optimal reference trajectory shaping and robust gain-scheduling for transition control of nonlinear...

Optimal Reference Trajectory Shaping and Robust Gain-Scheduling forTransition Control of Nonlinear Processes

Pietro Altimari,*,† Lucia Russo,‡ Erasmo Mancusi,§ Mario di Bernardo,| and Silvestro Crescitelli⊥

Dipartimento di Ingegneria Chimica e Alimentare, UniVersita di Salerno, Via Ponte Don Melillo,84084, Fisciano (SA), Italy, Istituto sulle ricerche sulla combustione, CNR, Piazzale Tecchio 80, Napoli,80125, Italy, Facolta di Ingegneria, UniVersita del Sannio, Piazza Roma, 82100, BeneVento, Italy, Dipartimento diInformatica e Sistemistica, UniVersita “Federico II” di Napoli, Via Claudio 21, 80125 Napoli, Italy, andDipartimento d’Ingegneria Chimica, UniVersita “Federico II” di Napoli, Piazzale Tecchio 80, 80125 Napoli, Italy

A novel method for transition control of nonlinear processes is presented. Gain scheduling is implemented toensure stability and desired output behavior over the operating region of interest while transition between theinitial and the final steady state is achieved by predictive reference control (or reference governor). In thisframework, the search of a feasible reference input sequence is performed so that the closed-loop systemmoves along an optimal curve of steady states. This curve is constructed off-line so that its points lie far fromthe border of the process constraint set and correspond to satisfactory controllability properties of theuncontrolled plant. Moreover, bifurcation analysis of the closed-loop system is performed by varying thereference signal according to the selected curve. In this way, regions of state multiplicity and/or instabilityare identified enabling the choice of the controller parameters values preventing transitions to undesired solutionregimes. The method is validated on a simulated problem of start-up control of a jacketed continuous stirredtank reactor.

1. Introduction

Large changes of the operating conditions typically occurduring the operation of chemical processes. In polymerizationreactors, for example, operating conditions are modified to attainproduct specifications responding to different market demands.1,2

Products obtained during these transients do not meet generallythe required specifications and are recycled or sold off atconsiderably lower market price. Hence, the period needed forthe plant to reach the desired steady state must be minimizedto reduce the amount of off-specification products. On the otherhand, moving nonlinear processes over wide regions of the stateand parameter space imposes severe control difficulties. Theseprimarily result from the need to account for system nonlin-earities during the design of the closed-loop system ruling outthe possibility to directly apply established linear control tools.Furthermore, obstacles may occur due to the presence of processconstraints. These impose bounds on state and/or manipulatedvariables and, if violated, may lead to undesired phenomenaas, for example, plant shut-down and thermal runaways.3,4

The design of controllers performing constrained transitionsof nonlinear processes has been the object of much researcheffort. Model predictive control is recognized, in this context,to provide an effective approach to handle input and/or stateconstraints within an optimal control setting.5 Currently, modelpredictive control schemes are available stabilizing nonlinearprocesses around one a priori known steady state (see, forexample, (Chen and Allgower)6). However, no stability guar-antees are provided when this strategy is implemented to controltransitions between different steady states. Furthermore, theimplementation of model predictive control laws relies on the

online solution of a constrained optimization problem whichcan be cumbersome or even impossible to solve. Hence, modelpredictive controllers are typically used, in industrial practice,to control nonlinear processes around a fixed steady state andare introduced only when plant stabilization has been prelimi-narily achieved via standard feedback regulators as, for example,PID controllers.

Alternatively, the problem of controlling process transitionscan be handled by adaptive control techniques.7 These arecharacterized by the use of control schemes involving a feedbackcontroller and an adaptation strategy and have been proved tobe effective for controlling nonlinear processes over wideoperating regions. Nevertheless, they can be difficult to imple-ment because of the complexity in the controller design.

A simpler approach relies on the implementation of gainscheduling8 (GS). In its standard formulation, this technique isbased on the use of a family of linear feedback controllers, eachof them guaranteeing stability and desired output behavioraround a different steady state. Plant steady states and localfeedback controllers are, in this context, parametrized by asuitable set of reference variables, typically function of outputvariables. Hence, the transition between two given steady statesis achieved by a step change of the reference signal issued tothe closed-loop system. This can be thought of as a switchbetween the local feedback controllers associated to the initialand the final steady state and may cause constraints violationdue to large changes in process dynamics. This can be preventedby adding to the closed-loop system a predictive reference (PR)controller (or reference governor (RG)) which modifies onlinethe reference signal so as to fulfill the constraints over a finitetime horizon.9-12

PR-control entails computing a reference input sequenceenforcing process constraints based on the prediction of thefuture evolution of the closed-loop system. This can be done,for example, by constrained minimization of a process costfunction over a finite time horizon. However, this may requiresignificant computational resources and does not ensure, in

* To whom correspondence should be addressed. E-mail: [email protected].

† Universita di Salerno.‡ CNR.§ Universita del Sannio.| Universita degli Studi di Napoli “Federico II”.⊥ Universita “Federico II”.

Ind. Eng. Chem. Res. 2009, 48, 9128–91409128

10.1021/ie9001553 CCC: $40.75 2009 American Chemical SocietyPublished on Web 08/31/2009

general, transition to the desired steady state. A more effectiveapproach relies on the parametrization of the reference inputs.In this context, the plant is constrained to move along aprescribed curve of steady states. This restricts the search of afeasible reference input sequence to a one-dimensional subsetof the reference variables domain. As a result, a significantreduction of the computational burden is achieved. Moreover,convergence to the desired steady state can be, in this context,proved provided that the steady states of the selected curve fulfillprocess constraints.9,10 However, it must be emphasized that,with this approach, the evolution of the controlled plant isprimarily determined by the choice of a specific curve of steadystates. In fact, if steady states close to the border of the processconstraint set are selected, undesired reductions of the allowedvariations of the manipulated variables are observed. This mayin turn lengthen the transient needed for the plant to reach thedesired steady state.

Despite the practical relevance of these arguments, no studieshave been performed of methodologies to adequately computecurves of steady states for PR-control applications. Typically,reference inputs are restricted to the segment connecting thepoints of the reference variables domain associated to the actualand the desired steady state.9,10 In general, no guarantee aboutthe optimality of the steady states identified by this segmentcan be provided. These steady states may result, indeed, nearor even crossing the boundaries of the process constraint setand/or correspond to unsatisfactory controllability properties ofthe uncontrolled plant.

A further obstacle to the combined application of GS andPR-control is the lack of methodologies for systematicallyassessing the global stability features of the GS-closed-loopsystem. In practice, the global performance evaluation of GS-controllers is based on simulation studies.8 However, thisapproach may fail to detect the presence of multiple stablesolution regimes or identify robustness margins of the desiredsteady states from instability boundaries. In these conditions,sudden variations of the reference signal may cause transitionto undesired regimes.

In this paper, a novel approach to control transitions ofnonlinear processes is presented. The proposed method makesuse of GS and PR-control ideas and extends approaches fromnonlinear dynamics and flight control which appeared in theliterature.13-15 In this context, feedforward control is imple-mented to force the trajectories of nonlinear systems to track aprescribed curve of steady states and parametric continuationis performed to compute a parametrized family of feedbackcontroller matrices ensuring closed-loop global stability. Inspiredfrom this idea, we here propose to implement PR-control byperforming the selection of a feasible reference input sequenceso that the plant moves along an optimal curve of steady states.In particular, as in Wang et al. and Richardson et al.,14,15 weuse parametric continuation to select the gains of the GS-controller but, at the same time, use optimization techniques toselect (or tailor) the curve of steady states to be tracked in orderto fulfill a set of control requirements. Thus, an optimal curveis computed so that its steady states lie far from the border ofthe process constraint set and correspond to satisfactory control-lability properties of the uncontrolled plant. Moreover, bifurca-tion analysis is performed by varying the reference signalaccording to the computed curve. In this way, regions of statemultiplicity and/or instability are identified providing guidelinesto select the controller parameters values preventing transitionsto undesired regimes.

The paper is structured as follows. In section 2, guidelinesto design PR and GS-controllers are summarized and limits ofthe current approach to the combined application of these twotechniques are analyzed. In section 3, a qualitative descriptionof the proposed approach is presented. In section 4, the problemof finding an optimal curve of steady states is formulated. Insection 5, the application of bifurcation analysis to the designof the GS-closed-loop system is discussed. In section 6, resultsof the application of the proposed approach to a simulatedproblem of start-up control of a jacketed continuous tank reactor(CSTR) model are described. Final remarks end the paper.

2. Control Structure Design

In this section, guidelines to design PR and GS-controllers aredescribed. In particular, the main elements of classical GS aresummarized in subsection 2.1. Then, the PR-control schemeproposed by Bemporad9 and its application to the class of problemsconsidered in this paper are discussed in subsection 2.2.

2.1. Gain Scheduling. Let the plant to be controlled bedescribed by the following nonlinear system:

where x(t) ∈ Rn is the state vector, y(t) ∈ Rp is the output vector,and u(t) ∈ Rm is the input vector. State and input variables areassumed to be constrained within the following compact subsetof the state-parameter space:

We define a plant steady state or operating point as a point(xs,us) such that f(xs,us) ) 0 and refer to the set

as the set of steady states of system 1. If f is continuouslydifferentiable and rank[Dxf, Duf] ) n, the set E is described byan m-dimensional manifold of the state-parameter space. In thefollowing, we will refer to this manifold as the bifurcationmanifold. To simplify notation, we introduce the single variablep ) (x,u) to denote the n + m state and input variables.

We assume that the following parametrization for the set Eis available:

with σ referred to as the vector of scheduling variables.Typically, the set E is parametrized in terms of plant outputs,that is σ ) σ(y). However, situations might occur where this isnot possible. In these cases, differential geometry can beexploited to construct a parametrization of E.16

Once the parametrization (4) is obtained, the typical methodof GS is to linearize system 1 about ps ) Ω(σ) leading to thefollowing linear parameter varying model:8

where δx ) x - xs(σ), δu ) u - us(σ) are the deviations ofstate and input variables from Ω(σ) ≡ (xs(σ),us(σ)). Then, alinear feedback control strategy can be implemented uponsystem 5 to guarantee stability and desired output behavior as

x(t) ) f(x(t), u(t))y(t) ) g(x(t), u(t))

(1)

C ) (x, u) ∈ Rn × R

m: xmin < x < xmax, umin < u < umax(2)

E ) (xs, us) ∈ Rn+m: f(xs, us) ) 0 (3)

Ω(σ):D ⊂ Rm f E ⊆ R

n+m (4)

δx·) [ ∂f

∂x(xs(σ), us(σ))]δx + [ ∂f

∂u(xs(σ), us(σ))]δu

δy ) [∂g∂x

(xs(σ), us(σ))]δx + [∂g∂u

(xs(σ), us(σ))]δu(5)

Ind. Eng. Chem. Res., Vol. 48, No. 20, 2009 9129

σ varies. In this paper, we consider the following state feedbackGS-control law:

where K(σ) denotes a parametrized family of feedback gainmatrices. Substituting (6) into (1) gives the governing equationsof the GS-closed-loop system:

Since closed-loop dynamics are only affected by the schedulingvariables σ, we will use p(t,x0,w) to denote the evolution of (7)when σ ) w and x(0) ) x0.

It is worth noting that the control law (6) is composed oftwo distinct terms: a feedforward uff(σ) and a feedback ufb(x,σ).From this perspective, the closed-loop system (7) can berepresented by the block scheme displayed in Figure 1a. Here,σ is the reference fed to the closed-loop system and identifies,based on the filtering action of the feedforward control unit,the desired steady state ps ) Ω(σ). Furthermore, feedbackcontroller gains are scheduled according to the evolution of σto enforce stability of the desired steady states as plant operatingconditions vary.

2.2. Predictive Reference Control. According to the blockscheme of Figure 1a, transition between an initial steady stateps0 ) Ω(σ0) and a final steady state psf ) Ω(σf) can be achievedby performing a step change of the reference signal σ betweenσ0 and σf. However, if large changes in process dynamics occurbetween ps0 and psf, such a step change may cause processconstraints violation. To avoid this, the evolution of the referencesignal can be adequately controlled. This can be done by addingto the GS-closed-loop system a discrete time device, calledreference governor (RG), which modifies online the referencesignal so as to enforce the fulfilment of the constraints.9

Reference control is, in this context, performed in a predictivemanner: a feasible sequence of reference inputs is selected basedon the prediction of the future evolution of the GS-closed-loopsystem. Since this requires a finite computational time, thereference governor operates in discrete time modifying thereference signal every period Tp. In particular, during the timeinterval [kTp, (k+1)Tp] a constant reference input σk is appliedguaranteeing that process constraints are not violated over thehorizon interval t ∈ [kTp, kTp+Th]. To reduce the computationalburden, the search of a feasible reference input sequence isrestricted to a one-dimensional subset of the reference variables

domain. In particular, the reference input σk provided to theGS-closed-loop-system is chosen as a function of σk-1 and σf(kTp)as follows:9,10

On the basis of eq 8, only reference inputs lying on thesegment S(σk-1,σf(kTp))⊂D connecting σk-1 and σf(kTp) areconsidered as candidates when selecting a feasible referenceinput σk. We here assume that σf(kTp) ) const:) σf. Under thisassumption, eq 8 restricts the search of a feasible reference inputsequence σkk∈N to the segment S(σ0,σf) connecting σ0 and σf.This segment can be parametrized as follows:

In this setting, a feasible reference input sequence σkk∈N

≡S(Rk)k∈N can be computed by solving at each instant t )kTp the following scalar nonlinear program:9

It is important to note that, when applying eq 10, only therestriction of the control law (6) to S(σ0,σf) is implemented.Therefore, closed loop dynamics can be described as follows:

Accordingly, the overall controlled plant can be represented bythe block scheme reported in Figure 1b. Here, Rf ) Γ-1(σf) isfed to the reference governor which computes online a feasiblereference input R.

Remark 1. In accordance with Bemporad,9 indispensibleprerequisite to achieve constrained transition to psf whenapplying the PR-control scheme (9-10) upon (11) is fulfillingthe following conditions:

Condition a is needed to prevent that the border of C isapproached in steady state while condition b ensures that notransition to undesired solution regimes takes place when thereference signal R is varied. Unfortunately, situations can likelyoccur where conditions a and b are not fulfilled. Concerningcondition a, it must be noted that steady states ps ∈ Ω(S(σ0,σf))violating process constraints may be found due to the nonlin-earity of Ω(σ). An example showing this possibility is reportedin Figure 2.

On the other hand, the presence of undesired solution regimescoexisting with ps ∈ Ω(S(σ0,σf)) may prevent the fulfilment ofcondition b. To prove it, observe that if undesired stable solutionregimes coexisting with ps ∈ Ω(S(σ0,σf)) and fulfilling processconstraints are found, sudden changes in R might cause thetrajectories of the GS-closed-loop system to leave the stabilityregions of ps ∈ Ω(S(σ0,σf)) without violating process constraints.In this case, application of the PR-control scheme (9-10) mightmove the GS-closed-loop system to an undesired solution regimeeven if an infinite horizon Th is used.

Figure 1. Block scheme of the controlled plant: (a) GS-closed-loop system;(b) GS-closed-loop system and PR-controller.

u(x, σ) ) us(σ) - K(σ)(x - xs(σ)): ) uff(σ) + ufb(x, σ)(6)

x ) f(x, u(x, σ))y ) g(x, u(x, σ))

(7)

σk ) σk-1 + R[σf(kTs) - σk-1], R ∈ [0, 1] (8)

Γ(R) ) σ0 + R[σf - σ0], R ∈ [0, 1] (9)

Rk ) maxR∈[Rk-1,1]

R

s.t. p(t, x(kTp), Γ(R)) ∈ C, ∀t ∈ [kTp, kTp + Th](10)

x(t) ) f(x(t), u(x(t), Γ(R)))y(t) ) g(x(t), u(x(t), Γ(R)))

(11)

(a)Ω(Γ(R)) ⊂ C ∀R ∈ [0, 1]

(b) if the reference signal R(t) is monotonically convergent toR*,

∀x(0) limtf+∞

p(t, x(0), Γ(R(t))) ) Ω(Γ(R*))

9130 Ind. Eng. Chem. Res., Vol. 48, No. 20, 2009

Remark 2. To prevent transition to undesired solutionregimes, Lyapunov functions are employed in Gilbert andKolmanovsky17 to get an explicit characterization of the stabilityregions of ps ∈ Ω(S(σ0,σf)). In this framework, only steady statesps ∈ Ω(S(σ0,σf)) whose stability regions include the current stateof the GS-closed-loop system are considered as candidates whenselecting a feasible reference input σk ∈ S(σ0,σf). However,Lyapunov functions can be derived only for few classes ofnonlinear systems restricting the range of applicability of thisapproach. Therefore, the global stability features of GS-controllers are frequently investigated by means of extensivenumerical simulations.8 Obviously, this approach cannot, ingeneral, reveal the presence of undesired solution branches.15

Remark 3. It must be stressed that even if a systematicanalysis of the global stability features of the GS-closed-loopsystem is possible based, for example, on the use of Lyapunovfunctions, the presence of multiple stable solution regimes maysignificantly restrict the stability regions of the desired steadystates ps ∈ Ω(S(σ0,σf)). In this case, the reference signal mustbe slowly varied to prevent transitions to undesired solutionregimes eventually lengthening the transient needed for the GS-closed-loop system to reach psf.

Remark 4. Steady states ps ∈ Ω(S(σ0,σf)) can presentcharacteristics preventing fast transition to psf even if conditionsa and b hold. In particular, steady states ps ∈ Ω(S(σ0,σf)) closeto the border of C may be found. In these conditions, undesiredreductions of the allowed variations of the manipulated variablesare observed forcing the reference governor to slowly vary thereference signal to ensure the fulfilment of the constraints.Moreover, the closed-loop system may exhibit undesired outputbehavior around ps ∈ Ω(S(σ0,σf)). This can happen even whenthe closed-loop system is carefully designed due to unsatisfac-tory controllability of the open-loop system about ps ∈Ω(S(σ0,σf)).

18-22 For example, unstable steady states ps ∈Ω(S(σ0,σf)) may be found requiring large control effort to bestabilized.

3. Problem Formulation and Outline of the SolutionProcedure

In accordance with remarks 1-4, the following limits ofapplication of the PR-control scheme (9-10) can be identified:(1)The steady states ps ∈ Ω(S(σ0,σf)) may be near or crossingthe border of C and the uncontrolled plant may exhibit

unsatisfactory controllability properties about ps ∈ Ω(S(σ0,σf)).(2)Undesired stable solution regimes coexisting with the steadystates ps ∈ Ω(S(σ0,σf)) may be found.

The first limit arises as a result of restricting a priori thereference inputs to S(σ0,σf) without taking into account thecharacteristics of the corresponding steady states ps ∈Ω(S(σ0,σf)). To overcome this limit, we extend the applicationof the PR-control scheme (9-10) to curves of the referencevariables domain D connecting σ0 and σf. Infinite curves Γ ⊂D connecting σ0 and σf are, indeed, available when dim(D) )m > 1 allowing to optimize the characteristics of the steady statesps ∈ Ω(Γ) around which the plant is operated during thetransition (see, for example, Figure 3). In particular, the curveΓ ⊂ D should be selected so that no steady states ps ∈ Ω(Γ)close to the border of C are found and the uncontrolled plantexhibits satisfactory controllability properties around ps ∈ Ω(Γ).

Here, we consider the set of continuously differentiable curvesZ(σ0,σf) ⊂ D connecting σ0 and σf. This means that it is alwayspossible to construct a continuously differentiable and invertibleparametrization mapping the interval [0,1] onto a curve Γ ∈Z(σ0,σf):

Once a parametrization of the type (12) is selected, a feasiblereference input sequence wkk∈N ) Γ(Rk)k∈N can be computedby solving the nonlinear program (10) at each period t ) kTp.

Figure 2. Projections of the segment S(σ0,σf) onto the bifurcation manifold. Because of the shape of the bifurcation manifold, steady states ps ∈ Ω(S(σ0,σf))violating process constraints are found.

Figure 3. Projections of curves of the reference variables domain onto thebifurcation manifold.

Γ(R):[0, 1] f D ⊆ Rm

Γ(0) ) σ0, Γ(1) ) σf(12)


The second limit imposes to account for the global stabilityfeatures of the GS-closed-loop system. In particular, both theselection of the curve Γ and the design of the GS-closed-loopsystem should be performed so that no undesired solutionregimes coexisting with the steady states ps ∈ Ω(Γ) are detectedas R varies. It is, indeed, clear that depending on how the GS-closed-loop system is designed, it might be impossible to finda curve of globally stable steady states Ω(Γ). This happens, forexample, when the GS-closed-loop system is designed so thatundesired solution regimes coexisting with ps0 and/or psf aredetected.

Therefore, Γ(R) and K(Γ(R)) should be computed so that thefollowing requirements are fulfilled as R varies:

• The steady states Ω(Γ(R)) lie as far as possible from theborder of C.

• No undesired solution regime coexisting with Ω(Γ(R)) isdetected.

• The closed-loop system exhibits desired output behaviorabout Ω(Γ(R)).

It must be noted that the requirement of satisfactory control-lability properties of the uncontrolled plant is here implicitlyimposed by demanding that the closed-loop system does exhibitdesired output behavior about Ω(Γ(R)). It is also worthobserving that computing Γ(R) and Κ(Γ(R)) so as to achievethe addressed control objectives results in tailoring the bifurca-tion diagram (the nature, the number, and the stability charac-teristics) of the GS-closed-loop system as R varies.

To solve the previous problem, information about closed-loop performance and distances of plant steady states from theborder of the process constraint set and from regions of theparameter space characterized by state multiplicity must bederived. In this framework, we observe that several methodsare available in literature to estimate closed-loop performanceand can be used to structure the formulated problem.22 Also,for the class of process constraints considered in this paper,functions providing information about the distance of plantsteady states from the border of the process constraint set canbe easily derived. On the contrary, difficulties might arise whenattempting to obtain information about the distance of plantsteady states from regions of the parameter space characterizedby state multiplicity. In principle, nonlinear constraints can beformulated imposing that the distances of ps ∈ Ω(Γ) from theboundaries of such regions do not exceed a prescribed value.23,24

This approach has been applied for the optimization baseddesign of nonlinear processes.25,26 Nevertheless, its applicationto compute a parametrized family of steady states, such as Ω(Γ),may become cumbersome. In this case, a set of nonlinearconstraints should be imposed for each ps ∈ Ω(Γ), significantlyincreasing the complexity of the problem. To overcome thisdifficulty, we separately compute Γ and K(Γ). In particular, thefollowing two-step solution procedure is proposed:

(1) Optimal reference trajectory shaping. A curve Γ iscomputed so that Ω(Γ(R)) lies as far as possible from the borderof C and the open-loop system linearization about Ω(Γ(R))exhibits satisfactory controllability properties as R varies.

(2) Robust GS-closed loop design. K(Γ(R)) is computed sothat the GS-closed-loop system does not exhibit undesiredsolution regimes and desired output behavior is observed aboutΩ(Γ(R)) as R varies.

In accordance with step 1, the curve Γ is computed beforedesigning the closed-loop system by only exploiting informationabout the uncontrolled plant. The selection of Γ is performedso to enforce satisfactory controllability of the open-loop systemlinearization about ps ∈ Ω(Γ). Fulfilling this requirement is,

indeed, necessary to ensure the existence of a family of feedbackgain matrices K(Γ(R)) ensuring desired output behavior aroundps ∈ Ω(Γ) as R varies. This approach to select Γ sidesteps theneed to compute the distances of the steady states Ω(Γ) frombifurcation boundaries. Conversely, step 2 aims at computinga family of feedback gain matrices K(Γ(R)) ensuring satisfactoryoutput behavior and ruling out the presence of undesired solutionregimes. In sections 5-6, a simple method to compute such aK(Γ(R)) is presented ensuring feasibility of the proposed solutionprocedure.

4. Optimal Reference Trajectory Shaping

In this section, the problem of computing an optimal referencetrajectory is described. A rigorous formulation of this problemis presented in subsection 4.1 while computational issuesconcerning its numerical solution are discussed in subsection4.2.

4.1. Optimization Problem. To solve the optimal referencetrajectory shaping problem, functions providing informationabout the distance of plant steady states from the border of Cand open-loop controllability must be defined. Integrating thesefunctions over the curve Ω(Γ) as Γ varies gives a functionaldefined in the set Z(σ0,σf). Thus, an optimal Γ can be obtainedby minimization of such a functional. Therefore, given the steadystates ps0 and psf ∈ E ∩ C, a vector function L(p):E ⊂ Rn+m fRl whose elements provide measures for open-loop control-lability about p, a function M(p):E ⊂ Rn+m f R quantifyingthe distance of p from the border of C, the optimal referencetrajectory shaping problem can be formulated as the problemof computing a parametrization Γ(R):[0,1]f D ⊂ Rm verifyingthat

where Ψ(R)) Ω(Γ(R)), Ψ′(R))DRΨ(R), w1 and w2 are weight-ing constants and η ∈ Rl is a vector of bounds.

It is worth noting that, with constant M and Li, the functionalΦ defined in eq 18 increases as the length of the curve spannedby Ψ(R) increases. This is in line with the objective of reducingthe transient needed for the plant to reach the final steady state.To prove it, consider the problem of performing a transitionbetween two steady states lying close to each other. In this case,the use of a short curve restricted to the bifurcation manifoldand connecting the two considered steady states would beprobably sufficient to guarantee fast transition while fulfillingprocess constraints. Nevertheless, if the steady states ps ∈ Ω(Γ)are selected only based on controllability measures, a longercurve may be found moving far away from the region sur-rounding the initial and the final steady state. As a result, thetransient needed for the plant to reach the final steady state maybe lengthened even though the open-loop system linearizationabout the selected steady states ps ∈ Ω(Γ) exhibits satisfactorycontrollability properties.

For the function L(p), the inverse of the minimum singularvalue of the process steady state gain matrix and the sum ofthe real parts of the right-half plane eigenvalues of the open-loop system linearization are used. A small value for the inverse

Γ(R) ) minΓ∈Z(σ0,σf)

Φ(Γ) )

minΓ∈Z(σ0,σf)

∫0

1(Ψ′(( ∑

i

l

w1,iLi(Ψ)) + w2M(Ψ))) dR (13)

subject toΨ(R) ∈ C ∀R ∈ [0, 1] (14)

L(Ψ(R)) e η ∀R ∈ [0, 1] (15)


of the minimum singular value indicates that the steady stategain matrix is not close to being singular and that the planteasily moves between different steady states27,28 while smallvalues for the real parts of the right-half plane eigenvalues ensurethat no large control efforts are needed to achieve stability.29

Therefore, L(p) is defined as follows:

where VS,min(p) and RHPEi denote the minimum singular valueof the process steady state gain matrix and the right-half planeeigenvalues of the open-loop system linearization at p, respec-tively. It must be remarked that more effective and sophisticatedmethods to estimate open-loop plant controllability are currentlyavailable and could be used to structure the problem (13-15).We refer the reader to Sakizlis et al.22 for a survey of theseapproaches.

Finally, the function M(p) is defined as the sum of thedistances of state and manipulated variables from the boundariesof the process constraint set C. This gives the followingexpression:

where xk,i⊥ and uk,i

⊥ denote the projections of x and u onto theplanes x ) xk,i and u ) uk,i of the state and parameter spaces,respectively. The function M(p), also known as inverse barrierfunction,30 takes positive values in the interior of the processconstraint set C and becomes infinitely large as p approachesthe border of C. Hence, replacing (17) in (13) has the effect ofpenalizing the selection of curves Γ corresponding to steadystates ps ∈ Ω(Γ) close to the border of C. This approachresembles the barrier method implemented to handle nonlinearconstraints when minimizing a given objective function.30

However, it is important to note that the spirit by which it ishere applied is completely different. The function M(p) is,indeed, introduced in (13) to prevent that steady states ps ∈ Ω(Γ)close to the border of C are found while the fulfilment of processconstraints is separately enforced by fulfilling (14).

4.2. Computational Issues. Solving problem (13-15) re-quires the finding of a parametrized curve of the referencevariables domain minimizing the functional Φ. This gives aninfinite dimensional optimization problem. Therefore, discreti-zation is necessary to make the problem (13-15) numericallyaffordable. To this aim, the parametrization Γ(R) is replacedby the sequence ΓΝ )σ1, ..., σN and the optimal referencetrajectory shaping problem is recast in the following form:

where the norm is defined by recourse to a weighting matrixQ. In particular, |p|Q ) pTQp, with Q positive definite.

Once the problem (18-20) is solved, the elements of theoptimal sequence ΓΝ )σ1, ..., σN can be interpolated to getan approximate solution Γ*(R) to (13-15). In this framework,the objective is to compute a parametrization Γ*(R) guaranteeingthat the corresponding steady states Ω(Γ*(R)) exhibit, as Rvaries, characteristics similar to those observed for the steadystates ΨN ) Ω(σ1), ..., Ω(σN). The choice of the steady statesin ΨN is, indeed, optimized by solving (18-20), whereas thereis no control on the steady states Ω(Γ*(R)) with Γ*(R)∉σ1,..., σN. However, solving (18-20) may lead to sequences ΓΝ

) σ1, ..., σN corresponding to steady states ΨN ) Ω(σ1), ...,Ω(σN) far from each other. In this case, the characteristics ofΩ(Γ*(R)) may likely result, as R varies, significantly differentcompared to those observed for ΨN ) Ω(σ1), ..., Ω(σN). Inparticular, steady states Ω(Γ*(R)) with Γ*(R)∉σ1, ..., σN maybe found violating process constraints or corresponding tounsatisfactory controllability properties of the open-loop systemlinearization. This can be prevented by interpolating ΓΝ at eachstep while solving (18-20) and checking that Ω(Γ*(R)) fulfillthe feasibility conditions (14-15) as R varies. Obviously, thissignificantly increases the computational resources needed tocompute Γ*(R).

Alternatively, the following constraint can be added to(18-20):

where δ ∈ Rn+m is a vector of bounds. This forces the steadystates ΨN )Ω(σ1), ..., Ω(σN) to be close to each other enablingus to ensure feasibility of the interpolating branch Ψ(Γ*(R))by adequately selecting δ. In particular, a feasible solution Γ*(R)can be obtained by solving (18-21) for decreasing values ofthe elements of δ until the obtained branch Ψ(Γ*(R)) is foundto fulfill (14-15) as R varies. It must be here observed thatreducing the elements of δ may require an increase in N in orderto get a feasible solution to (18-21). Therefore, the followingalgorithm is followed to compute Γ*(R):

1. select initial guesses for δ and N2. solve (18-21)3. if a solution to (18-21) is found, interpolate the computed

sequence ΓΝ to get Γ*(R) otherwise increase N and repeatstep 2

4. if steady states Ω(Γ*(R)) not fulfilling (14-15) are found,reduce the elements of δ and repeat the previous stepsuntil a feasible Ω(Γ*(R)) is found

We note that an initial guess for δ can be obtained based onthe physics of the process, namely, looking at the characteristicvariations of state and manipulated variables. Then, an estimateof the minimum N ensuring feasibility of (18-21) can becomputed as the minimum number of steady states ps,i ∈Ω(S(σ0,σf)) fulfilling -δ < ps,i - ps,i-1 < δ and ps,N ) psf. Aninitial guess for N can be, therefore, derived by slightlyincreasing this value.

The proposed algorithm enables one to get a feasible solutionΓ*(R). Nevertheless, it must be remarked that the sequence ΓN

solving (18-21) and, hence, its interpolation Γ*(R) approximatewith increasing accuracy the curve Γ solving (13-15) when Nis increased. Therefore, a rigorous way to get an interpolatingfunction close to the solution of (13-15) is, after obtaining afeasible Γ*(R), to solve (18-21) for increasing N values untilno significant changes in the observed solution occur. Therequired computational resources needed to perform this taskcan be reduced by employing the sequence ΓN solving (18-21)

L(p) ) [1/VS,min(p)

∑i

ReRHPEi ] (16)

M(p) ) ∑i)1

n

[|x - xmax,i⊥ |Q

-1 + |x - xmin,i⊥ |Q

-1] +

∑i)1

m

[|u - umax,i⊥ |Q

-1 + |u - umin,i⊥ |Q

-1] (17)

ΓN ) minΓN

ΦN(ΓN) )

minΓN

∑k)2

N

(( ∑i

l

w1,iLi(pk)) + w2M(pk))|Ω(σk) - Ω(σk-1)|Q

(18)

Ω(σk) ∈ C ∀k ∈ 1, ..., N (19)

L(Ω(σk)) e η ∀k ∈ 1, ..., N (20)

-δ e Ω(σk) - Ω(σk-1) e δ, ∀k ∈ 1, ..., N (21)


with N ) K to construct an initial sequence when solving(18-21) with N ) K + 1.

5. Bifurcation Analysis of the Gain-ScheduledClosed-Loop System

Bifurcation analysis has been extensively applied to charac-terize the behavior of nonlinear systems and can be consideredan established tool in the analysis of engineering processes. Thestudies conducted in this area have been traditionally aimed atimproving the understanding of the qualitative dynamics of theprocess units. A representative example of such an approach isthe seminal paper on the dynamics of an exothermic continuoustank reactor by Ray and his co-workers.31 Over the past fewdecades, the use of bifurcation analysis has been proposed toaddress synthesis problems in a rigorous manner. In thisframework, two different approaches to the application ofbifurcation analysis to control system design can be distin-guished. The first approach assumes that a control structure hasbeen preliminary selected and makes use of parametric continu-ation to identify regions of the controller parameter spacecharacterized by unstable plant operation and/or undesired globaldynamics as, for example, multistability.13,32-34 The secondapproach, also known as bifurcation control,35 incorporatesinformation about the bifurcation characteristics of the uncon-trolled plant into the synthesis of the control structure. In thiscontext, control strategies capable of tailoring the bifurcationcharacteristics of nonlinear processes are formulated.14,36,37

In what follows, we discuss the application of bifurcationanalysis and parametric continuation to compute a parametrizedfamily of feedback gain matrices K(Γ(R)). The presentedmethodology enables one to account for the global stabilityfeatures of the GS-closed-loop system and extends the approachproposed in Richardson et al.15 by providing a framework forthe design of the control requirements that improve the globalstability features of the GS-closed-loop system.

To compute K(Γ(R)), feedback gain matrices enforcing a setof control requirements (e.g., settling time, risetime, etc.) aboutspecified steady states of the curve Ω(Γ) are interpolated.38,39

With this approach, poor controller performance or eveninstability can occur due to hidden coupling terms.40,41 More-over, control requirements at specified steady states are generallydefined based on trial-and-error procedures aimed at finding agood compromise between closed loop performance and reli-ability of the control law. However, the effect of these choiceson the global stability features of the controlled plant is generallyignored. To overcome these limitations, bifurcation analysis canbe performed to investigate the effect of interpolating feedbackgains and/or modifying controller parameters on the occurrenceof multiple stable solution regimes of the GS-closed-loopsystem.

To account for the dependence of closed-loop dynamics oncontroller parameters, we recast eqs 11 in the following form:

with u(x(t),Γ(R), λ) described as follows:

Here, λ denotes the vector of parameters which is necessaryto fix to compute the feedback controller gains. For example, λdefines the desired poles of the closed-loop system linearizationwhen applying a pole placement procedure or the elements of

weighting matrices when making use of a linear quadraticregulation algorithm.29 Therefore, the effect of varying λ onthe global stability characteristics of the steady states ps ∈ Ω(Γ)must be assessed. To this aim, the evolution of the bifurcationsof the GS-closed-loop system (22) in the R-λ space is analyzed.In so doing, possible bifurcations responsible for the instabilityof the steady states ps ∈ Ω(Γ) and/or the occurrence ofmultistability are detected. Hence, λ values can be selected toguarantee that the steady states ps ∈ Ω(Γ) exhibit prescribedmargins from the bifurcation boundaries. Note that the rangeof λ over which the analysis should be performed must beselected according to the specific control problem to guaranteesatisfactory closed-loop behavior about the steady states ps ∈Ω(Γ).

6. Simulation of the Start-up of a Continuous StirredJacketed Tank Reactor

To prove the effectiveness of the proposed approach, wepresent in this section the results of its application to the problemof controlling the start-up of a CSTR where the series of twofirst order irreversible exothermic reactions Af Bf C takesplace. Depending on the reaction heat values and the activationenergies, these reactors can exhibit multiple stable solutionregimes and nonminimum-phase behavior.42

6.1. Closed-Loop System Design. Under standard modelingassumptions, reactor dynamics can be described by the followingnonlinear system:

where CA is the concentration of the species A, CB is theconcentration of the species B, T is the reactor temperature, Tc

is the coolant temperature, Q is the feed flow rate, and Qc isthe coolant flow rate. We assume, in the following, that Q andQc are the inputs to the system which can be modified. Theother parameters appearing in (24-27) have the usual meaningand are kept at constant values reported in Table 1.

The problem we address is to perform transition of (24)-(27)from ps0 to psf as described in Table- 2. ps0 corresponds toreaction extinction conditions while psf is characterized by highconversion of the intermediate product B. In particular, psf is

x(t) ) f(x, u(x(t), Γ(R), λ))y(t) ) g(x, u(x(t), Γ(R), λ))

(22)

u(x, Γ(R), λ) ) us(Γ(R)) + K(Γ(R), λ)(x - xs(Γ(R))) )uff(R) + ufb(x,R, λ) (23)

Table 1. Reactor Parameter Values

V (L) 100 k2 (min-1) 16632F (g/L) 1000 ∆Η1 (J/mol) 20596Cp (J/g ·K) 0.239 ∆Η2 (J/mol) 3913Cpc

(J/g ·K) 0.25 Tf (K) 300U (J/min ·K) 18462 Tcf (K) 293E1/R (K) 8355 CA,in (mol/L) 1E2/R (K) 3759 CB,in (mol/L) 0k1 (min-1) 7.46 × 1011

dCA

dt) Q

V(CA,in - CA) - k1 exp(-ER1

RT )CA (24)

dCB

dt) Q

V(CB,in - CB) - k2 exp(-ER2

RT )CB +

k1 exp(-ER1

RT )CA (25)

dTdt

) QV

(Tin - T) -∆H1k1

FCpexp(-ER1

RT )CA -

∆H2k2

FCpexp(-ER2

RT )CB - UVFCp

(T - Tc) (26)

dTc

dt)

Qc

Vc(Tc,in - Tc) +

UVFCpc

(T - Tc) (27)


unstable and coexists with a stable and an unstable steady state.In this framework, we assume that state and input variables areconstrained within the set C described in Table 2.

To avoid sudden variations of the control law, we assumethe control inputs to the plant to be generated through a low-pass filter. Therefore, we pose x ≡ (CA, CB, T, Tc ,Q, Qc) anddescribe the filter and plant dynamics by means of the followingnonlinear system:

Equations 32-33 describe the dynamics of the filter. Here, τ1

and τ2 are time constants while u1 and u2 are the inputs to thefilter and define the desired values of feed and coolant flowrates. In the following, we assume that τ1 ) τ2 ) 0.1. Therefore,we pick u1 and u2 as manipulated variables and apply themethodology described in the previous sections to implementthe transition between ps0 and psf.

The concentration of the species A and the reactor temperatureare chosen as scheduling variables, that is σ ) (x1, x3). Therefore,eqs 28-33 are solved under steady state conditions withunknown variables (x2, x4, x5, x6, u1, u2). This gives the followingmap Ω(σ) ) (xs1(σ), xs2(σ), xs3(σ), xs4(σ), xs5(σ), xs6(σ),us1(σ),us2(σ)):

where

with h1(σ) and h2(σ) defined as follows:

An optimal curve Γ of the scheduling variables domain canbe now constructed following the approach illustrated in section4. To this aim, the nonlinear program (18-21) is solved withparameters values reported in Table 3 and the spline toolboxof Matlab is used to interpolate the resulting sequence of steadystates. The interpolating curve Γ and its projection Ω(Γ) ontothe bifurcation manifold are described in Figure 4.

It is important to note that the evolutions of the manipulatedvariables u1(R) and u2(R) shown in Figure 4e,f provide the

Table 2. Process Constraints, Initial and Final Equilibrium

ps0 psf lower bound upper bound

CA (moli/L) 0.89 0.19 1 0CB (moli/L) 0.11 0.57 1 0T (K) 305 354 400 293Tc (K) 300.1 329.1 400 293Q (L/min) 70 100 110 40Qc (L/min) 51.19 51.2 70 30

dx1

dt)

x5

V(x1,in - x1) - k1 exp(- E1

Rx3)x1 ) g1(x, u)

(28)

dx2

dt)

x5

V(x2,in - x2) - k2 exp(- E2

Rx3)x2 +

k1 exp(- E1

Rx3)x1 ) g2(x, u) (29)

dx3

dt)

x5

V(x3,in - x3) -

∆H1k1

FCpexp(- E1

Rx3)x1 -

∆H2k2

FCpexp(- E2

Rx3)x2 - U

VFCp(x3 - x4) ) g3(x, u) (30)

dx4

dt)

x6

Vc(x4,in - x4c) +

UVFCpc

(x3 - x4) ) g4(x, u)

(31)

dx5

dt) - 1

τ1(x5 - u1) ) g5(x, u) (32)

dx6

dt) - 1

τ2(x6 - u2) ) g6(x, u) (33)

xs1(σ) ) σ1 (34)

xs2(σ) ) -σ1h(σ) (35)

xs3(σ) ) σ2 (36)

xs4(σ) )-k1(V/U) exp(- E1

Rσ2)σ1h2(σ)

(σ1 - x1,in)+

k2 exp(- E2

Rσ2)h1(σ)σ1∆H2(V/U) + σ2 (37)

xs5(σ) )-k1Vexp(- E1

Rσ2)σ1

(σ1 - x1,in)(38)

xs6(σ) )σ1UV(A - B)

CD + EF(39)

A ) k1 exp( E2

Rσ2)h2(σ)

B ) k2(σ1 - x1,in)∆H2 exp( E1

Rσ2)h1(σ)

C ) k2∆H2FCpcVσ1(σ1 - x1,in)

D ) exp( E1

Rσ2)h1(σ)

E ) FCpcexp( E2

Rσ2)

F ) (σ1 - x1,in) exp( E1

Rσ2)(σ2 - x4,in)U - k1σ1h2(σ)V

us1(σ) ) xs5(σ) (40)

us2(σ) ) xs6(σ) (41)

h1(σ) )k1(σ1 - x1,in - x2,in) exp( E2

Rσ2)

k1σ1 exp( E2

Rσ2) - k2(σ1 - x1,in) exp( E1

Rσ2)

(42)

h2(σ) ) (σ1 - x1,in)∆H1 + FCp(σ2 - x3,in) (43)

Table 3. Parameters Used to Compute the Optimal Γ

w1 w2 N η δ

(5, 6) 10 20 (2, 3) (0.1, 0.1, 5, 5, 5, 5, 5)


feedforward control law uff(R) which needs to be implementedto achieve transition to psf. Besides this control law, feedbackmust be implemented to guarantee stability and desired outputbehavior about Ω(Γ(R)) as R varies. To this aim, a parametrizedfamily of feedback gain matrices K(Γ(R),λ) is computed byapplying the linear quadratic regulation (LQR) algorithm(Kailath, 1996). This gives the following GS-control law:

where B ) Du[g(xs(Γ(R)),us(Γ(R))]| and P is the solution of thealgebraic Riccati equation:

Here, A ) Dx[g(xs(Γ(R)),us(Γ(R))] and R(λ) is a positivedefinite matrix whose elements are parametrized by λ. Inparticular, it is assumed that λ is a scalar and R is the diagonalmatrix defined by the vector V ) (5λ., 5λ., 5λ., 5λ., 1, 1), thatis R ) diag(V).

At this point, the PR-control scheme (10) can be applied toperform transition of the GS-closed-loop system to psf. Withthis objective, the control law (44) is implemented with λ ) 1,which is found to ensure satisfactory output behavior aboutΩ(Γ(R)) as R varies. Furthermore, an horizon interval Th ) 5min and a period Τp ) 0.2 min are selected and a bisectionalgorithm is used to solve the nonlinear program (10). Resultsof the implementation of (10) are described in Figure 5. Here,the reactor temperature simulation response (Figure 5a) and theevolution of the reference signal imposed by the PR-controller(Figure 5b) are described. The reactor temperature value xs3(1)corresponding to psf is also shown in Figure 5a. It is apparentfrom Figure 5 that the reference control scheme (10) with thecomputed Γ fails to bring the GS-closed-loop system to psf. Inparticular, after an initial rise, the reactor temperature starts todecrease and an undesired stationary value is eventually reached.

It is remarkable to note that the observed behavior occursirrespective of the horizon interval Th used to implement (10).To prove it, it must be observed that the inversion in the patterninitially followed by the reactor temperature occurs at t = 8.2min when the reference signal is switched between R ) 0.58and R ) 1. This switch is performed, in accordance to (10), asit enables reaching the desired value of the reference signal R) 1 and does not produce process constraint violation over thehorizon interval t ∈ [8.2, 8.2 + Th]. However, process constraintsare found to be fulfilled not only in the range t ∈ [8.2, 8.2 +Th] but ∀ t > 8.2 min. For this reason, the observed switch ofthe reference signal is performed, in accordance with (10), evenif an infinite Th is used. In these conditions, undesired transitionscan be prevented by modifying the design of the GS-closed-loop system so that undesired solution branches coexisting withΩ(Γ) are removed. With this objective, the approach describedin section 5 is applied.

Figure 6a shows steady state solution regimes of the GS-closed-loop system as R varies. The reactor temperature is

Figure 4. Parameterization Ω(Γ(R)) with Γ solving the optimal reference trajectory shaping problem.

u(x,R) ) us(x, Γ(R)) - K(Γ(R), λ)(x - xs(Γ(R)))

) uff(x, Γ(R)) - 12

BT(Ω(Γ(R)))P(Ω(Γ(R)), λ) ×(x - xs(Γ(R))) (44)

ATP + PA - PBBTP + R(λ) ) 0 (45)Figure 5. Evolution of the GS-closed-loop system under PR-control withthe optimal Γ and λ ) 1: (a) reactor temperature simulation response; (b)reference signal.


chosen as a variable representative of the state of the system. Itcan be observed that the steady states Ω(Γ(R)) are stable overthe entire range R ∈ [0,1]. Nevertheless, undesired solutionbranches are detected at large R values because of the presenceof the saddle node bifurcation point S1. In particular, a stablelow conversion and an unstable steady state coexisting withΩ(Γ(R)) are detected at R values larger than RS1.

To examine the effects of modifying controller parameterson such coexistence, the evolution of the saddle node bifurcationpoint S1 is analyzed by varying λ in the range [1, 4] Projectionsof the saddle node bifurcation points onto the plane R-λ areshown in Figure 6b. It can be observed that no saddle nodebifurcation is detected in the range R ∈ [0,1] when λ > 2.3. Forthese λ values, no undesired steady state coexisting with Ω(Γ)can be found. Therefore, the reference control scheme (10) isapplied by implementing the control law (44) with λ ) 3, whichis still found to ensure satisfactory output behavior aboutΩ(Γ(R)) as R varies. The reactor temperature simulationresponse observed in this case is described in Figure 7. It isapparent that the controlled plant moves to the desired steadystate psf.

6.2. Performance Analysis. To analyze the performance ofthe proposed approach, results of the implementation of thereference control scheme (10) with the optimal Γ computed inthe previous subsection and the segment Γ ) S(σ0, σf) arecompared. In this framework, we preliminarily describe inFigure 8 the curve Ω(Γ) and Ω(S(σ0, σf)).

It is apparent that the profiles of plant outputs (Figure 8a-d)for the considered curves stay close to each other as R varies.In particular, temperature profiles are barely distinguishable.Moreover, output values far from the bounds of the constraintset are found in both cases. On the contrary, significantdifferences between the profiles of plant inputs found for Ω(Γ)and Ω(S(σ0,σf)) are observed (Figure 8e-f). Feed flow ratevalues us1 (Figure 8e) for Ω(S(σ0,σf)) are invariably larger thanthose found for Ω(Γ), while the contrary holds for the coolantflow rate us2 (Figure 8f). However, while feed flow rate valuesus1 far from the bounds of the constraint set are observed forboth Ω(Γ) and Ω(S(σ0,σf), the us2 profile is, for Ω(S(σ0,σf)),close to its lower bound around R ) 0.2. This means that whenR is close to such value, reduced variations of the coolant flowrate are allowed if S(σ0,σf) is used in (10). The effects of thison transition control are shown in Figure 9. Here, the reactortemperature simulation response observed when the referencecontrol scheme (10) is implemented with Γ ) S(σ0, σf) (Figure9a) and the evolution of the reference signal imposed by thereference controller are reported (Figure 9b). The same value λ) 3 selected when implementing (10) with the Γ computed inthe previous subsection is used in (44). This value guaranteesuniqueness of the steady states ps ∈ Ω(S(σ0,σf)). With thissetting, differences between the behaviors observed whenS(σ0,σf) and Γ are used in (10) must be imputed to differentcharacteristics of the corresponding steady states ps ∈ Ω(S(σ0,σf))and ps ∈ Ω(Γ).

Figure 9 shows that the transition period found when Γ )S(σ0, σf) is used in (10) is about 30 min. This value issignificantly larger than that found with the optimal Γ computedin the previous subsection (=13 min) (Figure 7). The observeddifference is mainly due to the presence of steady states ps ∈Ω(S(σ0,σf)) close to the border of the process constraint set. Asshown in Figure 8, the coolant flow rate for ps ∈ Ω(S(σ0,σf))approaches its lower bound when R = 0.2. In this range, thereference governor is forced to slowly vary R in order toguarantee the fulfilment of the constraints. This clearly appearsin Figure 9b. It can be here observed that the reference controllertakes a long time to move R away from the range around R )0.2. On the contrary, a fast growth in R is found once this regionis crossed.

7. Summary and Future Directions

A novel method for transition control of nonlinear processeswas presented. In accordance with the proposed approach, gain-scheduling is implemented to ensure desired output behaviorover the operating region of interest while transition betweenthe initial and the final steady state is achieved by predictive

Figure 6. Bifurcation structure of the GS-closed-loop system: (a) solution diagram with R as bifurcation parameter (stable and unstable branches are denotedby solid and dashed line respectively); (b) projections of the saddle node bifurcations in the R-λ plane.

Figure 7. Evolution of the GS-closed-loop system under PR-control withthe optimal Γ and λ ) 3: (a) reactor temperature simulation response; (b)reference signal.


reference control. In so doing, a feasible sequence of referenceinputs is computed so that the closed-loop system moves alongan optimal curve of the bifurcation manifold. This restricts thesearch of a feasible reference input sequence to a one-dimensional subset of the reference variables domain, reducingthe required computational resources. Moreover, nonlinearitiesresponsible for transitions to undesired solution regimes can be,with this approach, promptly identified. It is, indeed, sufficientto trace the solution diagram of the closed-loop system byvarying the reference signal according to the selected curve todetect multiple stable solution regimes and/or bifurcationscausing instability of the selected steady states.

In this framework, the problem of tailoring a curve of thebifurcation manifold and a family of feedback gain matricesensuring desired output behavior and preventing the occurrenceof undesired solution branches was addressed. The followingtwo-steps procedure was proposed to tackle this problem: first,a curve of steady states is computed so that its points lie far

from the border of the process constraint set and correspond tosatisfactory controllability properties of the uncontrolled plant(optimal reference trajectory shaping); then, bifurcation analysisof the closed-loop system is performed enabling the choice ofthe controller parameters values preventing transitions toundesired solution regimes (robust GS-closed-loop design). Withthis approach, an optimal curve of steady states is computedbefore designing the closed-loop system by only exploitinginformation about the uncontrolled plant. At this stage, steadystates corresponding to satisfactory controllability of the open-loop system linearization are selected. Fulfilling this requirementis, indeed, needed to ensure the existence of feedback controllergains enforcing desired closed-loop behavior. On the other hand,bifurcation analysis is performed enabling a thorough examina-tion of the effect of varying feedback controller parameters onthe global stability properties of the overall controlled plant.Therefore, feedback controller parameters can be computedensuring satisfactory output behavior and ruling out the presenceof undesired solution branches.

The proposed method was validated on the problem ofcontrolling the start-up of a jacketed continuous tank reactor.In this context, the approach proved to effectively handle thepresence of multiple stable solution regimes enabling theremoval of a low conversion solution branch responsible forplant-shut down. Also, the importance of adequately shaping acurve of the bifurcation manifold to guarantee fast transition tothe final steady state was demonstrated by comparison with theclassical reference governor approach.9 In particular, PR-controlwas implemented by restricting the reference inputs fed to theclosed-loop system to (a) the segment connecting the referenceinputs associated to the initial and the final steady state and (b)to the curve obtained by solving the optimal reference trajectoryshaping problem. In the first case, the steady states spanned bythe considered segment were found close to the border of theprocess constraint set resulting in a large transition period. Onthe contrary, fast transition was achieved when restricting thereference inputs to the computed optimal curve.

Figure 8. Comparison between the parametrizations Ω(Γ(R)) obtained with Γ solving the optimal reference trajectory shaping problem (dashed line) and Γ) S(σ0,σf) (solid line).

Figure 9. Evolution of the GS-closed-loop system under PR-control withΓ ) S(σ0,σf) and λ ) 5: (a) reactor temperature simulation response; (b)reference signal.


It is important to remark that the advantage of the proposedmethod, compared to alternative existing control approaches as,for example, model predictive control, mainly resides in itscapability of fulfilling optimal control requirements and, at thesame time, accounting for the presence of critical operabilityboundaries during the design of the closed-loop system. Thisapproach is expected, in general, to ensure fast transition to thefinal steady state while preventing the occurrence of undesiredphenomena as, for example, secondary reactions or thermalrunaways.

It is worth remarking that the proposed method was hereapplied to a problem where the steady states of the uncontrolledplant can be parametrized in terms of internal plant variables.However, situations can be found where such a parametrizationcannot be used due, for example, to the occurrence of outputand/or input multiplicity. It must be stressed that, in this case,the application of the proposed methodology results unchanged.Obviously, an alternative parametrization must be initiallyconstructed. With this objective, an effective approach is tointroduce, based on the idea of pseudo-arc-length,43 a curvilinearcoordinates system onto the bifurcation manifold of the uncon-trolled plant.16 Here, each steady state is identified by a set ofcurvilinear coordinates on the bifurcation manifold. This ap-proach works disregarding the occurrence of input and/or outputmultiplicity providing the basis for a broad application of theproposed methodology. Nevertheless, the application of thisparametrization scheme to solve the optimal reference trajectoryshaping problem is not trivial and deserves to be thoroughlydiscussed. Therefore, we leave a detailed analysis of suchalternative approach to the parametrization of plant steady statesas the subject of a future work.

It must be finally stressed that significant improvementsof the proposed method are expected to be achieved bysimultaneously computing the curve of steady states to betracked and the feedback controller gains. In particular, notonly the feedback controller gains but also the curve of steadystates to be tracked should be selected to enforce desiredglobal stability features of the closed-loop system. To thisaim, nonlinear constraints can be formulated imposing thatthe distance of each steady state of the selected curve frombifurcation and performance boundaries do not exceed aprescribed value.26 This approach also enables the handlingof process constraints which are implicitly defined bynonlinear equations in the state and input variables. However,its application to the formulated problem becomes cumber-some when a large number of steady states is needed todescribe a curve connecting the initial and the final steadystate. In this respect, we are currently investigating reductiontechniques which allow the decomposition of the optimalreference trajectory shaping problem into a sequence ofsubproblems whose solution involves few steady states.

Acknowledgment

The authors kindly acknowledge professor Costin SorinBildea for the valuable suggestions provided during the prepara-tion of the manuscript.

Notation

Acronyms

GS ) gain-scheduledPR ) predictive referenceRG ) reference governorRHPE ) right-half plane eigenvalue

Notation

C ) constraint setD ) reference variables domainDx ) derivative with respect to xdiag ) diagonal matrix defined by vector vE ) set of steady states of the uncontrolled plantL ) controllability functionM ) barrier functionp ) state-input vectorr ) reference input vectorS(a, b) ) segment of the reference variables domain connecting a

and bTh ) time horizonTp ) sampling timeu ) input vectorVS ) singular valuex ) state vectory ) output vector

Greek symbols

R ) scalar reference inputη ) vector of bounds on plant controllabilityδ ) vector of bounds on the distance between plant steady statesλ ) vector of feedback controller parametersσ ) vector of scheduling variablesΩ ) parametrization of the set of steady statesΓ ) continuously differentiable curve of the reference variables

domainZ(a, b) ) set of the continuously differentiable curves of the

reference variables domain connecting a and b

Subscripts

0 ) initialf ) finalfb ) feedbackff ) feedforwardmax ) maximummin ) minimumN ) number of steady statesp ) periods ) stationary

Superscripts

m ) dimension of the input vectorn ) dimension of the state vectorp ) dimension of the output vectorl ) dimension of the controllability vector function

Notation: Exothermic CSTR with Reactions A f B f C

CA ) concentration of ACB ) concentration of BCp ) specific heat (J/g ·K)ER ) activation energy (J/mol)∆H ) heat of reaction (J/mol)k ) kinetic constant (min-1)Q ) volumetric flow rateR ) gas constant (J/(mol K))F ) density (g/L)T ) temperature (K)U ) heat-transfer coefficient (J/(min ·K))V ) reactor volume (L)

Subscripts: Exothermic CSTR with Reactions A f B f C

1 ) reaction A f B2 ) reaction B f Cc ) coolant


in ) inletX ) reactant A or B

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ReceiVed for reView January 29, 2009ReVised manuscript receiVed July 23, 2009

Accepted August 8, 2009

IE9001553


optimal reference trajectory shaping and robust gain-scheduling for transition control of nonlinear...

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