverse of the static nonlinearity exists and it isknown, then one could select g{} = h1{}, thus eectivelyremoving h{} from the closed-loop. In this case the opera-bility of the process P is only determined by its linear sub-system. However, even in the ideal case when the inverseh1{} exists and is readily available, the nonlinearitymay still impose restrictions on the dynamic operabilityof the overall process P. This issue arises primarily fromthe fact that the static nonlinearity yh = h{u} (see Fig. 2)may be dened only over a certain limited domain of theinput u 2 Rm and/or may span only a limited range ofyh 2 Rm, that is:hfug : Xu Rm ! Xyh Rm 9This is the result of physical limitations in the process, butit can also account for limitations in the actuators, outputconstraints, etc. Accordingly, the known inverse will be de-ned on the following range:

gfrhg , h1fg : Xyh ! Xu 10Under these conditions, the region Xyh places a constrainton the output rh(t) of the linear sub-system Qp(s) inFig. 2. An example of this situation is when the static non-linearity h{u} is a saturation function modelling actuatorlimitations. In this case, the inverse of h{u} is readily avail-able: in the linear range where u 2 [umin,umax] the inverse ofh{u} is the identity, whilst outside the linear range the in-verse is not dened. It is well known that input constraintsmay impose severe restrictions on the operability of anotherwise linear process [31,5054] and, in general, cannotbe ignored.

In many practical cases the static nonlinearity h{} is not

O.J. Rojas et al. / Journal of Pthe sake of completeness we present the following de-nitions:Denition 3.1 (Passive static nonlinearity). A static non-linearity yh hfug : Rm ! Rm is said to be passive if:hfugTuP 0 8u 2 Xu 11where h{u}T is the transpose of the vector h{u}.

Denition 3.2 (Incrementally passive static nonlinearity). Astatic nonlinearity yh hfug : Rm ! Rm is said to be incre-mentally passive if:

hfu Dug hfugTDuP 0 8u; u Du 2 Xu 12Based on these denitions we have the following result:

Theorem 3.3 (Desoer and Vidyasagar (1975)). Considera nonlinear mapping yh hfug : Xu Rm ! Xyh Rm.Assume that h{u} is incrementally passive and that it satisfiesthe following Lipschitz continuity condition:

khfu Dug hfugk 6 ckDuk 13for all u 2 Xu and u + Du 2 Xu with 0 6 c

Hence, we conclude that the Jacobian of the scaled nonlin-earity h 0{u 0} is positive denite for all u 0 inside the local re-gion Ku0 . Based on the denition of H

0u we have that

Ku0 H0u. Thus H0u is not empty. In addition, H0u containsthe point u 0 = u 0* since u0 2 Ku0 . h

Once the region of invertibility of h{} has been identi-ed, the problem of studying the dynamic operability ofa stable nonlinear process P becomes:

Dynamic operability analysis problem: Consider a stable

rocess Control 17 (2007) 157172khfu Dug hfugk ohouDu

6 ohou

kDuk 17

where we have omitted the explicit reference to the point atwhich the Jacobian is evaluated for ease of notation. If welet c k ohou k then (17) shows that the static nonlinearity isLipschitz continuous as per (13). Now, by means of (16)we have that:

hfu Dug hfugTDu DuTohou

T

Du 18

Thus, if there exists a nonempty and convex region Hu de-ned as in (14) in which the Jacobian is positive semi-def-inite, i.e. zT ohou zP 0 8z 2 Hu then the static nonlinearityh{u} is incrementally passive for all u 2 Hu andu + Du 2 Hu. Using the result in Theorem 3.3 we concludethat the static nonlinearity inverse h1{} exists and it is awell dened map from Hrh to Hu, where Hrh is given in(15). This concludes the proof. h

It is sometimes useful to consider a scaled version of thestatic nonlinearity h{}:

yh h0fu0g , hfKu0g 19where u = K u 0 and K is a constant matrix. The reason forthis is that by appropriately choosing the scaling matrix Kit is possible to guarantee that the region Hu in Theorem3.4 is never empty in the vicinity of a certain point of inter-est u = u* 2 Xu. This is shown in the following Lemma.Lemma 3.5. Consider a static nonlinearity yh hfug : Xu Rm ! Xyh Rm with the same assumptionsoutlined in Theorem 3.4. Consider the scaled nonlinearityyh h0fu0g : X0u Rm ! Xyh Rm defined in (19). Let

K ohou

uu

120

for a certain point of interest u = u* 2 Xu. Then the region ofinvertibility H0u of the scaled nonlinearity h

0{u 0} contains thepoint u 0* = K1u* and a non-empty neighbourhood ofu 0 = u 0*.

Proof. Let Ku0 be a small neighbourhood around the pointu 0 = u 0*. In the interior of Ku0 the nonlinear mapping h 0{u 0}can be approximated by the rst term of a Taylor seriesaround u 0 = u 0*, i.e.

yh h0fu0g yh ohou

uu

KDu0; 8u0 Du0 2 Ku0

21Thus,

oh0

ou0 oh

ou

uu

K; 8u0 2 Ku0 22

Substituting (20) in the above expression we have:

162 O.J. Rojas et al. / Journal of Poh0

ou0 I > 0; 8u0 2 Ku0 23nonlinear process P and assume that P can be modelledas in (6). For a given performance criterion, the dynamicoperability of P can be assessed by identifying the real-isable and stable transfer function Qp(s) that achievesthe best performance subject to the constraint:

rht 2 Hrh Xyh 8t 24

We conclude this Section with a brief discussion on howone can calculate the inverse g{rh(t)} = h

1{rh(t)} in a prac-tical set-up. However, we emphasise that the region ofinvertibility of the static nonlinearity h{} dened in Theo-rem 3.4 is independent of the specic method one may wishto use to calculate the inverse. In a practical set-up, onepossibility is to implement the static nonlinearity g{} viaa high-gain closed-loop conguration as shown in Fig. 3.It can be shown see Rojas et al. [21] that if rh(t) is equalto a constant value r* such that r* is contained inside theregion Hrh then the closed-loop in Fig. 3 is asymptoticallystable. Although obtained using a Lyapunov stability argu-ment, the result in Rojas et al. [21] relies on an incrementalpassivity condition for h{} similar to that required in The-orem 3.4 and 3.3. The asymptotic stability of the closed-loop in Fig. 3 implies that, in steady-state, y 0hss r, henceuss = h

1{r*}. Thus, the conguration in Fig. 3 achieves animplicit inverse of the static nonlinearity h{} in steady-state. Observe that with the implementation depicted inFig. 3, g{} is, strictly speaking, no longer a static nonlin-earity but becomes a dynamic nonlinear system. However,from an inputoutput point of view the relationu(t) = g{rh(t)} is, essentially, static provided the closed-loop in Fig. 3 can be made arbitrarily faster than thedynamic response of Qp(s). This can be achieved by usingan arbitrarily large gain e > 0 in Fig. 3.Fig. 3. Implementation of g{} via a high-gain feedback loop.

tifying in detail the operability of P. The denition of the

rocedynamic operability assessment problem requires the signalrh(t) to be constrained inside a certain pre-specied region.This constraint on rh(t) represents the inuence that the sta-tic nonlinearity h{} has on the operability of the overallnonlinear process P. Intuitively, we see that the smallerthe invertibility region Hrh the stronger the limitationimposed by the static nonlinearity on the operability ofthe process. Hence, a key issue in quantifying the dynamicoperability of P is to nd an eective and ecient way todeal with the constraint rht 2 Hrh Xyh .

One possible approach is to consider the constraintrht 2 Hrh explicitly by solving a constrained optimisationproblem in the time domain. This is similar to the optimi-sation approach adopted in the simultaneous process andcontrol design methodology of Perkins and co-workers[1517]. However, as mentioned earlier, the solution tothese type of nonlinear constrained optimisation problemsrequires a large number of computations and, therefore, islimited to relatively small-dimensional systems.

An alternative is to solve an optimisation problem in thefrequency domain, replacing the time-domain hard con-straint rht 2 Hrh by a related (though generally more con-servative) constraint on the system gain of the IMCcontroller Qp(s). This approach requires a limited amountof computations and can be easily implemented using com-mercial numerical computation software (such as Mat-lab). The key issue is to determine which system normis appropriate to approximate the hard time-constraintrht 2 Hrh .

To discuss this point in more detail we rst need todene a reference point against which all vector and systemnorm calculations are performed. Thus, without loss ofgenerality, it is assumed that the process P initially restsat a given operating point u; yh; y. Based on the chosenoperating point and the available information on the larg-est expected value of each process variable, we dene a newset of scaled variables given by

~ut , ut u

umax u

~yht , yht yh

yhmax yh~yt , yt y

ymax y

25

Accordingly, we also have that:

~rt , rt y

ymax y 264. Quantifying process operability via LMI synthesis

Given the dynamic operability analysis frameworkdescribed in Section 3, we now focus on the issue of quan-

O.J. Rojas et al. / Journal of P~rht , rht yhyhmax yhFrom this point onwards we assume that Qp(s), Gp(s), g{}and h{} have been scaled based on the above denitions.Similarly, the constraint region Hrh is also assumed to bescaled accordingly. We now consider the case when theoutput disturbance d(t) in Fig. 2 has zero mean and canbe scaled as follows:

~dt , dtdmax

27

Thus, with reference to the IMC closed-loop in Fig. 2, wehave that ~rt ~dt, since it is assumed that r(t) = y*.

A natural choice to express the constraint rht 2Hrh Xyh is to consider the innity norm of ~rht, i.e.

k~rhtk1 , maxs maxi j~rh;isj

28

To illustrate, assume that ~rht has only two dimensions.Thus, to guarantee ~rht 2 Hrh one needs to nd the largestsquare centred at the origin and completely inscribed inHrh . If 2b > 0 is the length of the sides of the square, thenwe require k~rhtk1 6 b. In addition, it is clear from thescaling of the disturbance in (27) that k~dtk1 6 1. We con-clude that ~rht 2 Hrh if we impose the following constrainton the system L1 norm of the IMC controller Qp(s):

kqptk1 ,Z 11

qptdt sup~dt

k~rhtk1k~dtk1

6 b 29

where qp(t) is the impulse response of Qp(s). Unfortunately,the results that can be obtained using the systemL1 normare, in general, very conservative [49]. Also, the system L1norm is not amenable to ecient and simple computation[56]. A second choice is to consider a slight variation ofthe innity norm of ~rht dened in (28). In particular, con-sider the following signal norm:

k~rhtk12 , supsk~rhsk 30

where, in this case, kk is the Euclidean vector norm, i.e.k~rhsk

~rhsT~rhs

q. To illustrate, consider again a

two dimensional example. Thus, to guarantee ~rht 2 Hrhone needs to nd the largest circle centred at the originand completely inscribed in Hrh . If a > 0 is the radius ofthe circle then we require k~rhtk12 6 a. Fig. 4 shows agraphical interpretation of the signal norms k~rhtk1 andk~rhtk12 discussed here when ~rht has only two dimen-sions and when ~rht is constrained inside an arbitrary re-gion Hrh Xyh . Fig. 4 shows that the innity-2 normk~rhtk12 is only slightly more conservative than the inn-ity norm k~rhtk1. Fig. 4 also shows that a time domainoptimisation approach that considers the constraint~rht 2 Hrh explicitly may, in principle, obtain dynamicoperability results that are more realistic. This is because~rht is allowed to vary inside the whole region Hrh andnot only inside its inner square or circle approximations.However, as discussed earlier, a full time-domain optimisa-

ss Control 17 (2007) 157172 163tion approach exhibits other sources of conservatism(e.g. nite horizon, xed disturbance prole, convex

roceapproximation of the constraints, etc.) and requires largeamount of computation. The degree of conservatism intro-duced by the innity-2 norm approximation in (30) of thetime domain constraint ~rht 2 Hrh will clearly depend onthe specic shape of the constraint regions and the specicoperating point u; yh; y chosen for operability analysis.

In a two dimensional case the required value of a > 0can be easily found by direct inspection. In a higher dimen-sional case a can be determined by calculating the vectornorm of each point (in practice, of a grid of points) onthe contour of the constraint regions Hrh and then assign-ing to a the smallest of these values. If it is assumed thatthe disturbance signal ~dt has bounded energy:

k~dtk2 ,Z 10

~dtT~dtdts

t 0, when the input signal~dt has bounded L2 norm. Thus, to guaranteek~rhtk12 6 a one can impose the following constrainton the generalised H2 norm of Qp(s):

kQpskg 6 at1 33One important advantage of the generalised H2 systemnorm is that it can be computed eciently using a LinearMatrix Inequality (LMI) formalism [38] in combinationwith convex optimisation routines [39]. Thus, we suggestto use the generalised H2 system norm to approximatethe hard time-constraint rht 2 Hrh Xyh .

The main performance criterion for operability that wewill consider in our study is the worst-case integral squareof the error signal ~et ~rt ~yt (wISE). When~rht 2 Hrh we have that g{} = h1{} and we can write:y^s I GpsQpsd^s 34where y^s is the Laplace transform of the scaled variable~yt and d^s is the Laplace transform of ~dt. Thus the er-ror signal is given by

e^s I GpsQpsd^s 35and the worst-case integral square error (wISE) is given bythe H1 norm of the sensitivity S(s) = I Gp(s)Qp(s), i.e.

wISE , kSsk1 kI GpsQpsk1 sup~dt

k~etk2k~dtk2

36

Both theH1 norm of the sensitivity S(s) and the general-ised H2 norm of the IMC controller Qp(s) can be com-puted if some knowledge about the energy of thedisturbance signal ~dt exciting the closed-loop is availableand provided this energy is bounded, i.e. k~dtk2 t

Qps KsI GpsKs1 39where K(s) is an equivalent closed-loop linear controller.The process linear dynamics Gp(s) are then extended usinga state-space generalised plant G(s) formalism given by[57]:

40where z(t) is a two component vector that contains the out-

O.J. Rojas et al. / Journal of Proceput signals that we wish to minimise or constrain (in ourcase the error ~et and the input ~ut to the linear plantGp(s)) whilst v(t) is the variable available for feedback tothe controller K(s). Fig. 5 shows a feedback interpretationof the parameterisation of Qp(s) required for LMI synthe-sis. The shaded region indicates the generalised plant G(s)considered in this case. In particular z(t) is given by

zt zet~ut

41

where ze(t) is a ltered version of the error ~et. The lterWe(s) is taken to be:

W es diag 1s u

u 1 42

This choice of We(s) is required to guarantee that the opti-mal IMC controller Qoptp s that minimises the wISE per-formance criterion in (37) satises:

Qoptp 0 Gp01 43so that, in steady-state, when r = y* we have:

rhss Qoptp 0y Gp01y yh 44as required by the choice of the operating point of interestu; yh; y. Next, assume that the linear dynamics Gp(s)have the following state-space representation:Fig. 5. Parameterisation of IMC controller Qp(s) for LMI synthesis.45

Observe that Gp(s) is required to be strictly proper in orderto guarantee the well-posedness of the closed-loop in Fig. 5[39]. Similarly, let a state-space model of the lter We(s) in(42) be:

46

Then, with reference to Fig. 5, the generalised plant G in(40) is readily seen to be:

47

It can be shown that theH1 norm in (37) is smaller thanc > 0 if and only if there exists a symmetric matrix P suchthat the following linear matrix inequalities are satised[39]:

ATP P A P B CTBTP cI DTC D cI

0B@

1CA < 0

P > 0 48where A; B; C; D is a state space representation of the sen-sitivity S(s). Similarly, the generalisedH2 norm of Qp(s) issmaller than a > 0 if and only if there exists a symmetricmatrix P such that the following linear matrix inequalitiesare satised [39]:

~ATP P ~A P ~B~BTP I

!< 0

P ~CT

~C a2I

!> 0

~D 0

49

where, in this case, ~A; ~B; ~C; ~D is a state space representa-tion of Qp(s). Using the LMIs in (48) and (49) in combina-tion with the generalised plant G(s) description in (47) andthe parameterisation of Qp(s) in (39) gives rise to a set of bi-linear matrix inequalities (BMIs). Unfortunately bi-linearmatrix inequalities cannot be solved eciently. However,the BMIs can be converted into a set of LMIs by usingthe following variable transformations similar to those pro-posed by Scherer et al. [39]:

A^ , NAKMT NBKCX YBCKMT YA B2DKC2XB^ , NBK YB2DKC^ , CKMT DKC2XD^ , D

ss Control 17 (2007) 157172 165K

50

and

Linear Matrix Inequalities are symmetric; thus the termswritten with * in the above expressions need to bereplaced by the transpose of the corresponding sub-matri-ces below the main diagonal. The variables in bold face arethe decision variables in the optimisation. Finally, we pro-

synthesis in Eqs. (51)(55) and compute the H1 normof the achieved sensitivity function S(s).

u; yh; y, repeat steps 16 with a new Hammersteinmodel for each new process design. However, to makethe results comparable use the same scaling for eachdesign in step 2.

8. Compare theH1 norm of the achieved sensitivity func-

B1 B2D^D21T YB1 B^D21T cI C1eX D12eC^ C1e D12eD^C2 D11e D12eD^D21 cI

B@ CA

AX XAT B2C^ B2C^T A^ A B2D^C2T ATY YA B^C2 B^C2T B1 B2D^D21T YB1 B^D21T I

0B@

1CA < 0 53

C1uX D12uC^ C1u D12uD^C2 a2I

0@

1A 54

D11u D12uD^D21 0 55

rocevide a step by step procedure that summarises the dynamicoperability analysis for stable nonlinear processes proposedin this paper.

Procedure 4.1 (Dynamic operability analysis for nonlinearsystems).

1. Approximate the nonlinear process P using a Hammer-stein-type model as in (6).

2. Select an operating point u; yh; y at which to performthe operability analysis.

3. Scale the process model based on (25) and (26).4. Consider the scaled static nonlinearityX I Y B C > 04.2. Dynamic operability analysis (wISE) via LMI synthesis

minA^;B^;C^;D^;X;Y

c 51

subject to the Linear Matrix Inequalities:

AX XAT B2C^ B2C^T A^ A B2D^C2T ATY YA B^C2 B^C2T

0BBBwhere (AK,BK,CK,DK) are the state-space matrices of thecontroller K(s) in (39), N and M are auxiliary variablesand X and Y are auxiliary decision variables. With thesevariable transformation, solving the dynamic operabilityanalysis (wISE) in (37) and (38) is equivalent to solvingthe following convex optimisation problem:

166 O.J. Rojas et al. / Journal of Pyh h0fu0g , hfKu0g 56tions S(s) for each operating point of interest or alterna-tive process design.

5. Illustrative examples and discussion

To illustrate the dynamic operability analysis for nonlin-ear processes described in this paper, we present two casestudies. The rst example is a neutralisation process similarto that studied by Lakshminarayanan et al. [25]. These typeof processes have been shown to be adequately described7. Repeat steps 46 for as many dierent operating pointsas required. Alternatively, if dierent process designsneed to be compared for the same operating point

1CCC < 0 52with K chosen as in (20). Identify the regionHrh h0fH0ug in which the static nonlinearity h 0{u 0} isguaranteed to be invertible based on the result in Theo-rem 3.4.

5. Calculate the required upper bound a > 0 for the gener-alised H2 norm of Qp(s) assuming k~dtk2 1.

6. Solve the convex optimisation problem based on LMI

ss Control 17 (2007) 157172by Hammerstein models such as that in (6). The secondexample considers a more complex multi-unit reactor-sep-

C 0:8054 0:8953 0:0219 0:0727 58

10.5

00.5

1

10.5

00.5

13

2

1

0

1

u1 Acidu2 Base 1

0.50

0.51

1 0.5

00.5

12

1

0

1

2

3

u1 Acidu2 Base

Fig. 6. Static nonlinearity h{} for the neutralisation example: (a) yh1 and (b) yh2.

O.J. Rojas et al. / Journal of Process Control 17 (2007) 157172 167while the static nonlinearity h{} is given byaration process similar to that investigated by Samyudiaet al. [58] and Lee et al. [59].

5.1. Neutralisation process

The state space matrices that dene the linear dynamicsGp(s) of the process are given by

A

0:0283 0:0083 0:0045 0:00160:0120 0:0345 0:0049 0:00480:3695 0:0700 0:0874 0:02180:8926 0:6055 0:0807 0:1669

266664

377775

B

0:0199 0:0059

0:0103 0:02690:2560 0:10720:6474 0:1380

266664

377775

57

1:8553 0:4552 0:1235 0:0078

2 1 0 1 21

0.5

0

0.5

1

1.5

2

2.5

3

y1

y 2

Fig. 7. Achievable output space (AOS) in the process output spacyh1 u1 0:3056u2 0:349u21 0:1719u22 0:5663u1u2 0:0322u31 0:0326u32 0:1987u1u22 0:2144u21u2;

yh2 0:0031u1 u2 0:2896u21 0:1983u22 0:5149u1u2 0:0356u31 0:0677u32 0:2032u1u22 0:1452u21u2;

59

The process inputs are the acid (u1) and base ow rates (u2),whilst the process outputs are the level (y1) and the pH (y2)of the liquid in the well stirred neutralisation tank. Allthese variables are deviation variables from the nominaloperating conditions. The time base is seconds. To performthe operability analysis we select the operating point ofinterest to be the origin (Step 2). The model has beenappropriately scaled, thus we can omit Step 3 in this exam-ple. Fig. 6 shows the static nonlinearity h{} when the in-puts u1 and u2 are contained in the available input space(AIS) dened by the following region:

AIS , fu1; u2j 1 6 u1 6 1 and 1 6 u2 6 1g60

The result of mapping the AIS into the process output space

and the static nonlinearity output space is shown in Fig. 7.

3 2.5 2 1.5 1 0.5 0 0.5 11.5

1

0.5

0

0.5

1

1.5

2

2.5

yh1

yh2

e (y1y2) and in the static nonlinearity output space (yh1yh2).

The static nonlinearity h{} in (59) is not amenable to asimple analytical inversion. Thus, we will use the result ofTheorem 3.4 to determine the regions Hu (dened in theinput space of the static nonlinearity, see (14)) and Hrh(dened in the output space of the static nonlinearity) inwhich an inverse of the static nonlinearity exists and is welldened. We nd that, in this example, the region Hu coin-cides with the available input space (AIS) in (60). Thus, theregion Hrh coincides with the achievable output space(AOS) shown in Fig. 7(b). We conclude that, in this case,the condition that guarantees the existence of the staticnonlinearity inverse does not impose any additional con-straint to that already imposed by the static nonlinearityoutput range Xyh , AOS since Hrh Xyh .

Next, we proceed with Step 5 and estimate the radius aof the largest circle centred at the origin that is entirelyinscribed inside the constraint region Hrh . This is shown

A + B! P + G takes place in the liquid phase. It isassumed that component B is completely consumed bythe reaction. The majority of the by-product G is extractedin a single stage liquidliquid extractor using solvent C.The ranate from the extractor is a mixture of A and P.A distillation column separates the product P as the over-head stream while the bottom ow of A is discarded. Thereactor-separation process considered in this example dif-fers from that in [58,59] in that the bottom ow of compo-nent A from the distillation column is not recycled into theCSTR. In addition, the process model does not include aash-drum unit used to purge the by-product G from theextract ow.

A detailed description of the nonlinear dynamics of thismulti-unit process can be found in [59], including all therelevant assumptions and parameters values. The processnonlinear model contains 17 states and up to 5 inputout-put pairs. For illustration we will consider only 2 inputs

168 O.J. Rojas et al. / Journal of Procein Fig. 8 where it is found that a = 0.67. Observe fromFig. 8 that using the generalised H2 norm of Qp(s) toapproximate the constraint ~rht 2 Hrh introduces a certaindegree of conservatism. However, if we were to use theinnity norm of ~rht, as in (28), to approximate the con-straint we would have to inscribe a square (instead of a cir-cle) centred at the origin inside the region Hrh . In eithercase, we see that the results would be comparable. Still,we prefer to use the generalisedH2 norm due to its ecientcomputation via LMIs.

Based on the estimated upper bound a of the generalisedH2 norm of Qp(s) we solve the dynamic operability analy-sis (worst-case ISE) via LMI synthesis in Eqs. (51)(55).The solution to the optimisation problem yields a worst-case ISE of:

kSsk1 kI GpsQpsk1 1:6663 61

Fig. 9 shows the largest singular value in the frequency do-main of the achieved sensitivity S(s) = I Gp(s)Qp(s). Thisfrequency domain response provides additional informa-

3 2.5 2 1.5 1 0.5 0 0.5 11

0.5

0

0.5

1

1.5

2

2.5

y

y h2

rh

h1

Fig. 8. The largest circle inscribed in Hrh .tion on the performance of the closed-loop. In particular,for a given frequency, it indicates the level of attenuation(or amplication) of the output disturbance ~dt achievedby the closed-loop. In addition, the plot in Fig. 9 providesan indication of the achieved closed-loop bandwidth (w0)

or, equivalently, the achieved closed-loop settling time

4w0

. In this example, we see that w0 2 102 rad/s;thus the closed-loop achieves a settling time of approxi-mately 200 s.

5.2. Reactor-separation process

The reactor-separation process considered in this exam-ple is shown in Fig. 10. A similar multi-unit process wasstudied by Samyudia et al. [58] and Lee et al. [59]. A briefdescription of the process follows, which is adopted from[59]: components A and B are fed into a jacketed CSTRwhere a rst order, endothermic, irreversible reaction

10-4 10 -3 10 -2 10 -1 10 0 10 1-50

-40

-30

-20

-10

0

10

Frequency (rad/s)

Max

imum

sin

gula

r Val

ue (d

B)

Fig. 9. Frequency response of the sensitivity S(s) = I Gp(s)Qp(s) for theneutralisation example.

ss Control 17 (2007) 157172and 2 outputs. The inputs are the reux owrate L(u1)and fresh feed F4 of component A into the CSTR(u2).

Steam Reactor

Solvent C Extract

Extractor

Column

rato

roceFig. 10. Reactor-sepaF4 B feed

O.J. Rojas et al. / Journal of PThe outputs are the composition of A in the reactor xA3(y1)and the distillate composition xD(y2). The time base ishours.

The process nonlinear dynamics are approximated usinga Hammerstein-type model as in (6) (Step1). Fig. 11 showsthe static nonlinearity h{} of the model. The lineardynamic part Gp(s) of the model is given by the followingtransfer function matrix:

The operating point selected for operability analysis (Step2) is given by

u 548:5183

; yh

40:67

56:9

; y 0:29

0:9

63

After appropriately scaling the model variables (Step 3) wecompute the region Hrh in which the static nonlinearity h{}

Gps 0 0:0068098s223:3s1:274s234:8

10:7518s48:68s148:5s227:9s106:2s43:12s4:051

2:2805s0:1355s23:862s3s43:12s4:051s1:897s1:27

24

540545

550555

560

6070

8090

10030

35

40

45

50

55

60

L (kmol/h)

yh1

F4 (kmol/h)

1

Fig. 11. Static nonlinearity h{} for the reactoSteam

r process owchart.Reflux L

ss Control 17 (2007) 157172 169is guaranteed to be invertible (Step 4). For convenience,only points inside the Desired Input Space (DIS) shownin Fig. 12(a) are considered to compute Hrh . The DIS inFig. 12(a) maps into a rectangular Desired Output Space(DOS) given by

DOS , fy1; y2j 0:2 6 y1 6 0:35 and 0:8 6 y2 6 1g 64

Fig. 12(b) shows the invertibility regionHrh obtained in thisexample. Observe that due to the scaling of the model vari-ables (Step 3) the selected operating point in (63) is nowmapped into the origin. The required upper bound a > 0for the generalisedH2 norm of Qp(s) is given by the radiusof the largest circle entered at the origin and inscribed inHrh (Step 5). This is also shown in Fig. 12(b). We have that

:8984s1:069

35 62

540545

550555

560

6070

8090

0020

30

40

50

60

70

80

L (kmol/h)

yh2

F4 (kmol/h)

r-separation process: (a) yh1 and (b) yh2.

95l/h)

-15 -10 -5 0 5 10 15

roce540 542 544 546 548 550 552 554 55665

70

75

80

85

90

u2

: Fe

ed fl

owra

te F

4 of

com

pone

nt A

(kmo

u*100Desired Input Space (DIS)

170 O.J. Rojas et al. / Journal of Pa = 9.9. Based on the estimated upper bound a of the gen-eralisedH2 norm of Qp(s) we solve the dynamic operabilityanalysis (worst-case ISE) via LMI synthesis in Eqs. (51)(55). The solution to the optimisation problem yields aworst-case ISE of:

kSsk1 kI GpsQpsk1 1:5938 65

Fig. 13 shows the largest singular value in the frequencydomain of the achieved sensitivity S(s) = I Gp(s)Qp(s).Observe that the achieved closed-loop bandwidth (w0) isw0 4 101 rad/h.

6. Conclusion

In this paper, we have described a dynamic operabilityapproach for stable nonlinear processes. The approach

u1 : Reflux flowrate L (kmol/h)

Fig. 12. Desired Input Space (DIS) and the invertibility region Hrh

104 103 102 101 100 101 102 10380

70

60

50

40

30

20

10

0

10

Frequency (rad/h)

Max

imum

sin

gula

r Val

ue (d

B)

Fig. 13. Frequency response of the sensitivity S(s) = I Gp(s)Qp(s) for thereactor-separation example.can be readily applied during process design and can besolved eciently using a limited amount of computations.By exploiting a Hammerstein/Wiener model approxima-tion to the process dynamics in combination with anextended IMC framework we have investigated the interac-tion between the static nonlinearities and the systemdynamics on the operability of the overall process. We haveshown that the eect of the static nonlinearities on opera-bility can be translated into a time domain constraint onthe signals that appear in the closed-loop. The dynamicoperability of the process is then quantied using a linearmatrix inequality (LMI) optimisation approach that mini-mises the required performance criterion subject to the con-straint imposed by the static nonlinearities.Scaled yh1

of the static nonlinearity h{} for the reactor-separation process.-15

-10

-5

0

5

10

Scal

ed y

h2

rh

ss Control 17 (2007) 157172Acknowledgements

The support of the Australian Research Council (GrantDP0558755) is gratefully acknowledged.

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172 O.J. Rojas et al. / Journal of Process Control 17 (2007) 157172

A dynamic operability analysis approach for nonlinear processesIntroductionA framework for dynamic operability analysisThe effect of the static nonlinearity on dynamic operabilityQuantifying process operability via LMI synthesisDynamic operability analysis for nonlinear processes (wISE)Dynamic operability analysis (wISE) via LMI synthesis

Illustrative examples and discussionNeutralisation processReactor-separation process

ConclusionAcknowledgementsReferences