Development of a Closed Form Nonlinear Predictive Control Law Based on a Class of Wiener Models

PROCESS DESIGN AND CONTROL

Development of a Closed Form Nonlinear Predictive Control Law Based on aClass of Wiener Models

Shraddha Deshpande,† Sachin C. Patwardhan,*,‡ Ravi Methekar,‡ and Raghunathan Rengaswamy§

Systems and Control Engineering and Department of Chemical Engineering, Indian Institute of TechnologyBombay, Powai, Mumbai 400076, India, and Department of Chemical Engineering, Clarkson UniVersity,Postdam, New York 13699-5705

Nonlinear model predictive control (NMPC) is increasingly being used for controlling microscale and system-on-chip devices, which exhibit complex and very fast dynamics. For effective control of such systems it isnecessary to develop computationally efficient approaches for solving the NMPC problem. In this work, aWiener type model has been used for capturing dynamics of multivariable nonlinear systems with fadingmemory. The resulting discrete nonlinear state space model is used to generate multistep predictions andformulate an unconstrained NMPC problem. A closed form solution to this problem is constructed analyticallyusing the theory of solutions of quadratic operator polynomials. The effectiveness of the resulting quadraticcontrol law is demonstrated by conducting simulation studies on a proton exchange membrane fuel cell(PEMFC) system, which exhibits fast dynamics and input multiplicity behavior. The quadratic control law isexpected to control the PEMFC at its optimum (singular) operating point. The proposed laws achieve a fastand smooth transition from a suboptimal operating point to the optimum operating point with significantlysmall computation time. The proposed law is also found to be robust in the face of moderate perturbation inthe unmeasured disturbances. The simulation results are validated by conducting experimental studies on asingle cell PEMFC system and a benchmark heater-mixer setup that exhibits input multiplicity behavior.Through the experimental studies, we demonstrate that the proposed quadratic control law is able to operatethe system at a singular operating point and establish the feasibility of employing the proposed control lawfor systems with very fast dynamics.

1. Introduction

With the availability of fast microprocessors and microcon-trollers, computationally complex control schemes, such asnonlinear model predictive control (NMPC), are increasinglybeing used for controlling systems with very fast dynamics.Application of embedded nonlinear predictive control formicroscale and system-on-chip devices gives rise to challengingproblems due to the fast transient behavior and limited comput-ing resources available with microcontroller systems.1 In recentyears, NMPC schemes have been applied for control of a dc-dcconverter,2 control of a spring-mass system,3 control of a twinpendulum system,4 control of temperature distribution alongwafer geometry, and temperature control of a fluid flowingmicrochannel.5 These are typical examples of systems that havevery fast and highly nonlinear dynamics. For effective controlof such systems it is necessary to develop computationallyefficient approaches for solving NMPC problem.

The NMPC applications mentioned above typically employcomputationally efficient solutions to the constrained NMPCformulations. The need to handle operating constraints makesit difficult to develop closed form solutions to the NMPCproblem. Even if the unconstrained version of the NMPCproblem is considered, highly nonlinear and complex model

structures seldom facilitate the construction of closed formsolutions. However, if a model can be constructed or locallyapproximated such that the resulting structure facilitates theconstruction of closed form solutions to the unconstrainedNMPC formulation, then the time required for online computa-tions can be reduced significantly. The development of suchunconstrained control laws can be traced back to the Newtontype controller developed by Economou6 and Economou andMorari7 under a nonlinear internal model control (NIMC)framework. They employed the concept of successive locallinearization to obtain the closed form control law. In addition,they reinterpreted that the contraction mapping principle canbe used to analyze the convergence of some iterative schemes.Patwardhan and Madhavan8 extended the Newton type approachusing successive quadratic approximations constructed basedon Taylor series expansion of the discrete nonlinear operator.The model approximation was developed using the solutionsof first and second order sensitivity equations. The controllersynthesis was carried out using the theory of solutions ofquadratic operator polynomials developed by Rall.9 A significantadvantage of the NIMC framework is that it facilitates analysisof an unconstrained NMPC scheme. However, the above-mentioned formulations are based on one-step-ahead predictions,which can give rise to deadbeat type control action. Moreover,these approaches, as proposed, are based on mechanistic models.The development of a reliable dynamic model of a system fromfirst principles is, in general, a difficult and time-consumingtask.

* To whom correspondence should be addressed. Fax: +91 2225726895. E-mail: [email protected].

† Systems and Control Engineering, Indian Institute of TechnologyBombay.

‡ Department of Chemical Engineering, Indian Institute of Technol-ogy Bombay.

§ Clarkson University.

Ind. Eng. Chem. Res. 2010, 49, 148–165148

10.1021/ie801284b CCC: $40.75 2010 American Chemical SocietyPublished on Web 11/23/2009

When compared with a mechanistic model, the developmentof a black box model requires much less time and effort. Thus,from a practical viewpoint, the development of an unconstrainedNMPC scheme based on a black box model appears to be anattractive option. The first step toward development of such ascheme is selection of an appropriate model structure that (i)can capture the dynamic behavior of commonly encounterednonlinear systems and (ii) facilitates the analytical treatment ofthe controller synthesis problem. Various black box models usedin the NMPC literature can be broadly classified into two classes:(i) nonlinear output error (NOE) or nonlinear moving average(NMAX) models and (ii) nonlinear ARX (NARX) Models.10,11

Some of the significant models belonging to the NOE class arethe recurrent neural networks (RNN),12 Volterra models,13

decoupled AB Net,14 and Laguerre-Wiener model.15 From theviewpoint of developing closed form control laws, the block-oriented models are particularly interesting as the dynamiccomponent of these models is linear. Wiener originally proposedthe use of orthonormal filters in combination with a memoryless nonlinear map for modeling of nonlinear dynamic systems.Boyd and Chua16 have later shown that any time-invariant,continuous, nonlinear operator with fading memory can berealized in terms of a finite dimensional linear dynamic systemcoupled with a nonlinear readout map. The block-orienteddecoupled AB Net14 and Laguerre-Wiener models15 belongto this class of Wiener type models where the linear dynamiccomponent of the model is parametrized using Laguerre/Kautzfilters. Recently, Srinivasrao et al.10,17 have proposed a Wienermodel where the linear dynamics is parametrized using general-ized orthonormal basis filters (GOBF). They chose a structurallysimple nonlinear state to output map, i.e., multivariable quadraticpolynomial, which can help in the construction of closed formsolutions. The main advantage of this OBF-Wiener model isthat it ensures good prediction capability over a large horizonwith respect to known (manipulated) and measured inputs, whichis of vital importance in any NMPC formulation. In addition,these OBF-Wiener models have also been shown to be boundedinput-bounded output (BIBO) stable10,13,15,17 when the stateto output map is bounded.

From the viewpoint of controlling systems with fast dynamics,the closed form control law developed by Patwardhan andMadhavan8 using one-step prediction appears interesting. How-ever, since an approximation of the mechanistic model was usedfor the controller synthesis, it was not possible to extend thatapproach to multistep predictions. In this work, the OBF-Wienertype model with NOE structure10,17 is used for capturingdynamics of multivariable nonlinear systems with fadingmemory. The resulting discrete nonlinear state space model isthen used to generate multistep predictions and formulate anunconstrained predictive control problem. The proposed for-mulation allows specification of arbitrary prediction and controlhorizons for shaping the closed loop behavior. A closed formsolution to this problem is constructed analytically using thetheory of solutions of quadratic operator polynomials developedby Rall.9 The effectiveness of the resulting multistep quadraticcontrol law is demonstrated by conducting (a) simulation studieson a proton exchange membrane fuel cell (PEMFC) system,which exhibits fast dynamics and input multiplicity behavior,and (b) experimental studies on a single cell PEMFC system (asingle input-single output (SISO) control problem with fastdynamics) and a benchmark heater-mixer setup (a multipleinput-multiple output (MIMO) control problem with inputmultiplicity). In the PEMFC simulations and the heater-mixersetup, the controller is expected to the maintain operation at an

optimum operating point, which is a singular point where thesteady state gain reduces smoothly to zero and changes its signacross this point. This challenging control problem is used as abenchmark to assess the performance of the proposed multistepquadratic control law (MQCL).

This paper is organized in six sections. The OBF-Wienermodel and its properties are briefly discussed in section 2. Thedevelopment of the closed form quadratic control law ispresented in section 3. The analysis of the simulation and theexperimental results are presented in sections 4 and 5, respec-tively. The main conclusions of the study are presented in section6.

2. OBF-Wiener Model

Consider a nonlinear process with fading memory16 andgoverned by a set of nonlinear differential equations

where z ∈ Rn represents the process state vector, UT ∈ Rm

represents the vector of manipulated inputs, d ∈ Rd representsunmeasured disturbances, Y ∈ Rr represents the vector ofmeasured outputs corrupted with measurement noise υy(t), andp ∈ Rν represents the parameter vector. It is further assumedthat U ∈ Rm represents known (or computed) value ofmanipulated input. The true input UT to the plant is related tothe computed input as follows

where υu(t) ∈ Rm represents, in general, a stationary colorednoise signal. In addition, the unmeasured disturbances evolveaccording to

where υd(t) ∈ Rm represents a stationary colored noise signaland dj represents the mean value of the unmeasured disturbance.

In practice, the operators F[ · ] and G[ · ] are seldom knownexactly and are too complex to be used for developing acontroller. Thus, the information available from the plant is thesampled sequence of input and output vectors ΣN ) (Y(k),U(k)) : k ) 1, 2, ..., N. Given the input and output data set ΣN

collected from a plant in the neighborhood of some steady stateoperating point of interest, say (Yj , Uj ), perturbation variablescan be defined as follows

Given the input and output data set ΣN collected from a plant,the problem of identifying a nonlinear time series model canbe stated as finding a nonlinear operator Θ[ · ]

such that a suitable norm of model residuals v(k) : k ) 1, ...,N is minimized with respect to parameter vector θ. Here, term(k) defined as

and represents a regressor function of the past inputs and/orthe past outputs at sampling time (k). The modeling problemcan be further decomposed as (a) choosing a suitable regressor

dzdt

) F[z, UT(t), d(t), p] (1)

Y(t) ) G[z, p] + υy(t) (2)

UT(t) ) U(t) + υu(t)

d(t) ) d + υd(t)

y(k) ) Y(k) - Y and u(k) ) U(k) - U

y(k) ) Θ[(k), θ] + v(k) (3)

(k) )def

[u(k - 1), ..., u(1), y(k - 1), ..., y(1)] (4)

Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010 149

function [ · ], (b) selecting a suitable nonlinear mapping Θ[ · ]from regressor space to the output space, and (c) estimatingthe model parameters by a prediction error method.

In this work, it is proposed to develop a MIMO model witha nonlinear output error (NOE) structure. This implies that (i)the regressor vector can be chosen as a function of pastmanipulated inputs alone, i.e.

and (ii) no assumptions are made regarding the structure of theunmeasured disturbance term, v(k), except that it is additiveoutput error.

As a consequence, an r × m MIMO model can be fragmentedinto r disjoint multiple input-single output (MISO) models ofthe form

where i ) 1, 2, ..., r. Since θ(i) is a function of past inputs andyi(k) : k ) 1, ..., N alone, the model development can becarried out by solving r disjoint the parameter estimationproblems. In particular, it is proposed to use a Wiener type statespace model of the form

where x(i)(k) ∈ Rni represents the state vector and Ω(i)[ · ]represents some nonlinear static map relating states with theoutputs for the ith MISO model. Matrices (Φ(i), Γ(i)) areparametrized using generalized orthonormal basis filters,18 whichrepresent an orthonormal basis for the set of strictly proper stabletransfer functions (denoted as H2). Ninness and Gustafson19

have shown that a complete orthogonal set in H2 can beconstructed as follows

where k : k ) 1, 2, ..., is an arbitrary sequence of polesinside the unit circle appearing in complex conjugate pairs.Given a set of real poles inside a unit circle, a method ofparametrization of matrices is given in Patwardhan and Shah.18

When some of the poles are complex, a method of constructinga state realization for such models is discussed in Heuberger etal.20 The nonlinear state output map Ω(i)[ · ] : Rnif R is chosento be simple multidimensional quadratic polynomial functionsof the form

Here C(i) represents a (1 × ni) vector and D(i) represents an ni

× ni symmetric matrix. The main advantage of choosingquadratic polynomial functions is that the resulting controllersynthesis problem can be solved analytically. The model (eqs7 and 8) can be looked upon as a truncated second order Volterraseries model. The estimation of OBF poles and the parametersof the state output map can be carried out using a nestedoptimization approach as proposed by Srinivasrao et al.10

After identifying r MISO models using data set ΣN, thesemodels are stacked to formulate a MIMO OBF-Wiener modelas follows

where

and n ) ∑i)1r ni. Note that D is a (r × n × n) bilinear matrix

representation of a three-dimensional array of the form

Details of bilinear matrix representation of three-dimensionalarrays is given in the Appendix (also see Patwardhan andMadhavan8). Srinivasrao et al.10 have shown that the OBF-Wiener model has following important properties:

(i) OBF-Wiener model has a nonlinear output error (NOE)structure.

(ii) OBF-Wiener model belongs to the class of nonlinearsystems with fading memory.

(iii) If the state to output map in the OBF-Wiener model isbounded on a compact set (or is continuous), then the model isbounded input-bounded output (BIBO) stable.

It may be noted that models with NOE structure have betterlong range prediction capabilities. Also, the open loop BIBOstability characteristics are of vital importance as the proposedNIMC scheme requires that the internal model be open loopstable.7

3. Development of Multistep Quadratic Control Law

In the present study, the following assumptions are madewhile developing the closed form control law: (a) the plant tobe controlled is internally asymptotically stable and (b) thesystem under consideration is square, i.e., number of inputs areequal to the number of outputs (i.e., m ) r).

3.1. Multistep Predictions. The MIMO model given in eq12 can be used to formulate an open loop state estimator asfollows

where y(k) represents measured output. At sampling instant k,given p future input moves

(k) ) [u(k - 1), ..., u(1)] (5)

yi(k) ) Θ(i)[u(k - 1), ..., u(1), θ(i)] + vi(k) (6)

x(i)(k + 1) ) Φ(i)x(i)(k) + Γ(i)u(k) (7)

yi(k) ) Ω(i)[x(i)(k)] + vi(k) (8)

Fk(z, ) )√1 - |k|

2

z - k∏i)1

k-1 1 - i*z

z - i(9)

Ωi[·] ) C(i)x(i)(k) + (x(i)(k))TD(i)(x(i)(k)) (10)

x(k + 1) ) Φx(k) + Γu(k) (11)

y(k) ) Ω[x(k)] (12)

) Cx(k) + [x(1)(k)TD(1)x(1)(k)

x(2)(k)TD(2)x(2)(k)l

x(r)(k)TD(r)x(r)(k)](13)

) Cx(k) + D(x(k), x(k)) (14)

x(k) ) [(x(1)(k))T (x(2)(k))T ... (x(r)(k))T ]T

Φ ) block diag[Φ(1) Φ(2) ... Φ(r) ]n×n

Γ ) [Γ(1)T Γ(2)T ... Γ(r)T ]n×mT

C ) block diag[C(1) C(2) ... C(r) ]r×n

x(k) ) Φx(k - 1) + Γu(k - 1) (15)

y(k) ) Ω[x(k)] + v(k) (16)

Uf(k) ) [u(k|k)T ... u(k + p - 1|k)T ]T

150 Ind. Eng. Chem. Res., Vol. 49, No. 1, 2010

(p + j)-step-ahead open loop state and output predictions canbe generated by recursively using the state observer as follows

Thus, (p + j)-step-ahead output predictions can be expressedas follows

When the model is used for online predictions outside thetraining region, a mismatch arises between the model predictionsand the plant behavior. To achieve offset closed loop behaviorin the face of the model plant mismatch, artificial states thatbehave as integrated white noise are introduced in the predictionmodel.21,11 In this work, it is proposed to use the deadbeatdisturbance estimator employed in conventional nonlinear modelpredictive control formulations,21 in combination with a unitygain robustness filter, to introduce the artificial states in themodel predictions. By this approach, an estimate d(k + i|k) offuture disturbance is generated as follows

for i ) 0, 1, ..., p + q -1, where

Here, the matrix can be chosen as follows

where 0 e Ri < 1 are treated as tuning parameters. This filter isequivalent to unmeasured disturbance filter (FI) in the feedbackpath of nonlinear internal model control (NIMC) structure withtwo degrees of freedom (see Figure 1). The predicted output is

corrected to incorporate feedback and integral action in thecontroller as follows

The prediction equation (24) can be rearranged as follows

where

The symbols “o” and “•” represent the circle product and rightdot product of a bilinear matrix with a matrix, respectively (seethe Appendix for details).

While developing the set of prediction equations (25), it wasassumed that p future input moves are available for manipula-tion. However, in a typical NMPC formulation, the degrees offreedom (q) for future trajectory manipulation are typically fewerthan the prediction horizon (q , p). Moreover, these degreesare spread across the horizon through input blocking as follows

where mj are selected such that

To accommodate such constraints, vector U(k) is defined as

The relation between U(k) and Uf(k) can be represented as

where matrix is a block matrix of block dimension (p × q)defined as follows

x(k + j + 1|k) ) Φx(k + j|k) + Γu(k + j|k)) Φjx(k) + Φj-1Γu(k|k) + ... +

Γu(k + j - 1|k) (17)

x(k + p + j|k) ) Φp+jx(k) + S(j)Uf(k) (18)

y(k + p + j|k) ) Cx(k + p + j|k) +D(x(k + p + j|k),

x(k + p + j|k)) ) [CΦp+jx(k) + D(Φp+jx(k),

Φp+jx(k))] + CS(j)Uf(k) +

2D(Φp+jx(k), S(j)Uf(k)) +

D(S(j)Uf(k), S(j)Uf(k))

(19)

S(j) ) [Φp+j-1 Φp+j-2 ... Φ I ]Γ

d(k + i + 1|k) ) d(k + i|k) (20)

d(k|k) ) vf(k) (21)

vf(k) ) ΦVvf(k) + [I - ΦV]v(k) (22)

v(k) ) y(k) - Ω[x(k)] (23)

ΦV ) diag[R1 R2 ... Rr ]

Figure 1. Nonlinear IMC structure with two degrees of freedom: schematic representation.

y(k + p + j|k) ) y(k + p + j|k) + d(k + p + j|k)(24)

y(k + p + j|k) ) y(k + p + j|k) + [Λ(j)(k)]Uf(k) +Ψ(j)(Uf(k), Uf(k)) (25)

y(k + p + j|k) ) vf(k) + CΦp+jx(k) +DoΦp+j•Φp+j(x(k), x(k))

Λ(j)(k) ) CS(j) + 2DoΦp+j•S(j)x(k)

Ψ(j) ) DoS(j)•S(j)

u(k + j|k) ) u(k|k) for j ) 1, ..., m1 - 1 (26)

u(k + j|k) ) u(k + m1|k) for j ) m1 + 1, ..., m2 - 1(27)

... ) ...

u(k + j|k) ) u(k + mq-1|k) for j ) mq-1 + 1, ..., p - 1(28)

0 < m1 < m2 < ... < mq-1

U(k) ) [u(k|k)T u(k + m1|k)T ... u(k + mq-1|k)T ]T

(29)

Uf(k) ) × U(k)


where Im is an identity matrix of size (m × m) and [0] is a nullmatrix of dimension (m × m). With respect to control inputmoves U(k), matrix S(j) appearing in eq 19 becomes

3.2. Development of Quadratic Control Law. Let r(k)represent the desired final set point target specified at instant k.Consider an unconstrained NMPC formulation formulated overprediction horizon [p, p + q - 1] as follows

The necessary condition for optimality for the minimizationproblem (31) can be written as

If there exists a vector U(k) such that the following set ofcoupled equations is satisfied

where 0j represents the zero vector, then U(k) can treated as theglobal optimum solution of the NMPC formulation at instantk. However, as shown later in this section, such a vector maynot exist at some sampling instants. Nevertheless, it is proposedto use eq 34 as the basis for the development of a closed formcontrol law. To develop a closed form solution to the set of qvector equations (34), these equations are arranged in a singlequadratic polynomial equation as follows

where

Here, vector Y(k) can be expressed as a single multidimensionalquadratic polynomial in in Rm×q as follows

With these definitions, the controller design equation reducesto a multidimensional quadratic polynomial of the form

in Rm×q. An analytical approach to solving such operatorpolynomial equations in Banach spaces has been developed byRall.9 Patwardhan and Madhavan8 have adopted this approachto develop a one-step-ahead control law under a nonlinear IMCframework. Following the controller synthesis approach sug-gested by Patwardhan and Madhavan,8 a multistep quadraticcontrol law can be derived as follows:

Let U0 denote some arbitrary vector such that the gradientmatrix

is nonsingular. Then, eq 18 can be transformed to a quadraticpolynomial of the second kind9 as follows

where

It may be noted that the transformation of eq 37 to eq 39 usingarbitrary vector U0 can be viewed as a change of origin for theconvenience of constructing the analytical solution. An analyticalsolution of the transformed multidimensional quadratic equation(39) can be written as9

where the matrix ∆(k) is defined as

Here, the symbol “•” denotes the left dot product between thematrix (∇U[Q(U0)])-1 and the bilinear matrix Ψ (see theAppendix). In general, a matrix has multiple square roots, and,consequently different values of U(k) will be obtained for everychoice of the square root of matrix ∆(k). In abstract form, thesystem of linear algebraic equation (42) can be represented as

where A is an (n × n) matrix with all real elements and vectorb ∈ Rn. Depending on the nature of A1/2, the vector κ ∈ Rn orκ ∈ Cn. Patwardhan and Madhavan8 have proved the followingresults that help in understanding the behavior of solutions ofthis equation.

) [Im [0] [0] ... ... [0]... ... ... ... ... ...Im [0] [0] ... ... [0][0] Im [0] ... ... [0]... ... ... ... ... ...[0] Im [0] ... ... [0]... ... ... ... ... ...[0] ... ... ... [0] Im

... ... ... ... ... ...[0] ... ... ... [0] Im

] (30)

S(j) ) [Φp+j-1 Φp+j-2 ... Φ I ]Γ

minU(k) ∑

j)p

p+q-1

[ef(k + j|k)]Tef(k + j|k) (31)

ef(k + j|k) ) r(k)-y(k + j|k) (32)

∑j)p

p+q-1 [∂ef(k + j|k)

∂U(k) ]T

ef(k + j|k) ) 0 (33)

[ ef(k + p|k)ef(k + p + 1|k)

lef(k + p + q - 1|k) ] ) [0

0l0

] (34)

Ef(k) ) R(k) - Y[U(k)] ) 0j (35)

R(k) ) [r(k)T ... r(k)T ]T

Ef(k) ) [ef(k + p|k)T ... ef(k + p + q - 1|k)T ]T

Y[U(k)] ) [y(k + p|k)T ... y(k + p + q - 1|k)T ]T

Y(k) ) Y(k) + [Λ(k)]U(k) + Ψ(U(k), U(k)) (36)

Y(k) ) [y(k + p|k)T ... y(k + p + q - 1|k)T ]T

[Λ(k)] ) [(Λ(1)(k))T ... (Λ(q)(k))T ]T

Ψ ) [Ψ(1)

.....

Ψ(q)]

Ef(k) ) Q(U(k)) ) Ψ(U(k), U(k)) + [Λ(k)]U(k) +[Y(k) - R(k)] ) 0j (37)

∇U[Q(U0)] ) 2[Ψ(U0)] + Λ(k) (38)

Ψ(k)(U(k) - U0, U(k) - U0) + (U(k) - U0) + E0(k) ) 0j(39)

Ψ(k) ) (∇U[Q(U0)])-1•Ψ (40)

E0(k) ) (∇U[Q(U0)])-1[Q(U0)] (41)

(U(k) - U0) ) -(12

[I + (∆(k))1/2])-1E0(k) (42)

∆(k) ) [I - 4Ψ(k)(E0(k))] (43)

(I + A1/2)κ ) b (44)


Theorem 1:8 If A is diagonalizable or nonsingular, then thereexists a matrix B such that B2 ) A.

Corollary 2:8 If A is a positiVe definite matrix, then it has aunique positiVe definite square root.

Theorem 3:8 If square roots of matrix A in eq 44 exist, thenit is always possible to select A1/2 such that (I + A1/2) isnonsingular.

Lemma 4:8 For the system of linear algebraic equationsgiVen by eq 44 where A is a diagonalizable matrix, considerthe set

The smallest bound for the diameter of set H is obtained bychoosing A1/2 such that its eigenValues haVe nonnegatiVe realparts.

From the control viewpoint, it is desirable to select input thatcan achieve the desired changes with the smallest |U(k)|2. Ineq 42, once a set point trajectory R(k) is specified and U0 isselected, E0(k) gets specified; i.e., vector b in eq 44 getsspecified. Since A1/2 can take multiple values, there are multiplesolutions to eq 42 (or eq 44). It is desired to find the set ofsolutions for which the sensitivity |(U(k) - U0)|/|E0(k)| isminimum. Lemma 4 helps in establishing that this set corre-sponds to choosing A1/2 such that its eigenvalues have nonne-gative real parts.8 Also, it may be noted that the matrix squareroot can have complex elements, and, consequently the resultingU(k) can be complex. Patwardhan and Madhavan8 have sug-gested that the real part of the complex solution vector can beused for manipulation when the solution vector becomescomplex. They have also shown that the situation where thesolution becomes complex arises when the specified set pointis rendered unattainable due to system nonlinearity. A detaileddiscussion regarding the rationale behind these recommendationscan be found in Patwardhan and Madhavan.8

Thus, incorporating the above suggestions, a modified solutionto the control problem can be stated as follows

This controller is implemented in a moving horizon frameworkand only the first element of U(k), i.e., u(k|k), is implementedon the system. Thus, the final form of the control law can bestated as follows

where matrix (1) represents the first block row of matrix defined by eq 30. While the choice of vector U0 does not changethe solution U(k), it can be conveniently selected either as thenull vector or set equal to REAL[U(k - 1)].

The control law (46) is an unconstrained formulation. Inpractice, however, the manipulated inputs are constrainedbetween upper and lower bounds. As a consequence, theproposed unconstrained control law (46) requires additionalmeasures to avoid integral windup caused due to input satura-tion. For a linear IMC controller, Morari and Zafirio22 haveshown that internal stability can be maintained if inputs to theplant as well as the internal model are constrained. Thus, thebounds on manipulated inputs are incorporated by constrainingthe controller output as follows

This constrained input is used in the control law calculations atthe subsequent sampling time.

If it is desired to reach a final target set point, say yr, ingradual manner, then a smooth set point trajectory can begenerated by filtering yr through a unit gain filter as follows

Here, matrix Φr can be chosen as follows

where 0 e γi < 1 are tuning parameters. Such a set point filtercan be effectively used to generate attainable targets and avoidcomplex solutions.

The proposed multistep quadratic control law (MQCL)together with the unmeasured disturbance filter and the set pointfilter represents nonlinear IMC structure two degrees of freedom.A schematic representation of this NIMC controller is shownin Figure 1. It may be noted that the nonlinear process, model,and controller are deliberately represented as nested squareblocks to emphasize the fact that usual block diagram manipula-tions do not apply.

Remark 5: Note that the deriVation of the quadratic controllaw requires only a Wiener type MIMO model of the form giVenby eq 12. It is not necessary that the linear dynamic part of themodel be deriVed from a realization of OBF network. Model ofthe form (12) deVeloped by any other approach can be used toderiVe the quadratic control law (46).

Remark 6: Consider a special case of quadratic control lawdeVeloped with q ) 1. Using the identity

leads to

Since all eigenValues of Φ are inside a unit circle, it followsthat Φp ≈ [0] for sufficiently large p and

Thus, for sufficiently large p eq 18 can be written as

Here, xj(k) represents the steady state attained by the modelpredictions if constant u(k) is applied oVer the entire futurehorizon. Also, for sufficiently large p, yj(k + p|k) = vf(k) andthe controller design equation reduces to

SolVing quadratic polynomial equation (50) analytically tocompute u(k) Values that achieVe the adjusted target set point[r(k) - vf(k)] amounts to inVerting the steady state input-outputmap of the model. Thus, when the prediction horizon p isselected sufficiently large, the proposed quadratic control lawbehaVes like a nonlinear analogue of “gain inVerse controller”.

Remark 7: Consider eq 35, which is later rewritten as themultidimensional quadratic equation (37). Let

H ) κ:κ ) (I + A1/2)-1b, | |b| |2 e 1

U(k) ) U0 - REAL[12

I + (∆(k))1/2]-1E0(k) (45)

u(k) ) (1)[U0 - REAL[12

I + (∆(k))1/2]-1E0(k)]

(46)

uL e u(k) e uH

r(k) ) Φrr(k - 1) + [I - Φr]yr (47)

Φr ) diag[γ1 γ2 ... γr ]

(I - Φl) ) [Φl-1 + Φl-2 + ... + I](I - Φ)

S(p) f S ) (I - Φp)(I - Φ)-1Γ

S = (I - Φ)-1Γ (48)

x(k + p + j) = x(k) ) Su(k) ) (I - Φ)-1Γu(k)(49)

Ψ(u(k), u(k)) + [CS]u(k) + vf(k) ) r(k) (50)

Ψ ) DoS•S

U(k) ) UR(k) + jUI(k)


represent the general solution of eq 37. Substituting for U(k)in eq 37 and rearranging lead to

If there exists a real solution UR(k) ∈ Rm×q such that UI(k) ) 0jand UR(k) solVes eq 37, then from eq 35, it is clear that

the corrected output predictions Y[UR(k)] attain the desired setpoint trajectory R(k). When both UR(k) * 0j and UI(k) * 0j,then from ER(k) ) 0j, it follows that

Thus, when only UR(k) is implemented, the corrected outputpredictions Y[UR(k)] do not attain the desired set pointtrajectory R(k). This situation arises when the specified desiredtrajectory, R(k), does not belong to the range of functionY[UR(k)]; i.e., the set point is unattainable.

3.3. Control Law Development by Conventional Appro-ach. Successive local linearization of the nonlinear modeloperator is often employed in the domain of NMPC to simplifythe task of controller synthesis. Taking motivation from thisapproach, the design equation (37) can be rearranged as follows

The matrix [Ψ U(k) + Λ(k)] appearing in eq 51 can beapproximated using the future input trajectory computed at theprevious sampling instant, i.e., U(k -1). Thus, using thissimplification, a successive approximation type control law canbe derived as follows

Alternatively, defining the prediction error vector Ef(k) at instantk as

the controller design problem can be formulated as minimizationof the following objective function

where WE and W∆U represent error weighting matrix and inputmoves weighting matrix, respectively. Here, vector ∆U(k) isdefined as

where

Using the necessary condition for unconstrained optimality, andusing eq 37, the control law assumes the form

where Q(k) ) [Ψ U(k -1) + Λ(k)]. This control lawdeveloped using successive local linearization, referred to asthe successive linearization control law (or SACL) in the restof the text, is viewed as a controller synthesized through theconventional approach.

3.4. Framework for Stability Analysis. Since a closed formcontrol law has been derived under a nonlinear IMC framework,it is possible to carry out the closed loop stability analysis (undernominal conditions) in the framework suggested by Economou.7

Let the plant dynamics given by equations be represented as adiscrete nonlinear operator equation

under the assumption that disturbances and parameters areconstant at some nominal values, i.e., d(t) ) dj and p ) pj, andυy(t) ) 0j. The quadratic control law (46) can be represented asan nonlinear operator equation of the form

The closed loop state dynamics can now be represented as anaugmented operator equation

Let (z*, x*, vf*, yr) represent the steady state of the closed loopoperator ( · ) such that yr ) G(z*). Then, the following theoremby Economou7 can be used to characterize the region ofattraction in the neighborhood of (z*, x*, vf*, yr).

Theorem:7 Consider the system defined by eq 58 and theequilibrium state (z*, x*, vf*). If

where Π[(z*, x*, vf*, yr), F] represents a closed ball of radius Fin the neighborhood of (z*, x*, vf*, yr), then

(a) The equilibrium state (z*, x*, vf*, yr) is unique in Π[(z*,x*, vf*, yr), F].

Ef[U(k)] ) ER(k) + jEI(k) ) 0j + j0j

ER(k) ) Ψ(UR(k), UR(k)) + [Λ(k)]UR(k) + [Y(k) -R(k)] - Ψ(UI(k), UI(k))

EI(k) ) Ψ(UI(k), UR(k)) + Ψ(UR(k), UI(k)) +[Λ(k)]UI(k)

Y[UR(k)] ) R(k)

Y[UR(k)] ) R(k) + Ψ(UI(k), UI(k))

[ΨU(k) + Λ(k)]U(k) ) -[Y(k) - R(k)] (51)

U(k) ) [ΨU(k - 1) + Λ(k)]-1[R(k) - Y(k)]

u(k) ) (1)U(k)

Ef(k) ) [R(k) - Y(k)] - [ΨU(k - 1) + Λ(k)]U(k)(52)

J ) Ef(k)TWEEf(k) + ∆U(k)TW∆U∆U(k) (53)

∆U(k) ) [ u(k/k) - u(k - 1)u(k + 1/k) - u(k/k)

...u(k + q - 1/k) - u(k + q - 2/k)

]) HU(k) - H0u(k - 1) (54)

H ) [I [0] [0] [0]-I I [0] [0]... ... ... ...[0] ... -I I

]; H0 ) [I[0]...[0]

]u(k) ) (1)[Q(k)TWEQ(k) + HTW∆UH]-1 ×

[Q(k)TWE(R(k) - Y(k)) + HTW∆UH0u(k - 1)]

z(k + 1) ) z(k) + ∫kT

(k+1)TF[z(τ), u(k), d, p] dτ )

F[z(k), u(k)] (55)

y(k) ) G[z(k)] (56)

u(k) ) Y [x(k), y(k), r(k)] (57)

[z(k + 1)x(k + 1)vf(k)r(k)

] )

[F [z(k), Y [x(k), Gz(k), r(k)]]Φx(k) + ΓY [x(k), Gz(k), r(k)]ΦVvf(k - 1) + [I - ΦV](Gz(k) - Ω[x(k)])Φrr(k - 1) + [I - Φr]yr

] (58)

) [z(k)x(k)vf(k - 1)r(k - 1)

] (59)

| ∂∂(z, x, vf)| e η < 1, ∀ (z, x, vf, yr) ∈ Π[(z*, x*, vf*, yr), F]


(b) The equilibrium state (z*, x*, vf*, yr) is stable and Π[(z*,x*, vf*, yr), F] is the region of attraction for (z*, x*, vf*, yr).

From eq 58, it is clear that the region of attraction can beinfluenced by the appropriate choice of the robustness filtermatrix (ΦV) as well as the set point filter matrix (Φr). Whenthe operators F [ · ] and G[ · ] are known, the numerical proce-dure proposed by Economou7 can be employed for finding theregion of attraction (stability) in the neighborhood (z*, x*, vf*,yr).

The nominal stability analysis framework discussed abovecan be extended to deal with the perturbations in disturbancesand/or model parameters. The relevant stability results andextensive discussion on this robust stability of NIMC underdifferent conditions can be found in Economou.6 It may be notedthat these results provide only sufficient conditions for thenominal and the robust stability.

4. Simulation Studies

The effectiveness of the proposed approach is demonstratedby simulating control of a proton exchange membrane fuel cell(PEMFC). From a control viewpoint, this system has fastdynamics and it exhibits input multiplicity behavior (i.e.,identical steady state output for multiple sets of input values).Input multiplicities, in general, occur due to the presence ofcompeting effects in a process. Such systems are difficult tocontrol as they exhibit change in the sign of steady state gain(s)in the desired operating region. The phenomenon of change inthe sign of the steady state gain poses a difficulty even fornonlinear control schemes. From a system theoretic viewpoint,the relative degree of the system becomes undefined and theinvertibility is lost at the optimum (singular) point where thesteady state gain is reduced to zero. As a consequence, the globalinput-output linearization based approaches cannot be applied(in a straightforward manner) in the regions of state space wherethe relative degree is not well-defined and invertibility is lost.23

Thus, in the case of almost all the nonlinear control strategiesbased on exact linearization, an assumption, often not statedexplicitly, is that the steady state gain of the model cannotchange sign in the operating region.24 Biegler and Rawlings25

observe that use of an unconstrained nonlinear controller maynot be sufficient in such cases, even when the model is perfect.When a zero process gain situation is encountered, the perfectmodel cannot be inverted and the model inversion basedcontroller can become ill-conditioned. Thus, controlling PEMFCsystem at the optimum operating point is a challenging controlproblem and is used as a benchmark to assess the performanceof the proposed control laws.

It is desired to develop the proposed closed form control lawsto control PEMFC at the peak power operating point. The stepsinvolved in the controller development can be summarized asfollows:

1. Offline model development:(a) The first step involves perturbing the system by

injecting multilevel random input signals to excite thesystem dynamics in the desired operating region.

(b) The input-output data are then used to constructmultiple MISO OBF-Wiener models and stack thesemodels to generate a MIMO state space model.

2. Off-line computations for controller law: Once theMIMO state space model is developed, the matrix [Λ(k)]and bilinear matrix Ψ appearing in the control lawcomputations can be precomputed for later use in theonline calculations.

3. Online computations: Online computations at eachsampling instant involve(a) estimating the current state x(k) using equations 15

and 16,(b) filtered disturbance signal vf(k) using eqs 22 and 23,

and(c) computing the controller output u(k) using eq 46.

To begin with, a brief description of the mechanistic modelused for simulating the PEMFC (or the process) behavior ispresented. The details of the OBF-Wiener model developmentand controller implementation are presented in the latersubsections.

4.1. Proton Exchange Membrane Fuel Cell (PEMFC)Simulation. In this work, a reduced order mechanistic modelof PEMFC developed by Golbert and Lewin26 is used tosimulate the process dynamics. A set of differential algebraicequations that can be used for process simulation, nominal modelparameters, and operating conditions can be found in Golbertand Lewin.26 All simulations were performed in MATLABenvironment, and the details of the solution procedure can befound in Bedi.27 It is assumed that the average power density(P) defined as

and the average solid/stack temperature (Ts,avg) are the measured(and controlled) process outputs while the cell voltage (Vcell)and coolant inlet temperature (Tw,in) are the process inputs thatcan be manipulated. Figure 2 shows steady state behavior ofcontrolled outputs with respect to the manipulated inputs. Asevident from this figure, the power density exhibits inputmultiplicity with respect to both manipulated inputs. Theoptimum value for power density is 1.09 W/m2 (at solid averagetemperature of 338.8 K), which represents the desired (nominal)optimum operating point in this study. This is a singularoperating point where the steady state gains with respect to bothmanipulated inputs is reduced to zero.

4.2. OBF-Wiener Model Development. It is assumed thatthe measurements of power density and average solid temper-ature are corrupted with zero mean Gaussian white noise signalswith the standard deviations σ1 ) 2.5 × 10-3 W/cm2 and σ2 )0.25 K, respectively. The data required for parameter estimationof the OBF-Wiener model was obtained by perturbing the fuelcell model in open loop by simultaneously introducing square

Figure 2. PEMFC simulation, model validation: comparison of steady statebehavior.

P ) IavgVcell (60)


pulse sequences in Vcell and Tw,in of random magnitudes boundedbetween

and in the frequency range [0 0.02ωN ] where ωN ) π/Trepresents the Nyquist frequency. A total of 2000 data pointswere collected at sampling time of 0.1 s. Two MISO OBF-Wiener models were developed relating both process outputs,i.e., P and Ts,avg, to both inputs. The results of the modelidentification (optimum GOBF pole locations) are summarizedin Table 1. It may be noted that, in this table, instead of reportingdiscrete pole locations (i), equivalent continuous time polelocations (ai) have been reported such that

where T represents sampling time. Figure 4 presents thecomparison of model simulation with the process output on thevalidation data set. The corresponding input variations arepresented in Figure 3. Comparison of the steady state behaviorof the identified model with that of the plant is presented inFigure 2. From these figures, it is clear that the identified MISOOBF-Wiener models are able to capture the dynamic as well asthe steady state behavior of the fuel cell system with a reasonableaccuracy in the desired operating region.

4.3. Servo Response. The MIMO servo control problem isformulated in such a way that it is desired to shift the fuel celloperating point from a given suboptimal initial steady state tothe optimum operating point as is given in Table 2.

The filter tuning parameters used in MQCL and SACL are

To generate stable closed loop responses, SACL additionallyrequired introduction of input move suppression. Thus, theweighting matrices in SACL formulations are selected as WE

) W∆U ) I, where I represents the identity matrix. Themanipulated input moves are computed subject to the followingconstraints:

The controller performance is reported in terms of (a) averagecomputation time required for computing control moves (IntelPentium IV, 3 GHz with 512 MB RAM), (b) settling time, (c)integral square error (ISE), and (d) sum of square of manipulatedinput move variations PIui

) ∑[∆ui(k)]2.To begin with, the prediction horizon p is selected as 20,

which is close to the open loop settling time. Performances ofMQCL and SACL are assessed for the various values of thecontrol horizon ranging from q ) 1 to q ) 20. In particular,consider the following six cases of input blocking:

1. case q ) 1: single input move held constant over the entireprediction horizon

2. case q ) 4: four blocks of 1, 2, 5, and 12 samples each3. case q ) 8: three blocks of 1 sample, three blocks of 2

samples, one block of 4 and 7 samples each4. case q ) 12: five blocks of 1 sample, six blocks of 2

samples, one block of 3 samples5. case q ) 16: twelve blocks of 1 sample, four blocks of 2

samples6. case q ) 20: control horizon is set equal to the prediction

horizonFigure 5 shows a comparison of PEMFC responses for q )

8 and q ) 20. As evident from these figures, MQCL generatesfast and excellent closed loop behavior in both cases. The SACL,on the other hand, generates acceptable closed loop behavioronly for q ) 8. For q ) 20, the controller shifts operation to ahigher value of Tw,in due to input multiplicity. Figure 6 showsthat the power density exhibits oscillatory behavior under theseconditions.

The performances of MQCL and SACL in terms of thecomputation time and the settling time are presented in Tables3 and 4, respectively. Clearly, the average computation timerequired for the control law computations are significantly smallin both cases. Moreover, the computation times for SACL and

Table 1. PEMFC Simulation, Model Identication: GOBF PoleLocations

Vcell Tw,in

power density (P) [0.7238 79.5259] [0.8425 0.8425]average stack temperature (Ts,avg) [0.6150 0.6150] [0.5864 0.5850]

0.23 V e Vcell(k) e 0.83 V and 287 K e Tw,in e 347 K

i ) exp(-aiT)

Figure 3. PEMFC simulation, dynamic model validation: manipulatedinputs.

Φr ) diag[0.9 0.9 ]; ΦV ) diag[0.98 0.9 ] (61)

Figure 4. PEMFC simulation, model validation: comparison of process andsimulated model input.

Table 2. PEMFC Simulations: Intitial and Optimum OperatingConditions

initial conditions optimum conditions set point

P ) 1.02 P ) 1.09 P ) 1.092Vcell ) 0.63 Vcell ) 0.53Ts,avg ) 322 Ts,avg ) 338.8 Ts,avg ) 338.8Tcool,in ) 306 Tcool,in ) 317

0 e u1 e 1.5 V

290 K e u2 e 337 K


MQCL are comparable for higher values of the control horizon.However, MQCL clearly outperforms SACL in terms of thesettling time.

The MQCL and SACL performances in terms of ISE andPIui

values are presented in Tables 5 and 6, respectively. TheMQCL performs significantly better than SACL in terms of ISE

values of temperature. In the case of average power density,slightly higher values of ISE values are obtained using MQCL.The output trajectory for MQCL, in fact, overshoots the filteredset point trajectory. However, due to better prediction abilityof the quadratic model and nonlinear control action, the MQCLis able to restrict the overshoot. This results in better speed ofresponse and smaller settling time. However, for a large controlhorizon, the short rise time for control variables is achieved atthe cost of sudden and large variations in the manipulatedvariables. To penalize manipulated inputs, the only explicithandle in MQCL is the choice of control horizon. The valuesof PIui

reported in Table 5 and comparison of manipulatedvariables in Figure 5 indicate that shortening the control horizonhas an effect similar to penalizing the input moves.

Figure 7 shows the effect of choosing different predictionhorizons for the prediction horizon (p ) 10, 20 and 30) whilekeeping the control horizon at 8. As can be expected, inputvariations are significantly reduced by increasing the predictionhorizon.

4.4. Regulatory Response. Occurrences of sustained changesin unmeasured disturbances invokes peculiar steady statebehavior in the systems that exhibit input multiplicities. Theoptimum operating point is typically a function of unmeasured

Figure 5. PEMFC simulation: controlled outputs and manipulated inputmoves obtained using QCL.

Figure 6. PEMFC simulation: controlled outputs and manipulated inputmoves obtained using SACL.

Table 3. PEMFC Simulation, MQCL Performance: Computationand Settling Time

settling time (s)

control horizon case average computation time (ms) power Ts,avg

q ) 1 7.7033 3.15 4.325q ) 4 19.617 1.502 2.752q ) 8 74.017 1.292 2.798q ) 12 193.18 1.271 2.87q ) 16 421.55 1.26 2.95q ) 20 794.57 1.25 3.12

Table 4. PEMFC Simulation, SACL Performance: Computation andSettling Time

settling time (s)

control horizon case average computation time (ms) power Ts,avg

q ) 1 5.6167 3.1 5.31q ) 4 18.163 3.3 5.7q ) 8 66.813 3.4 8.5q ) 12 182.49 3.16 5.3q ) 16 400.97 3.15 4q ) 20 758.41 - 6.15

Table 5. PEMFC Simulation, MQCL Performance: ISE Values andControl Efforts

ISE PIui

controlhorizon case

power(×10-3)

Ts,avg

(×103)Vcell

(×10-3)Tw,in

(×102)

q ) 1 5.0253 1.2653 1.1854 0.2356q ) 4 6.4304 0.3565 2.7949 1.4562q ) 8 6.2802 0.3106 2.2016 1.7109q ) 12 6.2517 0.2289 1.531 2.4871q ) 16 7.1912 0.1941 1.4736 3.0221q ) 20 6.5772 0.1501 1.1257 4.2097

Table 6. PEMFC Simulation, SACL Performance: ISE values andControl Efforts

ISE PIui

controlhorizon case

power(×10-3)

Ts,avg

(×103)Vcell

(×10-3)Tw,in

(×102)

q ) 1 4.8262 1.4890 0.7356 0.1839q ) 4 5.23 1.7818 0.8944 0.1439q ) 8 5.7905 2.1577 0.8814 0.1091q ) 12 5.8464 1.9778 0.8921 0.1173q ) 16 5.7449 1.8483 0.9236 0.1379q ) 20 25.804 1.9184 0.9455 0.1415

Figure 7. PEMFC simulation, QCL performance: effect of increasingprediction horizon.


disturbance level, and sustained changes in unmeasured distur-bances change the location of the peak. From the analysis ofthe relative locations of the peaks with respect to the nominaloptimum point (set point), the following two types of controlproblems can be identified:8

(a) Suboptimal operation: In this case, the optimum operat-ing point shifts to a higher value than the nominal value. Insuch a situation, operation at the set point based on a nominalparameter value will become suboptimal.

(b) Unattainable set point: In this case, the optimumoperating point shifts to a lower value than the nominal value.In such a situation, the set point based on a nominal parametervalue will become unattainable.

Patwardhan and Madhavan8 point out that the situationleading to infeasibility of the set point is potentially moredangerous as the final steady state error cannot be eliminatedand does not change its sign. In the presence of such a persistenterror, a controller with integral action can cause instability inthe unconstrained case or input saturation/haunting of manipu-lated input(s) and loss of control in the constrained case. Sincethe later case is more difficult, results presented are only forthe case when the average power density set point becomesunattainable.

In order to investigate the effect of unmeasured disturbances,a simultaneous step jump is introduced in the model parameterwhen the set point change is introduced. Figure 8 compares theclosed loop behavior of the MQCL (p ) 20, q ) 8) with SACL(p ) 20, q ) 8) when Ta,in ()Tc,in) is changed from 353 to 343°C. In this case, the optimum operating point shifts to a lowervalue than the nominal value. Therefore, the step jumps inunmeasured disturbances renders the specified set point unat-tainable. The SACL fails to attain either of the set point targets.In the case of MQCL, on the other hand, the temperature setpoint is attained without any difficulty and the average powerdensity settles at a value close to the changed optimum. It wasobserved that, as the measured outputs approach close to theirrespective maximum attainable values (under given parameterperturbation), the solutions of the controller design equation (37)become complex. For large values of k, the vector [(1/2)I +(∆(k))1/2]-1E0(k) tends to a vector with only complex elementsand zero or negligible real part. As only the real part of thiscomplex vector is used in the quadratic control law, the inputsattain a steady state without getting saturated large for large k.Thus, the occurrence of complex solutions of the controllerdesign equation (37) have a clear interpretation in this case.

The complex values indicate that the specified set point isunattainable under the existing circumstances. In other words,there does not exist a real input vector that can take the outputsto the desired set points.

5. Experimental Studies

Experimental studies were carried out on two setups: (a) asingle cell PEMFC based experimental setup at the Departmentof Chemical Engineering, Clarkson University, and (b) a

Figure 8. PEMFC simulation: behavior of QCL and SACL under step jumpin unmeasured disturbance.

Figure 9. Single cell PEMFC: schematic diagram of experimental setup.

Figure 10. Single cell PEMFC experimental setup: dynamic modelvalidation.

Figure 11. Single cell PEMFC experimental setup: servo response of QCL.


benchmark heater-mixer setup28 in the Automation Lab,Department of Chemical Engineering, Indian Institute ofTechnology, Bombay.

The first system is a system with fast dynamics. However,only one input is available for manipulation in this case. Thesecond system has considerably slower dynamics. However, thisis a 3 × 4 multivariable system with considerable interactionsamong the variables. In addition, the complexity of the controlproblem is increased by deliberately introducing bell-shapednonlinearities that induce input multiplicity behavior.

5.1. Control of Single Cell PEMFC. The single cell PEMFCused in experimental work is a commercially available fuel cellfrom Fuel Cell Technologies, Inc. The experimental setup

consists of a humidifier, a fuel cell, and an electronic load. Themain component of the fuel cell system, i.e., the membraneelectrode assembly (MEA), is a high performance membranefrom PEAMEAS (E-TEK), Inc., which has an active area of10 cm2. The whole assembly consisting of MEA, gaskets, andflow channels is supported by stainless steel plates on both sides.The electron collector plates are placed in between the gas flowchannel plates and the supporting plates to facilitate the transferof generated electrons from the PEMFC to the electronic load.The hydrogen and oxygen flows are passed through an externalhumidifier (ElectroChem, Inc.) before injection in the fuel cell.The two separate mass flow controllers (ElectroChem, Inc.) are

Table 7. Single Cell PEMFC Experimental Setup: Comparison ofQCL and IMC Based PI Performancesa

performance index

controller mean ISE mean PIui

mean settlingtime (s)

MQCL 25.916 (5.1682) 8.5905 × 106

(2.1606 × 105)25.57 (5.56)

PI (IMC tuning) 32.059 (1.343) 9.02 × 106

(4.16 × 104)39 (6.54)

a Standard deviation values in parentheses.

Figure 12. Single cell PEMFC experimental setup: regulatory response ofQCL.

Figure 13. Heater-mixer setup: schematic diagram.

Table 8. Heater-Mixer Setup: Nominal Operating Conditions andParameter Values

variable description nominal value

m1 heat input Q1 (% input) 76.875%m2 heat input Q2 (% input) 87.5%m3 flow 2 input 37.5%m4 input recycle pump 50%d1 flow 1 input 37.5%Tc cooling water temperature 31.5 °CTa atmospheric temperature 27 °CT1 steady state temperature (tank 1) 60.2 °CT2 steady state temperature (tank 2) 54.9 °Ch2 steady state level 0.354 m

Figure 14. Heater-mixer setup: relationship between manipulated inputs(u) and process inputs (m).

Figure 15. Heater-mixer setup: dynamic model validation.

Table 9. Heater-Mixer Setup, Model Identication: GOBF PoleLocation

u1 u2 u3

T1 [0.0290 0.0290] [4.0923 0.0035] [0.5264 0.0181]T2 [0.0189 0.0189] [0.0149 0.0149 ] [0.8648 0.0084]h2 - - [3.9231 0.0070]


used for manipulating hydrogen and oxygen flow rates to fuelcell. The mass flow controllers have the range of 0-200 standardcm3/min with back-pressure adjustment facility from 0 to 60psi. The system has a provision to connect the fuel cell to a

variable electronic load (Agilent N 3300 A dc electronic load),which is operated under constant voltage mode (between 0 and6 V for voltage and 0-12 A) in the present study. The fuel celltemperature is controlled using a separate electronic controller,which manipulates electric heating to the fuel cell. The systemis connected to a PC using a data acquisition system (Compu-tating Measurement Inc.) through the USB port. A schematicdiagram of the entire setup is shown in Figure 9.

5.1.1. Model Identification and Validation. In this experi-mental evaluation, PEMFC is treated as an SISO system inwhich the inlet flow rate of hydrogen is manipulated forcontrolling power drawn from the PEMFC. To estimateparameters of the OBF-Wiener model, the inlet flow rate ofhydrogen is perturbed using multilevel random signals withstandard deviation of σu ) 10.1485 standard cm3/min andswitching time of 10 s. The identification data set contained

Figure 16. Heater-mixer setup: multilevel PRS input for model valida-tion.

Figure 17. Heater-mixer setup, model validation: comparison of steadystate behavior of OBF-Wiener model with that of mechanistic model.

Figure 18. Heater-mixer setup: output response using QCL with p ) 60and q ) 5.

Figure 19. Heater-mixer setup: controller moves using QCL with p ) 60and q ) 5.

Figure 20. Heater-mixer setup, QCL performance: effect of increasingcontrol horizon.

Table 10. Heater-Mixer Setup: Effect of Control Horizon onComputation Time and Controller Error Variance

PIui

control horizon (q) average computation time (ms) T1 T2

1 9.5 4.0322 1.94315 123.3 1.6760 1.124410 311.6 1.4637 0.9310


3000 data samples, whereas the validation data set contained1000 data samples. The standard deviation of the output is σy

) 0.3525 W, which is used for scaling output data used inidentification. An OBF-Wiener model with five poles wasidentified using the identification data set. The optimum GOBFpoles i reported in terms of continuous time poles ai are asfollows

To validate the identified model, the model simulations arecompared with outputs obtained from the plant (see Figure 10).

From Figure 10, it can be inferred that the identified modeladequately captures the output variability caused by manipulatedinput variations.

5.1.2. Servo and Regulatory Control. In closed loop studies,a quadratic control law was developed with p ) 20 and q ) 1.Figure 11 shows typical closed loop behavior of MQCL whiletracking set point changes between 4.5 and 6 V. The servocontrol experiment was repeated seven times, and the averageperformance of MQCL is reported in Table 7. This table alsoreports the average performance of a proportional-integral (PI)controller developed based on a linear transfer function model

Figure 21. Heater-mixer setup: change in steady state behavior in response to variations in F1.

Figure 22. Heater-mixer setup: regulatory response using QCL for negative step change in F1, controlled outputs.

ai ) 0.0799, 0.2621, 0.999, 0.999, 0.999


identified from the input-output data and tuned using IMCtuning rules. The IMC tuning parameters for the PI controllerloop are as follows

When compared with the performance of IMC based PIcontroller, MQCL requires significantly less settling time andgenerates smaller ISE values. Figure 12 presents results forregulatory performance when the cell voltage was increased by0.05 V. As can be observed from this figure, the MQCLsatisfactorily reject this disturbance within 40 s, which is closethe open loop settling time of the system.

5.2. Control of Heater-Mixer Setup. The heater-mixersetup consists of two stirred tanks in series as shown in Figure13. The contents in the tanks are well stirred by using variablespeed agitators. A cold water stream is introduced in the firsttank through CV-2. The contents of the first tank are heatedusing a 4 kW-h heating coil. The hot water that overflows thefirst tank is mixed with a cold water stream entering the second

tank through CV-1. The contents of the second tank are heatedusing another 3.5 kW-h heating coil. The water from the secondtank is also recycled back to the first tank. The heat supplied toboth tanks can be varied continuously using thyristor powercontrollers, which take 4-20 mA as inputs. The cold water inletflow to both tanks can be manipulated using pneumatic controlvalves. The temperatures in the first tank (T1) and in the secondtank (T2) and the liquid level in the second tank (h2) aremeasured variables and controlled variable. The heat inputs tothe first tank (m1) and to the second tank (m2) and the coldwater flow to the second tank (m3) are treated as manipulatedinputs. The cold water flow to the first tank and the cold watertemperature (Tw,in) are treated as unmeasured disturbances. Therecycle flow rate is kept constant. The system is interfaced witha control computer using a data acquisition system (Advantech,ADAM 5000 series hardware) through LabVIEW-8 andMATLAB.

The steady state operating condition for the process is givenin Table 8. The cooling water temperature changes slowly during

Figure 24. Heater-mixer setup: regulatory response using QCL for positive step change in F1, controlled outputs.

Figure 23. Heater-mixer setup: regulatory response using QCL for negativestep change in F1, manipulated inputs and disturbance.

kc ) 5.226; τi ) 5 s

Figure 25. Heater-mixer setup: regulatory response using QCL for positivestep change in F1, manipulated inputs and disturbance.


the experimentation and acts like a drifting disturbance. A gray-box model for this system can be found in Thornhill et al.28

Since the STH is mildly nonlinear, bell-shaped nonlinearities(see eq 62 and Figure 14) have been deliberately introducedbetween manipulated inputs (u1, u2) used for modeling andcontrol and the current inputs (m1, m2) to the thyristor powercontrollers. The Gaussian type nonlinear function relating u1,u2 to m1, m2 introduces input multiplicity behavior betweenu1, u2 and T1, T2. The optimum operating point is at u1 ) 0 andu2 ) 0.

This makes the control problem difficult and stronglynonlinear and introduces to input multiplicity behavior betweenthe manipulated inputs (u1, u2) and the controlled outputs (T1,T2). Thus, the resulting system has an optimum operating point,where the steady state gains with respect to manipulated inputsare reduced to zero and change their respective signs acrossthe optimum.

5.2.1. Development of the OBF-Wiener Model. The op-erating point is chosen as the peak steady state point u1 ) 0and u2 ) 0 (i.e., m1 ) 16.3, m2 ) 18), which is a singularpoint. In order to generate data for model identification andvalidation, simultaneous input perturbations (multilevel RBS)are introduced in u1, u2, and u3 the frequency ranges [0 0.05ωN],[0 0.05ωN], and [0 0.03ωN], respectively. The resulting multi-level RBS signal bounded between

A total of 2000 data points were collected at sampling timeof 5 s. Out of the total 2000 data points that were collected, thefirst 1200 data points were used for model identification andthe remaining 800 data points were used for model validation.Two MISO OBF-Wiener models were developed relating to theprocess outputs, i.e., T1 and T2, with all inputs. A model for thelevel is developed as an SISO model, as the level can bemanipulated using cold water flow (F2 or u3) only. The resultsof the model identification (equivalent continuous time LHPpole locations) are summarized in Table 9.

The results of dynamic model validation are plotted in Figure15, and the corresponding input variations are reported in Figure16. Figure 17 compares the steady state behavior of OBF-Wienermodels with that of the mechanistic model.28 It is evident fromthese figures that the identified MISO OBF-Wiener models areable to capture the dynamic as well as the steady state behaviorof the system with a reasonable accuracy.

5.2.2. Servo and Regulatory Studies. The prediction horizon(p) for MQCL and SACL is chosen as 60. The effect of thecontrol horizon on the closed loop performance is assessedexperimentally for three different control horizons (1, 5, and10). The set point filter and robustness filter parameters in bothcontroller formulations are chosen as

The manipulated input moves are computed subject to thefollowing constraints:

The MIMO servo control problem is formulated in such a waythat it is desired to shift the operating point to the optimumoperating point (T1 ) 60.2, T2 ) 54.9, h2 ) 0.354 m) from agiven suboptimal operating point (T1 ) 56.2, T2 ) 50.9, h2 )0.354 m).

Figure 18 shows output responses obtained using MQCL (q) 5). The corresponding manipulated input moves are presentedin Figure 19. The controller satisfactorily moves the processfrom suboptimal to the optimal point, and the process is tightlycontrolled at the singular peak operating point. Figure 20compares responses of the quadratic control law with q ) 1and q ) 10. Table 10 compares performances of MQCL fordifferent choices of control horizon.

As can be expected, the controlled outputs reach the optimumpoint faster when the degrees of freedom in MQCL formulation(i.e., q) are increased to 10. However, if average computationtimes for quadratic control law implementations are compared,then the controller with q ) 10 required 311.6 ms while thecontroller with q ) 1 required only 9.5 ms. The averagecomputation time is small even for q )10 and is quite acceptablefor a sampling time of 5 s. It may be noted that the controllerwas implemented through a MATLAB window created inLabVIEW. The MQCL (q ) 1) was also implemented byconverting MATLAB code to a C++ code and using theresulting controller DLL file through LabVIEW.29 This resultedin a drastic reduction in the mean computation time requiredfor control law computations (0.184 ms). Thus, it is possible toemploy the proposed quadratic control law for systems withvery fast dynamics if q , p.

As in the case of PEMFC, the optimum operating in thepresent case is a function of unmeasured disturbances (F1 andTw,in) and sustained changes in the unmeasured disturbanceschanges the location of the peak. Figure 21 show effects ( 10%variations in F1 on the steady state behavior of the mechanisticmodel for the process. The effect of changes in Tw,in on thesteady state behavior is qualitatively similar. Figure 22 showsthe closed loop behavior when a negative step change ofmagnitude 1.729 × 10-6 m3/s (i.e., 7.25% of the base value)was introduced in F1. When F1 is reduced, the steady state peakshifts upward. The set point based on the nominal model remainsfeasible in this case and MQCL (p ) 60, q ) 5) manages toreject the disturbance within 5 min. The corresponding inputmoves and variations of Tw,in are shown in are shown in Figure23.

When F1 was increased by 7.25% of the base value, peakpoint shifted in the downward direction and the set point basedon nominal model parameters became unattainable. This,resulted in a constant offset that could not be eliminated (seeFigure 24). The corresponding input moves and variations ofTw,in are shown in Figure 25. As was expected from simulationstudies, the offset did not lead to input saturation even thoughMQCL has in-built integral action.

6. Conclusion

In this work, a Wiener type model has been used for capturingdynamics of multivariable nonlinear systems with fadingmemory. The resulting discrete nonlinear state space model isused to generate multistep predictions and formulate an uncon-

m1 ) 16.3 exp(-(u1/6)2); m2 ) 18 exp(-(u2/6)2)(62)

-6 e u1 e 6, -6 e u2 e 6, and -1.5 e u3 e 1.5

Φr ) diag[0.9 0.9 0.9 ]; ΦV ) diag[0.9 0.9 0.92 ](63)

-6 e u1 e 6

-6 e u2 e 6

8 mA e u3 e 20 mA


strained NMPC problem. A closed form solution to this problemis constructed analytically using the theory of solutions ofquadratic operator polynomials developed by Rall.9 The ef-fectiveness of the resulting quadratic control law is demon-strated by conducting simulation studies on a proton exchangemembrane fuel cell (PEMFC) system, which exhibits fastdynamics and input multiplicity behavior. The quadratic controllaw is expected to control the PEMFC at its optimum (singular)operating point. Unlike other nonlinear control schemes thatbecome ill conditioned in the neighborhood of such a singularpoint, the proposed laws achieve a fast and smooth transitionfrom a suboptimal operating point to the optimum operatingpoint with significantly small computation time. The proposedlaw is also found to be robust in the face of moderateperturbation in the unmeasured disturbances. In particular, whenthe parameter perturbations render the specified set pointunattainable, the quadratic control law is able to maintain stableclosed loop operation without causing input saturation orhaunting. The simulation results are validated by conductingexperimental studies on a single cell PEMFC system and abenchmark heater-mixer setup that exhibits input multiplicitybehavior. The experimental studies demonstrate that the pro-posed quadratic control law is able to operate the system at asingular operating point and establish the feasibility of employ-ing the proposed control law for systems with fast dynamics.

Appendix: Bilinear Matrix Operations

Definition A.1 (Bilinear Matrix): A bilinear matrix B ofdimensions (r × n × m) is an ordered collection of numbersbRγ, R ) 1, 2, ..., r; ) 1, 2, ..., n; γ ) 1, 2, ..., m. It ishighlighted by inclusion in curly brackets as B or bRγ.

Definition A.2: An (r × n × m) bilinear matrix B operatingon an (n × 1) vector v is represented as A ) B(v), where Ais an (r × n × m) matrix with elements

Definition A.3: An (r × n × m) bilinear matrix B operatingon an (n × 1) vector v and a (m × 1) vector w is representedas z ) B(v, w) where z is a (r × 1) vector with elements

An (r × n × n) bilinear matrix is called symmetric if B(v,w) ) B(w, v) for every v, w ∈ Rn.

Definition A.4 (Left Dot Product): The left dot product of an(r × n × m) bilinear matrix B with a (k × r) matrix A isrepresented as

where D is a (k × n × m) bilinear matrix with elements

Definition A.5 (Right Dot Product): The right dot product ofan (r × n × m) bilinear matrix B with an (m × k) matrix Ais represented as

where D is an (r × n × k) bilinear matrix with elements

Definition A.6 (Circle Product): The circle product of an (r× n × m) bilinear matrix B with an (n × k) matrix A isrepresented as

where D is an (r × n × k) bilinear matrix with elements

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aRγ ) ∑)1

n

bRγV

zR ) ∑γ)1

m

∑)1

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bRγVwγ

D ) A•B

dRγ ) ∑η)1

r

aRηbηγ

D ) B•A

dRγ ) ∑η)1

m

bRηaηγ

D ) BoA

dRγ ) ∑η)1

n

bRηγaη


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ReceiVed for reView August 23, 2008ReVised manuscript receiVed September 7, 2009

Accepted September 9, 2009

IE801284B


Development of a Closed Form Nonlinear Predictive Control Law Based on a Class of Wiener Models

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