Dynamic shaping of transport–reaction processes with a combinedsliding mode controller and Luenberger-type dynamic observer design

Davood Babaei Pourkargar, Antonios Armaou n

Department of Chemical Engineering, The Pennsylvania State University, University Park, PA 16802, United States

H I G H L I G H T S

� The dynamic shaping of transport–reaction processes is investigated.� The spatiotemporal shaping is addressed via model order reduction.� The desired spatiotemporal behavior is described by a target PDE.� A sliding mode controller design is applied to track desired dynamics.� A dynamic observer is employed to estimate the system dominant modes.

a r t i c l e i n f o

Article history:Received 4 March 2015Received in revised form26 June 2015Accepted 29 July 2015Available online 15 August 2015

Keywords:Dynamic shapingDistributed parameter systemsModel order reductionSliding mode controlDynamic observerProcess control

a b s t r a c t

We focus on shaping the long-term spatiotemporal dynamics of transport–reaction processes which canbe described by semi-linear partial differential equations (PDEs). The dynamic shaping problem isaddressed via error dynamics regulation between the governing PDE and a target PDE which describesthe desired spatiotemporal behavior. A model order reduction methodology is utilized to construct therequired reduced order models (ROMs) for governing and target dynamics via Galerkin's method. Wesubtract the governing from the target ROMs to obtain reduced offset dynamics error. Then an outputfeedback sliding mode control structure is synthesized to stabilize the reduced error dynamics andcorrespondingly synchronize the system and the target spatiotemporal behaviors. A Luenberger-typedynamic observer is applied to estimate the states of the governing ROM required by the sliding modecontroller. The proposed approach is applied to address the thermal spatiotemporal dynamic shapingproblem in a tubular chemical reactor.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Recently there has been an increasing focus on modeling andcontrol of distributed parameter systems (DPSs) in chemical processand advanced material production industries. Such type of systemsfrequently arise in a wide range of chemical processes, e.g., fixed andfluidized bed reactors, polymerization and crystallization processes,chemical vapor deposition systems and semiconductor manufacturingprocesses, due to the existence of diffusion, dispersion and convectionmechanisms (Adomaitis, 2003; Christofides, 2000; Lin and Adomaitis,2001; Ray, 1981; Theodoropoulou et al., 1998). It is imperative totightly control these processes so that there are zero product qualityexcursions, even when the process objectives dynamically changewhich is a usual occurrence in such industrial applications. WhileDPSs can be mathematically described by partial differential equations

(PDEs) and the control problem is a difficult task due to the spatialdistribution of the system states (Bohm et al., 1998; Christofides,2000; Curtain and Zwart, 1995; Krstic and Smyshlyaev, 2008; Ng andDubljevic, 2012; Smyshlyaev and Krstic, 2010), it becomes even morecomplicated in the case of chemical DPSs where chemical reactionstake place leading to nonlinearities in the governing equation.

Focusing on transport–reaction processes with significant diffu-sive mechanisms and their mathematical description, we note thatthey can be described by semi-linear dissipative PDEs whoseinfinite-dimensional representation in an appropriate functionalsubspace can be partitioned into two subsystems: slow (andpossibly unstable) and fast (and stable), with a time scale dynamicseparation (Christofides, 2000). Considering such property, modelorder reduction (MOR) methodologies have been extensively used inmodeling and control of chemical DPSs (Babaei Pourkargar andArmaou, 2014b, 2015b,c; Balas, 1991; Bentsman and Orlov, 2001;Christofides, 2000; Dubljevic et al., 2004; El-Farra et al., 2003;Hanczyc and Palazoglu, 1995). Galerkin's method is one of thetypical approaches to implement MOR. The required basis functions

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Chemical Engineering Science

http://dx.doi.org/10.1016/j.ces.2015.07.0540009-2509/& 2015 Elsevier Ltd. All rights reserved.

n Corresponding author. Tel.: þ1 814 865 5316; fax: þ1 814 865 7846.E-mail address: armaou@engr.psu.edu (A. Armaou).

Chemical Engineering Science 138 (2015) 673–684

in such approach can only be computed analytically if and only if thespatial differential operator is linear and the process operates overregular domains. Statistical approaches can be employed as analternative solution to compute the empirical basis functions of ageneral class of DPSs from an ensemble of solution profiles. Properorthogonal decomposition is one of the commonly used statisticaltechniques to find the optimal set of empirical basis functions for arepresentative set of solution data which has been widely applied inmodel reduction, optimization and control of DPSs (Armaou andChristofides, 2002; Babaei Pourkargar and Armaou, 2013b, 2014a;Izadi and Dubljevic, 2013; Sirovich, 1987) where geometric andLyapunov based approaches have been used.

Sliding mode control is a variable structure nonlinear controlmethod which changes the nonlinear system dynamics by applyinga discontinuous control signal (Khalil, 2002; Slotine and Li, 1991). Thesliding mode controller forces the system dynamics to slide along theboundaries of the system normal behavior called “sliding surface”(Edwards and Spurgen, 1998; Utkin, 1992). The discontinuous natureof the controller structure causes insensitivity to parameter variationsand complete disturbance rejection (Bandyopadhyay and Janardhanan,2006). Sliding mode optimization and controller designs have beenapplied in a wide range of chemical, mechanical and electrical systems(Bartolini et al., 1997; Bartoszewicz et al., 2008; Fridman, 2003;Hanczyc and Palazoglu, 1995; Misawa and Utkin, 2000).

To implement the model-reduced controller design for DPSswe need an accurate estimation of the states of the governingreduced order models (ROMs). Static observer designs, whichwere widely employed to estimate such desired states(Christofides, 2000; Dubljevic et al., 2004; Varshney et al.,2009), require the number of measurement sensors to be super-numerary to the number of ROM states. One of the solutions tocircumvent such requirement is applying dynamic observerswhich theoretically need only one measurement sensor(Babaei Pourkargar and Armaou, 2013a,b; 2014c). While dynamicobserver synthesis has reached an extensive maturity forlumped parameter systems described by ordinary differentialequations (ODEs) (Gauthier et al., 1992; Karafyllis and Kravaris,2005, 2012; Kazantzis and Kravaris, 1998; Keller, 1987; Michalskaand Mayne, 1995; Soroush, 1997; Thau, 1995), the synthesisproblem remains challenging for DPSs (Curtain et al., 2003;Fuji, 1980; Smyshlyaev and Krstic, 2005; Xu et al., 1995; Yangand Dubljevic, 2014).

In this paper we consider the spatiotemporal dynamic shaping oftransport–reaction processes via MOR. The dynamic shaping pro-blem is addressed by regulating the error dynamics between thegoverning PDE and a desired spatiotemporal dynamics which aredescribed by a target PDE with the same spatial differential operator.The governing target PDEs are discretized by applying Galerkin'smethod to obtain ROMs in the form of low-dimensional modal ODEswhen the required dominant basis functions are computed analyti-cally by solving the eigenproblem of the linear part of the spatialdifferential operator. The error dynamics between the governing andtarget systems are derived by subtracting the ROMs in the form oflow-dimensional ODEs which describe the spatiotemporal errordynamics. Then an output feedback control structure is synthesizedto stabilize the error dynamics. The control structure is consideredas a combination of a Lyapunov-based sliding mode controller(Khalil, 2002; Slotine and Li, 1991) and a Luenberger-type dynamicobserver to estimate the system modes.

The remainder of the paper is organized as follows: a mathe-matical description of the studied class of semi-linear DPSs andtheir properties are presented in Section 2. Section 3 presents ashort description of the used MOR method. The spatiotemporaldynamic shaping problem is addressed via an output feedbacksliding mode control structure synthesis in Section 4. Finally, theproposed dynamic shaping method is successfully illustrated on

thermal dynamic shaping inside a tubular chemical reactordescribed by a semi-linear PDE in Section 5.

2. Preliminaries

2.1. Problem formulation

To formulate the spatiotemporal dynamic shaping problem weconsider a 1D transport–reaction process which can be describedby a semi-linear PDE:

∂∂tXðz; tÞ ¼AnðzÞXðz; tÞþF z; Xðz; tÞð ÞþBðzÞuðtÞ;

yðtÞ ¼ZΩsðzÞXðz; tÞ dz;

q X;∂X∂z

;…;∂n�1X∂zn�1

� �¼ 0 on ∂Ω;

Xðz;0Þ ¼ X0ðzÞ; ð1Þ

where Xðz; tÞAR is the spatiotemporal state of the system, zAΩthe 1D spatial coordinate, t the time, Ω the process domain, ∂Ωthe process boundaries, AnðzÞ the linear spatial differential opera-tor of order n, F z; Xðz; tÞð Þ the smooth Lipschitz nonlinear function,uðtÞARl the vector of manipulated inputs, BðzÞ the spatial distribu-tion of manipulated inputs, yðtÞARp the vector of contentiousmeasurements, sðzÞ the vector of measurements' spatial distribu-tion, qðX; ∂X=∂z;…; ∂n�1X=∂zn�1Þ the vector of linear homogeneousboundary conditions, and X0ðzÞ the initial spatial profile of thesystem state. The dissipative PDE of (1) is linearly dominant, i.e.,the spatial differential operator is purely linear and the nonlinear-ity only appears as a Lipschitz function in the system dynamics.Such equation arises in the majority of transport–reaction pro-cesses in the chemical process industries (Christofides, 2000; Ray,1981), where the linear term of AnðzÞXðz; tÞ indicates the transport(diffusion, dispersion and convection) component and the non-linear term of F z; Xðz; tÞð Þ expresses the reaction dynamics.

Remark 1. According to the Lipschitz property of the nonlinearfunction of F z; Xðz; tÞð Þ which makes it to be sufficiently smooth,the Picard–Lindelöf theorem can be applied to guarantee theexistence and uniqueness of the solution (Teschl, 2012).

2.2. System representation

The studied DPS, which is described by the PDE of (1), can berepresented in the abstract infinite-dimensional form of

_xðtÞ ¼ AxðtÞþ f xðtÞð ÞþbuðtÞ; xð0Þ ¼ x0;

yðtÞ ¼ SxðtÞ; ð2Þ

by defining the functional state of xðtÞAW:

xðtÞ ¼ Xð�; tÞ;the linear differential operator:

AxðtÞ ¼AðzÞXð�; tÞ;the nonlinear function:

f xðtÞð Þ ¼ F z;Xð�; tÞð Þ;and the manipulated input operator:

buðtÞ ¼ BðzÞuðtÞ;in an appropriate Sobolev subspace of W:

WðΩÞ ¼ H;∂H∂z

;…;∂n�1H∂zn�1 AL2ðΩÞ : q H;

∂H∂z

;…;∂n�1H∂zn�1

� �¼ 0

� �;

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684674

equipped with the following inner product and norm:

H;Gð Þ ¼ZΩrðzÞHTG dz;

JHJ2 ¼ H;Hð Þ1=2;where H and G are elements of W, and HT denotes the transpose.Note that the inner product weighting function, r(z), is consideredto be 1 to simplify the analysis.

Assumption 1. We assume that the DPS described by the PDE of(1) and its infinite-dimensional representation, (2), is approxi-mately observable and controllable (Curtain and Zwart, 1995).

3. Model order reduction via Galerkin's method

The infinite-dimensional functional representation of (2) can beprojected into an infinite set of ODEs of the system vectorizedeigenmodes using standard Galerkin's method. The requiredeigenfunctions to discretize the states of the Sobolev subspaceare the solution of the following eigenproblem:

Aϕi ¼ λiϕi;

q ϕi;dϕi

dz;…;

dn�1ϕi

dzn�1

!∂Ω

¼ 0; ð3Þ

for i¼ 1;…;1, where λi and ϕi denote the ith eigenvalue and itscorresponding orthogonal eigenfunction, respectively. Note thatthe resulting countable set of eigenfunctions is a strong generatorof the defined Sobolev subspace, i.e., W9spanfϕig1i ¼ 1. To applyGalerkin's method we also require defining the adjoint eigenfunc-tions which satisfy the orthonormal property:

ϕi;ϕn

j

� �¼ δij; ð4Þ

where ϕn

i AW indicates the ith adjoint eigenfunction and δij theKronecker delta.

For the majority of transport–reaction processes where diffu-sion and dispersion play an important role we can assume a time-scale separation in the eigenspectrum of the linear operator of A.This assumption is formally stated as follows:

Assumption 2. Assume the eigenspectrum of linear operator of Adenoted by ΛðAÞ9fλ1; λ2;…g, where Reðλ1ÞZReðλ2ÞZ⋯ZReðλmÞZReðλmþ1ÞZ⋯, satisfies Reðλmþ1Þo0, σ ¼ jReðλ1Þj=jReðλmþ1Þj , where Reð�Þ denotes the real part and σ is a smallpositive number. According to such separation the eigenspectrumand corresponding Sobolev subspace, W9spanfϕig1i ¼ 1, can bepartitioned into the following subsets and subspaces:

1. finite subset of first m eigenvalues, ΛsðAÞ9fλ1; λ2;…; λmg,which are slow and possibly unstable, and its correspondingslow Sobolev subspace, Ws9spanfϕigmi ¼ 1,

2. complement infinite subset of the remainder eigenvalues,Λf ðAÞ9fλmþ1; λmþ2;…g, which are fast and stable, and itscorresponding fast Sobolev subspace, Wf 9spanfϕig1i ¼ mþ1,

where W¼Ws [ Wf .

Parameter σ needs to be a small positive number to ensure aproper separation between the finite dimensional slow andpossibly unstable, and infinite dimensional fast and stable sub-systems. A common value used in transport–reaction processesdynamics area is σC0:1. The condition of small σ in Assumption2 is satisfied by the majority of transport–reaction processes(Adomaitis, 2003; Christofides, 1998; Lin and Adomaitis, 2001;Lu et al., 2005; Theodoropoulou et al., 1998). The small value for σis not generally satisfied for

1. Convection–reaction processes described by first-order hyper-bolic PDEs where the eigenvalues cluster along vertical (or nearlyvertical), asymptotes in the complex plane (see Chapter 2 inChristofides, 2000 for more details).

2. Diffusion–reaction processes described by parabolic PDEs forwhich the spatial coordinate is defined in the infinite domain,where wavy behavior is usually exhibited as the result ofcontinuous eigenspectrum (see Christofides, 2000; Marquardt,1990) for more details).

In such cases, a modal separation in the system dynamics can beenforced by appropriate controller design (Christofides, 2000;Christofides and Daoutidis, 1996a,b).

Taking advantage of Assumption 2, the Galerkin integralprojectors can be defined,

P : W-Ws; Pð�Þ ¼ ð�;ΦsÞ; Φs ¼ ½ϕ1 ϕ2 ⋯ ϕm�T ;Q : W-Wf ; Qð�Þ ¼ ð�;Φf Þ; Φf ¼ ½ϕmþ1 ϕmþ2 ⋯�T ; ð5Þ

to project the infinite-dimensional system representation of (2) topartitioned sets of ODEs of the vectorized slow and fast eigen-modes:

_xsðtÞ ¼ AsxsðtÞþ f s xsðtÞ; xf ðtÞ� þbsuðtÞ; xsð0Þ ¼Px0;

_xf ðtÞ ¼ Af xf ðtÞþ f f xsðtÞ; xf ðtÞ� þbf uðtÞ; xf ð0Þ ¼Qx0; ð6Þ

where xðtÞ ¼ xsðtÞ � xf ðtÞ, As ¼PA¼ diagfλigmi ¼ 1, Af ¼QA¼diagfλig1i ¼ mþ1, f s ¼Pf , f f ¼Qf , bs ¼Pb, bf ¼Qb and diagf�gdenotes the diagonal matrix with diagonal elements. Then thepartitioned infinite dimensional ODEs of (6) can be approximatedby

_xsðtÞ ¼ AsxsðtÞþ f s xsðtÞ;0ð ÞþbsuðtÞ; xsð0Þ ¼Px0; ð7Þafter a short period of time, tb, when xf-0, by applying singularperturbation analysis (Babaei Pourkargar and Armaou, 2013b,2014b, 2015a; Christofides, 2000) and considering Tykhonov'stheorem for solution convergence of systems that consist of slowand fast subsystems (Lobry et al., 1998). The time-scale separationbetween slow and fast dynamics of the DPS modeled by PDEsensures that a controller which exponentially stabilizes the closed-loop finite dimensional approximation also exponentially stabilizesthe closed-loop infinite-dimensional system (Christofides, 2000).

Remark 2. We may replace the approximate observability andcontrollability assumption of the infinite-dimensional system of(2) formally addressed in Assumption 1, by observability andcontrollability of the system approximation (slow subsystem) of(7) (Curtain and Zwart, 1995) when the fast subsystem in (6) isexponentially stable and the control input is bounded and changesat a rate that is in the slow time-scale.

Remark 3. A lower bound for the relaxation time, tb, required bythe fast dynamics of the system to relax, can be identified bysingular perturbation analysis (Babaei Pourkargar and Armaou,2013b, 2015a; Christofides, 2000).

4. Spatiotemporal dynamic shaping using sliding modecontroller designs

4.1. Dynamic shaping error formulation

To address the spatiotemporal dynamic shaping problem of theDPSs described by PDE system of (1), we consider a desiredspatiotemporal dynamics described by a target PDE with the samespatial differential operator:

∂∂tXdðz; tÞ ¼AnðzÞXdðz; tÞþF 0 z; t;Xdðz; tÞð Þ;

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684 675

q Xd;∂Xd

∂z;…;

∂n�1Xd

∂zn�1

� �¼ 0 on ∂Ω;

Xdðz;0Þ ¼ Xd0ðzÞ: ð8Þ

According to same spatial differential operator and boundaryconditions, both the system and target PDEs have the same setof dominant eigenfunctions which can be applied to approximatethe spatiotemporal states as follows:

Xðz; tÞ � xTs ðtÞΦsðzÞ;Xdðz; tÞ � xTdsðtÞΦsðzÞ: ð9ÞNote that such approximations are quite accurate after a shortrelaxation time when the fast dynamics stabilize. Then thespatiotemporal dynamic shaping error with respect to slow systemdynamics can be formulated by

Eðz; tÞ ¼ Xðz; tÞ�Xdðz; tÞ � xTs ðtÞ�xTdsðtÞ�

ΦsðzÞ ¼ eT ðtÞ ΦsðzÞ; ð10Þwhere eðtÞ ¼ xsðtÞ�xdsðtÞ indicates the vector of modal errors.

By applying MOR via Galerkin projection as it was discussed inSection 3, we can discretize the target PDE to the following set ofODEs:

_xdsðtÞ ¼ AsxdsðtÞþ f 0s t; xdsðtÞð Þ; xdsð0Þ ¼Pxd0; ð11Þwhich describes the slow modal dynamics of the desired spatio-temporal behavior. The modal tracking error dynamics can then bederived by subtracting the ODEs of (7) and (11),

_eðtÞ ¼ AseðtÞþ f s eðtÞ; xdsðtÞð Þ� f 0s t; xdsðtÞð ÞþbsuðtÞ; eð0Þ ¼Pe0;ð12Þ

where e0 ¼ x0�xd0. Thus the spatiotemporal dynamic shapingproblem of the transport–reaction processes described by (1) canbe addressed via error dynamics regulation of the low-dimensional ODEs of (12).

Remark 4. Note that the fast dynamics must be exponentiallystable for the specific method to be applicable even for unstabletarget PDEs.

Remark 5. Wemay directly apply the proposed MOR to the PDE ofthe spatiotemporal dynamic shaping offset which is formulated bysubtracting the governing PDE of (1) from the target of (8). Suchalternative approach results to the same modal tracking errordynamics as (12).

Remark 6. In theory, we can reshape the system to any desiredspatiotemporal response, in essence by canceling out the originaldynamics of the system and replacing them with the intendeddynamics of target. There are some considerations with allowableoriginal and target dynamics. The target spatiotemporal dynamicsmust be bounded to avoid possible issues that arise in designingcontroller structures with unbounded actions. Since in the proposedcontrol approach the desired spatiotemporal behavior described bythe target PDE must have the same spatial differential operator andboundary conditions as the process PDE, the nonlinear functions ofF and F 0 are the only sources of differences in the governing andtarget dynamics. The most important restriction of the proposedapproach is choosing a bounded target nonlinear functions F 0 and astable target PDE. In addition, the locations of the actuators mustsatisfy the controllability requirements of error dynamics. Evenwhen bounded, F 0 should not be introducing an oscillatory behaviorin the target system with frequency that falls in the separation gapof the slow and fast dynamics; otherwise it will introduce a modelreduction error that can also affect the tracking of the desiredspatiotemporal dynamics.

4.2. Sliding mode controller design

Consider the nonlinear dynamical modal error system of (12):

_e ¼ Aseþ f sðe; xdsÞ� f 0sðt; xdsÞþbsu;

where e¼ ½e1; e2;…; em�T ARm and u¼ ½u1;u2;…;ul�T ARl. Theobjective is designing an output-feedback control law, u(t), whichregulates the error dynamics at the origin. By stabilizing the error,such regulator enforces the system dominant eigenmodes tofollow the target eigenmodes. According to the dynamical modalerror system dimension, we require at least m manipulated inputs(i.e., lZm) to stabilize the tracking error and force the systemspatiotemporal dynamics to follow the target dynamics with ageneral nonlinear term (Friedland, 1986; Gruyitch, 2013).

To reach such objective a sliding mode control approach isapplied. The controller synthesis includes two steps: (1) Choosinga manifold (reduced-order subspace), also known as sliding surface,which describes the desired dynamic behavior. (2) Designing thefeedback control law which forces the tracking error trajectory toconfine to the sliding surface and slides along it (Edwards andSpurgen, 1998; Utkin, 1992). Thus the problem of dynamic modalresponse shaping a- ~a is equivalent to “approaching to the slidingsurface and remaining on it”. The time-varying sliding surface, S(t), isdefined by the scalar equation of sðeÞ ¼ 0 which must guarantee thesystem control objective in regulating the tracking error dynamics:

S¼ feARm : sðeÞ ¼ 0g: ð13ÞThe switching function of s(e) is a distance measurement whichindicates how far the system eigenmodes are from the targeteigenmodes. It can be simply defined as the following proportional-integral form:

sðeÞ ¼ eþΠZ

e dt; ð14Þ

where Π ¼ diag fπigmi ¼ 1. To assure the stability of sliding modedynamic, the diagonal components of Π must have negative realparts, i.e., ReðπiÞo0 for i¼ 1;…;m. The places of such components inthe left half-plane determine the performance of the sliding modecontroller in stabilizing the tracking error dynamics.

Due to the discontinuous nature of the resulting sliding modecontrol action, the existence and uniqueness of the closed-loopsystem solution cannot be verified by the Picard–Lindelöf theorem(Teschl, 2012). Such switching discontinuous closed-loop dynamicsmust be analyzed using Filippov theorem (Filippov, 1988; Zinober,1990) which states that the resulting closed-loop system that slidesalong sðeÞ ¼ 0 can be approximated by a smooth dynamics which isdescribed using

_sðeÞ ¼ 0

under a continuous control design (Khalil, 2002; Slotine and Li,1991). To formulate the sliding mode controller design we considerthe Lyapunov function in the quadratic form of

VðsÞ ¼ 12 s

Ts: ð15Þ

By considering the above Lyapunov function, the asymptotic stabi-lity can be obtained by

_V ðsÞ ¼ sT _sr0: ð16ÞThen using Filippov's construction of the equivalent dynamics, thecontrol action can be computed by considering the smoothdynamics of _s ¼ 0, which

_s ¼ _eþΠe¼ Aseþ f sðe; xdsÞ� f 0sðt; xdsÞþbsueqþΠe¼ 0: ð17ÞAs a result the equivalent control law takes the following form:

ueq ¼ �b?s ðAsþΠÞeþ f sðe; xdsÞ� f 0sðt; xdsÞ�

; ð18Þ

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684676

where b?s ¼ bTs ðbsbTs Þ�1 identifies the Moore–Penrose pseudo-

inverse (Penrose, 1995). Note that b?s ¼ b�1

s for m¼ l. In order tosatisfy the sliding condition we consider the control law as

u¼ ueq�b?s η signðsÞ; ð19Þ

where η40 and signð�Þ denotes the sign function. Then we obtain

_V ðsÞ ¼ sT _s ¼ �η sT signðsÞ; ð20Þ

where _V ðsÞo0 and _V ð0Þ ¼ 0, thus the closed-loop system is locallyasymptotically stable in the Lyapunov sense (Khalil, 2002; Slotineand Li, 1991).

Remark 7. The control law of (19) was derived by Lyapunov directmethod which is a typical controller design approach for generalnonlinear systems (Khalil, 2002; Slotine and Li, 1991). First weintroduced a Lyapunov function candidate in the form of (15).Using Fillippov's construction of the equivalent dynamics wefound an equivalent controller formula for which the time deri-vative of the Lyapunov function candidate is equal to zero. Thenwe obtained the control law by subtracting a positive term whichcontains a tuning parameter from the equivalent control formulato obtain the negative sign for Lyapunov candidate time derivativewhich guarantees the closed-loop process stability.

Remark 8. The positive controller tuning parameter of η allows acertain degree of flexibility in spatiotemporal dynamic shaping ofthe studied transport–reaction process. Large values of η force theLyapunov function time derivative, _V , to be more negative andtherefore generate a faster transient response at the cost of largercontrol action.

Remark 9. In practice, implementation of the sliding mode con-troller with the discontinuous nonlinearity in the form of signð�Þ,presents the chattering phenomenon (“zig-zag” motion) due tofast switching fluctuations across the sliding surface. Chatteringwhich involves high control activity is undesirable in practice.Such high activities may excite high frequency dynamics neglectedin the process modeling step (Khalil, 2002; Slotine and Li, 1991).

4.3. Dynamic observer design

For the desired spatiotemporal dynamics which is described by(8) and approximated by the slow dynamics of (11) we have accessto the dominant eigenmodes, xds. However, to compute thetracking error vectors, e, and implement the control law of (19),we need to estimate the values of the system dominant eigen-modes of xs. For such estimation purpose, a Luenberger-type

dynamic observer is synthesized based on the system ROM:

_x s ¼ Asxsþ f sðxsÞþbsuþΘðy�CxsÞ;y¼ Cxs; ð21Þwhere

C ¼

ϕ1ðω1Þ ϕ2ðω1Þ ⋯ ϕmðω1Þϕ1ðω2Þ ϕ2ðω2Þ ⋯ ϕmðω2Þ

⋮ ⋮ ⋱ ⋮ϕ1ðωpÞ ϕ2ðωpÞ ⋯ ϕmðωpÞ

266664

377775;

and xs denotes the estimated dominant eigenmodes of the system,Θ presents the dynamic observer gain matrix and ω¼½ω1 ω2 ⋯ ωp�T is the vector of locations of continuous measure-ments, y. The modal observation error dynamics can then beformulated as follows:

_eo ¼ ðAs�ΘCÞeoþ f sðeoþ xsÞ� f sðxsÞ; ð22Þwhere the vector of observation error is defined by eo ¼ xs� xs.Assuming the principle of separation between control and obser-vation holds (Atassi and Khalil, 2001; Brezinski, 2002), we con-sider the observation Lyapunov function (OLF) in the standardquadratic form

Vo ¼ 12 e

ToP eo; ð23Þ

where P is a symmetric positive definite matrix with a boundednorm, JP JrK1. The time derivative of the OLF is obtained asfollows:

_V o ¼ 12

_eToP eoþeToP _eo� �

¼ 12

ðAs�ΘCÞeoþ f sðeoþ xsÞ� f sðxsÞ �TP eoþeToP ðAs�ΘCÞeoþ f sðeoþ xsÞ� f sðxsÞ

�� �

¼ 12

eTo ðAs�ΘCÞT þ f sðeoþ xsÞ� f sðxsÞ� Th i

P eoþeToP ðAs�ΘCÞeo�

Fig. 2. (a) Temperature dominant eigenfunctions and (b) their adjoints.

Fig. 1. Tubular flow reactor with l independent cooling jackets.

Table 1Dominant eigenvalues

Eigenvalues i¼1 i¼2 i¼3 i¼4 i¼5 i¼6

λi �1.94 �4.80 �10.97 �20.93 �34.78 �52.56

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684 677

þ f sðeoþ xsÞ� f sðxsÞ�Þ

¼ 12

eTo ðAs�ΘCÞTPþPðAs�ΘCÞh i

eoþeToP f sðeoþ xsÞ� f sðxsÞ ��

þ eToP f sðeoþ xsÞ� f sðxsÞ �� T�

: ð24Þ

Then the dynamic observer gain matrix, Θ, must be identified subjectto _V oo0 which indicates the observation stability. Due to the Lipschitzproperty of the nonlinear function of F in the DPS of (1) and boundednature of the eigenfunctions and their adjoints, the nonlinear functionof fs in the system approximation of (7) is also Lipschitz continuous:

J f sðeoþ xsÞ� f sðxsÞJrK2 Jeo J ; ð25Þ

where K2 denotes the Lipschitz upper bound gain. From (24) and (25)and using Cauchy–Schwarz inequality we obtain that

eToP f sðeoþ xsÞ� f sðxsÞ �þ eToP f sðeoþ xsÞ� f sðxsÞ

�� Tr2JeToP f sðeoþ xsÞ� f sðxsÞ

�J

r2JeTo J JP J J f sðeoþ xsÞ� f sðxsÞJr2K1K2 JeTo J Jeo J

¼ K Jeo J2 ¼ KeToeo; ð26Þ

where K ¼ K1K2. By applying the inequality of (26), we conclude that if

eTo ðAs�ΘCÞTPþPðAs�ΘCÞh i

eoþKeToeoo0 ð27Þ

then _V oo0 and _V ð0Þ ¼ 0. Thus

eTo ðAs�ΘCÞTPþPðAs�ΘCÞþKIh i

eoo0; ð28Þ

where ðAs�ΘCÞTPþPðAs�ΘCÞþKIo0 guarantees the asymptoticstability of the observation error dynamics. We can then address thedynamic observer synthesis problem via a standard linear matrixinequality (LMI) problem:

ðAs�ΘCÞTPþPðAs�ΘCÞþKIo�PYP�PΘZΘTP; ð29Þ

where Y and Z are the symmetric positive definite weighting matrices.From the inequality of (29) we obtain

ðATs �CTΘT ÞPþPðAs�ΘCÞþKIþPYPþPΘZΘTPo0

) ATs PþPAs�ðPΘCÞT �PΘCþPYPþPΘZΘTPþKIo0

) ATs PþPAs�ðUCÞT �UCþPYPþUZUT þKIo0

) ATs PþPAsþðUT �Z�1CÞTZðUT �Z�1CÞ�CTZ�1CþPYPþKIo0;

ð30Þwhere U ¼ PΘ. Then we reduce the degrees of freedom in theinequality by setting U ¼ CTZ�T ,

ATs PþPAs�CTZ�1CþPYPþKIo0: ð31Þ

Using the Schur complement lemma we represent the inequality of(31) in the following standard form (Boyd et al., 1994):

PAsþATs P�CTZ�1CþKI P

P �Y �1

" #o0: ð32Þ

The observer gain matrix can be computed by minimizing the trace ofP�1 subject to the LMI constraint of (32):

Θ¼ P�1CTZ�1: ð33ÞA detailed discussion on the LMI-constrained optimization problem canbe found in Scherer et al. (1997).

5. Application to thermal dynamic shaping in a tubular reactor

In this section, we illustrate the effectiveness of the proposedoutput feedback sliding mode control on spatiotemporal dynamicshaping of a typical transport–reaction process example. In the firstpart we present the mathematical model of the thermal dynamics ina tubular chemical reactor. The desired spatiotemporal behavior isdescribed in the second part. Then the tailored MOR and controlstructure are presented for the specific dynamic shaping problem inthe third part of this section. Finally, the closed-loop simulation

Fig. 3. Open-loop (a) spatiotemporal profile of the dimensionless temperature and (b) temporal profile of temperature L2-norm.

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684678

results are provided to assess the system performance under theproposed output feedback sliding mode control structure.

5.1. System description

We consider a tubular chemical flow reactor (Christofides,2000) where an irreversible exothermic reaction of the zero-thorder takes place. As presented in Fig. 1, a limited set of l coolingjackets are employed as the manipulated inputs to remove the

heat from the reactor and manage the thermal energy along thereactor length as time evolves. The spatiotemporal thermaldynamics in the presence of temperature-dependent reactionrate is derived from the energy balance which takes the form ofthe following dissipative PDE:

∂T∂t

¼ kρCp

∂2T∂z2

�v∂T∂z

þð�ΔHÞρCp

ro exp�ERT

� ��Xlj ¼ 1

BjðzÞhAc

ρCpðT�Tc;jÞ;

Fig. 4. (a) Spatiotemporal profile of the desired dimensionless temperature and (b) temporal profile of its L2-norm.

Fig. 5. Open-loop (a) spatiotemporal profile of the shaping error and (b) temporal profile of its L2-norm.

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684 679

z¼ 0 :∂T∂z

¼ ρCpvk

ðT�Tf Þ;

z¼ L :∂T∂z

¼ 0;

t ¼ 0 : T ¼ T0; ð34Þ

where T is the temperature of the fluid inside the reactor, t thetime, zA ½0; L� the spatial coordinate, L the reactor length, k thethermal conductivity, ρ the density, Cp the heat capacity, v theaxial velocity, ð�ΔHÞ the heat of reaction, r0 the pre-exponentialreaction constant, E the activation energy, h the heat transfercoefficient between reactor and cooling jacket, Ac the coolingsurface area, Tc;j the cooling jackets temperatures, Tf the feedtemperature, l the number of cooling jackets, BjðzÞ the spatialdistribution of the jth cooling jacket, and T0 the initial tempera-ture profile. We reformulate the PDE set of (34) in dimensionlessform and homogenize the left boundary condition to employ theproposed MOR. To homogenize the boundary condition weinduce the non-homogeneous part in the PDE by standard Diracfunction. The resulting dimensionless PDE takes the following

form:

∂T∂t

¼ 1Pe

∂2T∂z2

�∂T∂z

þBt expγT

1þT

!þBc

Xlj ¼ 1

BjðzÞðui�T Þþδðz�0ÞT f

z ¼ 0 :∂T∂z

¼ PeT ;

z ¼ 1 :∂T∂z

¼ 0;

t ¼ 0 : T ¼ 0; ð35Þwhere

t ¼ tvL; z ¼ z

L; T ¼ T�T0

T0; Pe¼ ρCpvL

k;

γ ¼ ERT0

; uj ¼Tc;j�T0

T0; T f ¼

Tf �T0

T0;

Bt ¼ð�ΔHÞro exp � E

RT0

� �L

ρCpT0v; Bc ¼

hAcLρCpv

:

A typical diffusion–convection–reaction process with the samespatial differential operator is considered as the target PDE which

Fig. 6. Temporal profiles of (a) dominant eigenmodes of the system, (b) desired dominant eigenmodes and (c) modal errors during the open-loop period.

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684680

describes the desired spatiotemporal dynamics:

∂T d

∂t¼ 1Pe

∂2T d

∂z2�∂T d

∂zþα3T

3dþα2T

2dþα1T dþα0þβ2 cos ð0:5πt Þ

þβ1 sin ð0:5πt Þ;

z ¼ 0 :∂T d

∂z¼ PeT d;

z ¼ 1 :∂T d

∂z¼ 0;

t ¼ 0 : T d ¼ 0: ð36Þ

Remark 10. The time-varying target PDE of (36), which is set bychoosing the bounded time-varying nonlinear term, presents apermanent oscillatory behavior in spatiotemporal thermal dynamicsof the tubular reactor.

Fig. 7. (a) Spatiotemporal profile of the dimensionless temperature and (b) temporal profile of its L2-norm for the entire process operation.

Fig. 8. (a) Spatiotemporal profile of the shaping error and (b) temporal profile of its L2-norm for the entire process operation.

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684 681

5.2. Model order reduction and output feedback control structureFor MOR of the system and target PDEs we require the eigen-

functions which must be computed by solving the followingeigenproblem of the system and target spatial differential operator:

1Pe

d2ϕi

dz2�dϕi

dz¼ λiϕi;

z ¼ 0 :dϕi

dz¼ Peϕi;

z ¼ 1 :dϕi

dz¼ 0: ð37Þ

where i¼ 1;2;…;1 . The solution of above eigenvalue–eigenfunc-tion problem takes the following form (Christofides, 2000; Lao et al.,2014; Liu et al., 2014):

λi ¼ � α2i

PeþPe

4

� �; tan ðαiÞ ¼

Peαi

α2i �

Pe2

� �2;

ϕiðzÞ ¼ ξi expPez2

� �cos ðαizÞþ

Pe2αi

sin ðαizÞ�

;

ξi ¼Z 1

0cos ðαizÞþ

Pe2αiz

sin ðαizÞ� 2

dz

!�ð1=2Þ

: ð38Þ

The resulting eigenfunctions are not self-adjoint because the spatialdifferential operator of ð1=PeÞð∂2=∂z2Þ�ð∂=∂zÞ is non-self-adjoint;then to apply the Galerkin projection we must define the adjointeigenfunctions:

ϕn

i ðzÞ ¼ expð�PezÞϕiðzÞ; ð39Þwhich satisfy the orthonormal property of (4). By considering the setof m dominant eigenfunctions of Φs ¼ ½ϕ1 ϕ2 ⋯ ϕm�T , we approx-imate the system and desired dimensionless temperatures:

T ðz; t Þ �Xmi ¼ 1

aiðt ÞϕiðzÞ;

T dðz; t Þ �Xmi ¼ 1

~aiðt ÞϕiðzÞ; ð40Þ

where σ ¼ jλ1 j=jλmþ1 j has a small positive value. Then by employ-ing Galerkin's method, the ROMs of the system and target PDEs takethe following modal forms:

_ak ¼ λkakþBt

Z 1

0ϕn

k expγPm

i ¼ 1 aiϕi

1þPmi ¼ 1 aiϕi

� �dzþϕn

kð0ÞT f �Bc

�Xlj ¼ 1

Z 1

0Bjϕ

n

k

Xmi ¼ 1

aiϕi dz

!þBc

Xlj ¼ 1

Z 1

0Bjϕ

n

k dz

!uj;

_~ak ¼ λk ~akþZ 1

0α3

Xmi ¼ 1

~aiϕi

!3

þα2

Xmi ¼ 1

~aiϕi

!20@

þα0þβ2 cos ð0:5π t Þþβ1 sin ð0:5π t Þϕn

k dzþα1 ~ak; ð41Þ

for k¼ 1;2;…;m. Then the above ROMs can be summarized in thefollowing abstract forms:

_a ¼ Aaþ f ðaÞþBu;_~a ¼ A ~aþ f 0ðt ; ~aÞ; ð42Þwhere

a¼

a1a2⋮am

26664

37775; ~a ¼

~a1

~a2

⋮~am

26664

37775; A¼

λ1 0 ⋯ 00 λ2 ⋯ 0⋮ ⋮ ⋱ ⋮0 0 ⋯ λm

266664

377775; u¼

u1

u2

⋮ul

266664

377775;

f ðaÞ ¼

BtR 10 ϕ

n

1 expγPm

i ¼ 1aiϕi

1þPm

i ¼ 1aiϕi

� �dzþϕn

1ð0ÞT f �Bc

Xlj ¼ 1

Z 1

0Bjϕ

n

1

Xmi ¼ 1

aiϕi dz

!

BtR 10 ϕ

n

2 expγPm

i ¼ 1aiϕi

1þPm

i ¼ 1aiϕi

� �dzþϕn

2ð0ÞT f �Bc

Xlj ¼ 1

Z 1

0Bjϕ

n

2

Xmi ¼ 1

aiϕi dz

!

⋮

BtR 10 ϕ

n

m expγPm

i ¼ 1aiϕi

1þPm

i ¼ 1aiϕi

� �dzþϕn

mð0ÞT f �Bc

Xlj ¼ 1

Z 1

0Bjϕ

n

m

Xmi ¼ 1

aiϕi dz

!

2666666666666664

3777777777777775

;

B¼

BcR 10 B1ϕ

n

1 dz BcR 10 B2ϕ

n

1 dz ⋯ BcR 10 Blϕ

n

1 dz

BcR 10 B1ϕ

n

2 dz BcR 10 B2ϕ

n

2 dz ⋯ BcR 10 Blϕ

n

2 dz

⋮ ⋮ ⋱ ⋮BcR 10 B1ϕ

n

m dz BcR 10 B2ϕ

n

m dz ⋯ BcR 10 Blϕ

n

m dz

2666664

3777775;

f 0ðt ; ~aÞ ¼

R 10 α3

Xmi ¼ 1

~aiϕi

!3

þα2

Xmi ¼ 1

~aiϕi

!2

þα0þβ2 cos ð0:5πtÞþβ1 sin ð0:5πtÞ0@

1Aϕn

1dzþα1 ~a1

R 10 α3

Xmi ¼ 1

~aiϕi

!3

þα2

Xmi ¼ 1

~aiϕi

!2

þα0þβ2 cos ð0:5πtÞþβ1 sin ð0:5πtÞ0@

1Aϕn

2dzþα1 ~a2

⋮

R 10 α3

Xmi ¼ 1

~aiϕi

!3

þα2

Xmi ¼ 1

~aiϕi

!2

þα0þβ2 cos ð0:5πtÞþβ1 sin ð0:5πt Þ0@

1Aϕn

mdzþα1 ~am

26666666666666664

37777777777777775

:

By considering the modal error as e¼ a� ~a, the error dynamics canbe formulated by

_e ¼ Aeþ f ðe; ~aÞ� f 0ðt ; ~aÞþBu: ð43ÞThen the sliding mode controller and dynamics observer structurestake the following form:

u¼ �B? ðAþΠÞeþ f ðe; ~aÞ� f 0ðt ; ~aÞþη sign eþΠZ t

0e dt

� �� �;

_a ¼ Aaþ f ðaÞþBuþΘðy�CaÞ;e ¼ a� ~a; ð44Þwhere the dynamic observer gain matrix,Θ, can be computed usingthe LMI-constrained optimization problem described in Section 4.3.

5.3. Simulation results

To simulate the system we set the following typical values forthe process parameters: Pe¼5, γ ¼ 3, Bt¼0.4, Bc¼0.5 and T f ¼ 0.The dominant eigenvalues of the system are presented in Table 1.We can easily recognize an order of magnitude separation

Fig. 9. Required control actions for the entire process operation. The zero in ½0;8� denotes the open-loop period.

D.B. Pourkargar, A. Armaou / Chemical Engineering Science 138 (2015) 673–684682

between the 3 dominant eigenvalues and the remainder,σ ¼ Reðjλ1 j Þ=Reðjλ4 j Þ ¼ 0:093. The corresponding dominanteigenfunctions and their adjoints are presented in Fig. 2.

When considering the first three dominant eigenfunctions todiscretize the system and desired PDEs, the resulting system andtarget approximations are of dimension 3 (m¼3). We set l¼3 coolingjackets along the reactor length to shape the spatiotemporal tem-perature dynamics of the system. The spatial distributions of thecontrol actuators are also described by B1ðzÞ ¼Hðz�0:1Þ�Hðz�0:2Þ,B2ðzÞ ¼Hðz�0:4Þ�Hðz�0:5Þ and B3ðzÞ ¼Hðz�0:7Þ�Hðz�0:9Þ,where Hð�Þ indicates the standard Heaviside step function. We alsoconsider two point sensors to measure the reactor temperature atω¼ ½0:3 0:6�T which can be employed by the dynamic observer toestimate the dominant eigenmodes of the system.

The entire system operation was partitioned into two timeperiods: (1) open-loop process operation, to ¼ ½0;8� when thecontroller was inactive, and (2) closed-loop process operation,t c ¼ ð8;12� under the proposed controller design. The open-loopspatiotemporal profile of the dimensionless temperature and thetemporal profile of its spatial L2-norm while the controller wasinactive ðui ¼ 0 for i¼ 1;2;3Þ are presented in Fig. 3. It is observedthat the dimensionless temperature converges to a nonuniformsteady state profile. Fig. 4 also shows the desired spatiotemporalprofile and its L2-norm where α0 ¼ 0:15, α1 ¼ 0:05, α2 ¼ �0:1,α3 ¼ 0:15, β1 ¼ 0:2 and β2 ¼ �0:05. We observe a permanentoscillatory behavior in the desired temperature spatiotemporaldynamics. The open-loop spatiotemporal profile of the shapingerror and its spatial L2-norm are presented in Fig. 5.

The dynamic observer gain matrix was computed as follows:

Θ¼0:20 0:640:31 �0:48�0:27 �2:12

264

375

by solving the LMI-constrained optimization problem. Fig. 6 pre-sents the open-loop temporal profiles of the estimated dominanteigenmodes of the system, the desired dominant eigenmodes andthe open-loop modal errors where the nondominant system anddesired eigenmodes were negligible.

To implement the proposed sliding mode controller we setη¼ 0:5 for the controller parameter which guarantees the closed-loop system stability and performance. The spatiotemporal profileof the dimensionless reactor temperature and the temporal profileof its spatial L2-norm is illustrated in Fig. 7 for the entire processoperation. The spatiotemporal profile of the shaping error and itsL2-norm are also presented in Fig. 8. Note that the controller wasonly active during the closed-loop process operation of t c ¼ ð8;12�.We observe that the shaping error converges to zero and thesystem follows the desired spatiotemporal behavior under theproposed control approach.

The required control actions to stabilize the shaping error areshown in Fig. 9. The zero control actions in the period of t o ¼ ð0;8Þillustrate the open-loop process operation. A sharp initial changein the control action is observed when the controller is activated att ¼ 8, but that is expected since this is a static controller designand there is significant offset from the desired behavior. We do notobserve any significant chattering in the temporal profile of thecontrol actions. An oscillatory behavior is observed in the con-troller signals due to the permanent oscillatory behavior of thetarget spatiotemporal dynamics. Finally, the temporal profile ofthe estimated dominant eigenmodes and the modal errors arepresented in Fig. 10 which illustrates the effectiveness of thesliding mode controller in regulating the shaping error andtracking the desired spatiotemporal dynamics.

6. Conclusion

The spatiotemporal dynamic shaping of semi-linear distributedparameter systems was investigated by regulating the errordynamics between the reduced order models (ROMs) of governingand target partial differential equations (PDEs). The required ROMswere derived by applying Galerkin projection to the describingPDEs. The spatiotemporal error dynamics between the governingand target ROMs were stabilized using a sliding mode controllerdesign combined with a Luenberger-type dynamic observer toestimate the required states. The effectiveness of the proposedcontrol structure was illustrated on thermal dynamic shaping of atubular flow reactor.

Acknowledgment

Financial support from the National Science Foundation, CMMIAward #13-00322 is gratefully acknowledged.

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