86 ieee transactions on control systems technology…annastef/papers/constrainthandling.pdf · 86...

86 IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 15, NO. 1, JANUARY 2007

Constraint Handling in a Fuel Cell System: A FastReference Governor Approach

Ardalan Vahidi, Ilya Kolmanovsky, and Anna Stefanopoulou

Abstract—A compressor-based air supply system in a fuel cellis susceptible to saturation during fast transients in load. Aggres-sive control of the air compressor may result in compressor surgeor choke, disrupt the flow of air into the cathode of a fuel cellstack, and negatively impact the fuel cell power generation. Lowpartial oxygen pressure in the cathode caused by rapid increasein load may also damage the stack and reduce its life. A load gov-ernor can be added to the air supply control system to monitorthe load and prevent violation of constraints (compressor surge,choke, and partial oxygen pressure) by modifying the load com-mand to the fuel cell system. In this paper, we present the stepsinvolved in the design of such a load governor for the fuel cell ap-plication. To reduce the online computations, we utilize the “fast”reference governor (FRG) approach which has been developed forlinear systems. FRG utilize a “maximal constraint admissible set”calculated offline and, thus, require fewer online calculations. Wepropose a modification to the FRG design to make it applicable tothe nonlinear fuel cell plant. The scheme is then implemented ona real-time simulation platform and it is shown that the computa-tions of the load governor can be performed in real-time.

Index Terms—Compressor surge, fast reference governor(FRG), fuel cell, load governor, oxygen starvation.

I. INTRODUCTION

I N FUEL CELL powered vehicles, one of the performancebottlenecks is posed by the air supply system. In a high pres-

sure proton exchange membrane (PEM) fuel cell, a compressorsupplies air to the cathode. The compressor itself consumesup to 30% of fuel cell generated power and, therefore, itssize has direct influence on overall system efficiency. Moreimportantly the compressor performs the critical task of pro-viding the oxidant into the stack. It is known in the fuel cellcommunity that low partial oxygen pressure in the cathodereduces the fuel cell voltage and the generated power, andit can reduce the life of the stack [1]. The challenge is thatoxygen reacts instantaneously as current (load) is drawn fromthe stack, while the air supply rate is limited by the mani-fold dynamics, compressor surge, and choke constraints [2],

Manuscript received November 22, 2005. Manuscript received in final formJuly 18, 2006. Recommended by Associate Editor Y. Jin. The work of A. Vahidiwas supported in part by the National Science Foundation under Grant 0201332and through the Automotive Research Center under the U.S. Army under Con-tract DAAE07-98-3-0022.

A. Vahidi is with the Department of Mechanical Engineering, Clemson Uni-versity, Clemson, SC 29634-0921 USA (e-mail: [email protected]).

I. Kolmanovsky is with Ford Research and Advanced EngineeringDepartment, Ford Motor Company, Dearborn, MI 48124 USA (e-mail: [email protected]).

A. Stefanopoulou is with the Walter E. Lay Auto Lab, University of Michigan,Ann Arbor, MI 48109 USA (e-mail: [email protected]).

Color versions of Figs. 1–17 are available online at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TCST.2006.883242

[3]. Surge causes large variations in flow and sometimes flowreversal through the compressor. Large amplitude surge mayeven damage the compressor [4], [5]. Reference [6] developsan active surge control approach for centrifugal compressors.Choke happens at sonic mass flow and is an upper limit tothe amount of air the compressor can provide. In the fuelcell system there is a potential for compressor choke during astep-up in current demand. In additional, there is a potentialfor compressor surge during a step-down in current demand.For instance, the air flow controller reduces the compressormotor voltage during a step-down in current demand. A suddendecrease in compressor motor voltage is followed by a fastdecrease in the compressor rotational speed. Since the manifoldpressure cannot drop as quickly, surge may occur.

Low-pass filtering of the current demand to a fuel cell duringsteps in current can prevent constraint violation. The design ofthese filters is usually conservative to ensure satisfactory opera-tion under various operating conditions (see, e.g., [7]). The tran-sients may be managed with a load governor which modifies thecurrent drawn from the fuel cell by only as much as needed forconstraint enforcement. In principal, the design of such a loadgovernor is based on online constraint optimization.

Sun and Kolmanovsky [7] have developed a load governorfor oxygen starvation prevention in a fuel cell using a nonlinearreference governor (RG) approach. The constraints of the com-pressor are not considered in their work. Their RG searches ateach sample time instant for the optimal and constraint admis-sible current demand to the fuel cell based on online optimiza-tion of a scalar parameter and online simulations of a nonlinearfuel cell model. Their approach ensures robustness against pa-rameter variations, but may be computationally demanding. Toimplement the load governor in memory- and chronometric-constrained automotive microcontrollers, it is desirable to re-duce the online computational effort as well as RAM and ROMrequirements.

Our goal in the present paper is to develop a computationallyefficient load governor which enforces the compressor surge,compressor choke, and oxygen starvation constraints in a fuelcell system. We start by employing the fast reference governor(FRG) approach proposed in [8] for linear systems for reducingthe size of the optimization problem and computational effort.To this end, we formulate a load governor for a linearized modelof the fuel cell system. We then present modifications so that theload governor can be applied to the nonlinear fuel cell model.Furthermore, we demonstrate that the order of the load governorcan be reduced without much sacrifice on the performance byusing a reduced-order fuel cell model. Finally, we explain im-plementation of the load governor on a real-time simulation plat-form and discuss real-time computational requirements.

1063-6536/$20.00 © 2006 IEEE

VAHIDI et al.: CONSTRAINT HANDLING IN A FUEL CELL SYSTEM: A FAST REFERENCE GOVERNOR APPROACH 87

Fig. 1. Schematic of air supply control system.

Fig. 2. Compressor map with approximated surge and choke boundaries.

II. PROBLEM DESCRIPTION

Constraint management of the high pressure air supplysystem for a 75-kW PEM fuel cell with 350 cells is addressedin this paper. A schematic of the fuel cell system is shown inFig. 1.

Hydrogen is supplied from a hydrogen tank and its flow is di-rectly controlled by an inlet valve. A compressor supplies highpressure air to the cathode. The control input to the system isthe compressor command , which influences the speed ofthe compressor and, consequently, the amount of air that is sup-plied to the cathode. The current taken from the fuel cell de-termines the rate at which hydrogen and oxygen are consumed,i.e., larger currents require more oxygen and hydrogen to besupplied. Given the current taken from the fuel cell , and thevoltage supplied to the compressor motor (and also the am-bient conditions), the state of species in the stack can be deter-mined. In this paper, we will keep track of the compressor flow

and the pressure downstream of the compressor sincethey, together, indicate if the compressor is near choke saturationor surge instability as shown in Fig. 2. The dashed lines shown inthis compressor map represent boundaries beyond which com-pressor surge and choke can occur. Later, in this paper, we en-force pointwise-in-time constraints to ensure operation of thecompressor inside the bounded region and away from the surgeand choke regions.

As mentioned in the introduction, low oxygen concentrationin the cathode (oxygen starvation) negatively affects the fuel cellvoltage response and can even permanently damage the cells. Aparameter called oxygen excess ratio , is defined as the ratio

between oxygen supplied and the oxygen reacted in the cathode.Low values of lead to oxygen starvation which should bestrictly prohibited. Thus, a lower limit on is another con-straint that we should enforce during operation of the fuel cell.

A phenomenological lumped parameter model of the fuelcell is used in this work to predict and prevent violation of thesurge, choke, and starvation constraints. The fuel cell modelused here is the nonlinear spatially-averaged model of the fuelcell stack together with its auxiliaries presented in [9] based onelectrochemical, thermodynamic, and fluid flow principles. Thestack model represents membrane hydration, anode and cathodeflow, and stack voltage. This stack model is augmented with themodels of ancillary subsystems including the compressor, man-ifold dynamics, cooling system, and the humidifier to obtain anonlinear model of the overall fuel cell system. Since the focusof this paper is on the load governor, we just provide a summaryof the model equations in the appendix of this paper; details ofthe model can be found in [9] and [10].

Due to the complexity of the model, all the governing equa-tions cannot be easily shown in a concise closed form. However,the general state-space form is

(1)

where the nine dynamic states in the model are shown in theequation at the bottom of the page, representing, respectively,oxygen mass in the cathode, hydrogen mass in the anode, ni-trogen mass in the cathode, air compressor speed, supply man-ifold pressure, mass of air in the supply manifold, water massin the anode, return manifold pressure, and water mass in thecathode. Three outputs of interest are compressor air flow ,manifold pressure , and oxygen excess ratio .

Fig. 3 shows the nonlinear simulation results for the modelduring a series of step changes in current. During the currentsteps, the compressor motor input is calculated as a function ofcurrent drawn from the fuel cell based on a feedforward map.This feedforward map is designed such that at steady-state theoxygen excess ratio is regulated at the value of 2.1 The Figureshows current, compressor input, compressor flow, manifoldpressure, and oxygen excess ratio. It can be shown that whilethe compressor flow and manifold pressure increase when thecurrent increases, an undesirable rapid drop in oxygen excessratio still occurs during sudden changes in current levels.

Fig. 4 shows the evolution of the compressor flow2 and pres-sure ratio in the compressor map during the transients of Fig. 3.During a step-up in command, the compressor flow in-creases faster than the pressure downstream from the com-pressor. As a result the compressor operates near the choke

1It is shown in [9] that oxygen stoichiometry of 200% (� = 2) results inmaximum fuel cell net power. It also provides a good margin of safety againststarvation.

2In all the compressor maps in this paper, the x-axis shows the corrected com-pressor flow. The steps for correcting the compressor flow are shown in Table IVin the appendix.


Fig. 3. Fuel cell model response to step changes in current demand.

Fig. 4. Compressor response to step changes in current demand.

boundary. During a step-down in , the operating trajectorynears the surge boundary. Larger steps in the current requirelarger compressor commands that, if applied instantaneously,may result in surge or choke.

It can be shown from the previous results that rapid transientsin current can send the compressor into surge or choke and sat-urate the air supply system or result in oxygen starvation in thecathode. However, once the compressor control system is de-signed to meet basic control design requirements (for example,closed-loop stability, overshoot, etc.) a load governor can beadded for constraint enforcement. A load governor, shown inFig. 5, is an add-on device that slows the transitions in currentdemand , so that the constraints are not violated. A load gov-ernor, can be as simple as a first-order filter which is designedfor the worst case current demand. The drawback of such a pas-sive filter is that it slows down the system response even undersmall transients. The transients can be managed less conserva-tively with an active load governor which modifies the referencecommand only when needed to avoid constraint violation.

Fig. 5. Schematics of fuel cell air supply control.

Modification of the reference command to a close-loop con-trol system for enforcing constraints has been studied under themore general title of RG. A good body of literature has beendeveloped in the past fifteen years to establish theoretical prop-erties of reference governors.

In principle, two different strategies have been used for designof RGs. One approach is based on the ideas of model predic-tive control. The problem is posed as an optimization problem,one in which the difference between the reference commandand actual command is minimized over a future horizon, sub-ject to pointwise-in-time constraints. Bemporad et al. [11] usethis approach and formulate the RG in a quadratic program-ming framework. Bemporad [12] extends this receding horizonstrategy to a nonlinear system and use a bisectional search ap-proach to solve the optimization problem. Casalova and Mosca[13] address the robustness with the governor when a linearsystem is subject to bounded input disturbance.

The other strategy is based on characterizing a set of ini-tial conditions and references that would satisfy the constraints.This idea of maximal output admissible sets has been developedby Gilbert and Tan [14] for design of error governors and fur-ther expanded by Gilbert and Kolmanovsky to design of RGs forlinear [8] and nonlinear [15] systems. In [8], it is shown that forlinear systems the RG can be constructed largely offline, whilethe online computational effort can be systematically reducedat the expense of the increased conservatism in the RG opera-tion. Because of the reduced online computations, the governoris called a fast reference governor (FRG). Disturbances are ad-dressed based on forming disturbance invariant sets [16].3

In this paper, we use the FRG approach introduced in [8] forthe fuel cell problem.

III. LOAD GOVERNOR DESIGN

A. Linearized Model

For the design of the load governor, the nonlinear model of thefuel cell (1) is linearized at a representative operating point. Wechoose nominal stack current as 192 A. The nominal valuefor oxygen excess ratio is selected at 2.0, which provides

3The first strategy assumes that the future reference signal remains constant.Infeasibility of solution may arise because of the finite prediction horizon as-sumption. The second strategy does not make any assumption on future valuesof the reference. Moreover, if the initial conditions lie in the constraint admis-sible set, infeasibility would not arise, since by construction the admissible setis based on satisfying the constraints over a semi-infinite future horizon. A briefcomparison of the two strategies can be found in [17].


maximum fuel cell net power for the nominal current [9]. Thecompressor motor voltage needed, to supply the optimum airflow that corresponds to and 2.0, is 164 V. Thelinearized system has eight dynamic states4 and is described by

(2)

where

and stands for the deviation from the operating point. Thecompressor control input is determined by a combination offeedforward and proportional plus integral (PI) feedback controlaction. The feedback controller is designed to ensure a closed-loop stable system. The discrete-time model of the closed-loopsystem is used for the design of the load governor

(3)

B. Constraints

Fig. 2 shows a compressor map with superimposed surgeand choke constraint lines. In this map, each solid line curverepresents a compressor rotational speed. The surge and chokeboundaries are represented by dotted lines. We introduce con-straints that confine operation of the compressor between thesurge and choke lines and prevent stack starvation. The non-linear surge boundary can be approximated by a straight line forthe most part of the operating region as shown in Fig. 2. Bothcompressor flow and pressure ratio are functions of states of thesystem and are relatively easy to measure. The choke limit canbe expressed similarly. The constraints can then be representedby two linear inequalities defined by the linear approximation

(4)The lower limit of oxygen excess ratio is set to 1.9 to avoid largevariations in the cathode oxygen partial pressure and, hence,stack oxygen starvation

(5)

Equations (4) and (5) can be combined as follows:

(6)where is the predicted value of the outputs at instant

based on information available at instant .

4One of the states, the mass of water in the cathode, is unobservable duringlinearization and, therefore, does not appear in the linearized model.

C. Linear Load Governor

Our goal is to determine the current , which is as close aspossible to the current demand , and does not violate the con-straints. The reference modification can be accomplished via afirst-order linear filter with a scalar adjustable bandwidth pa-rameter

(7)

and . Ideally, 1 meaningand the current command, only suffer a unit delay. When thereis a large change in and a possibility of future constraint vi-olation exists, is reduced to avoid constraint violation. In theextreme case when 0, we have . Theparameter is maximized at each sample time , subject tothe condition that maintaining for allguarantees constraint satisfaction. The assumption at each in-stant is that measured current demand , stays constant overthe future horizon.

The optimization can be solved in a few different ways. Itcan be arranged as a linear programming (LP) problem withthe single variable and the constraints (6) for a sufficientlylarge horizon, and solved online. Bisectional search for max-imum constraint-admissible is another possible online solu-tion and is applicable to nonlinear systems as well [7]. Suchonline solutions may be computationally intensive for systemswith more than a few states.

Fortunately, for linear systems, a large portion of calculationscan be performed offline, thereby reducing the online computa-tional effort. Specific procedures for such FRGs are detailed in[8] and are used here for the fuel cell application. We providea brief summary of the methodology and refer the interestedreader to [8] for details.

Future constraint violations can be predicted by checking ifthe state of the system belongs to a maximal output admissibleset, called . The is the set of all initial states , andthe modified reference , which with 0 guaranteessatisfaction of future constraints. It is defined as

(8)

where is the constraint set described by (6) and the state dy-namics are those of (3). The set, , is positively invariant forthe system defined by (3) with 0. Thus, if the system startsin this set and the current is kept constant into the future, thetrajectory will remain in and the constraints will be satis-fied.

The goal is to find the maximum value of which maintainsthe state in

(9)The set does not, in general, admit a characterization bya finite set of linear inequalities (i.e., it may not be finitely de-termined). It does, however, have a computable approximation

, which is finitely determined, see [8].


The set is derived from using the following proce-dure. Let denote the steady-statevalue of the state for a constant and let

be the dc gain of the plant from the inputto the output . Then the set admits the following charac-terization:

Then, is constructed as

where 1 0. Under the assumptions that is asymp-totically stable, is compact and it can be shownthat , and that it is positively invariant and finitelydetermined. In other words, can be represented by a finitenumber of linear inequalities and there exists a finitesuch that

The reason is finitely determined is that due to the stabilityof , , and, thus, implies thatthe inequality constraintbecomes inactive for all sufficiently large.

For a system with states and linear constraints,is a polytope5 with faces represented as a set of solutions toa system of linear inequalities of the general form

(10)

For systems with large state dimension and high samplingrates, the number of inequalities in the representation ofcan grow large. This is undesirable for two reasons: The effortto compute increases with the number of inequalities inthe representation of and ROM size to store a representa-tion of also increases with the number of inequalities in itsrepresentation. Often, however, some inequalities in the repre-sentation of are almost redundant, i.e., if they are elimi-nated from the representation of the resulting polytope isonly slightly larger than . Since the polytope resulting fromsuch constraint elimination may not be a constraint-admissibleset of initial conditions itself, it is scaled down uniformly in the

-direction (but not in the direction) until it is contained in. After this process of inequality elimination and shrinkage,

we obtain a polytope , which is constraint admissible and hasfewer inequalities compared to . At the same time, one hasto keep in mind that is only a subset of and, thus, canresult in more conservative performance.

5A polytope may be defined as the convex hull of a finite set of points, or asa bounded intersection of a finite set of half-spaces.

Fig. 6. ~O projected onto the state-current planes.

The set may not be positively invariant. Thus, the situationmay arise that a feasible does not exist. In the eventthat the trajectory leaves the set , in agreement with the theo-retical results in [8], the RG sets 0.

Once or are determined, the online evaluation ofis relatively simple. A computationally efficient method forfinding is given in [8] which involves a fixed number ofadds, subtracts, divides, multiples, max, and min operations ineach sampling interval.

We next demonstrate the performance of the fast loadgovernor in linear simulations. Once constraint satisfactionis shown in linear simulations, we explain the procedure forapplication of the linear load governor to the nonlinear plant.

IV. LINEAR SIMULATION ANALYSIS

A. Governor Performance

In this section, we analyze the performance of the fast loadgovernor based on both the and admissible sets. The sam-pling frequency is fixed at 100 Hz. Constraints are surge, choke,and starvation constraints given in (6). The set ischaracterized by 348 linear inequalities ( 348) and is de-termined offline. Due to its large dimension, the admissible set

cannot be geometrically visualized. Instead, Fig. 6 showsits projection onto two dimensional state-current planes. In allgraphs, the x-axis represents the current taken from the fuel celland the y-axis shows the fuel cell states. We can see that the con-straints of the system can be satisfied for a current up to 430 A.The limiting factor is the rate by which the current is increased.If the current is increased too quickly, the states leave the poly-gons shown and constraints may be violated.

Constraint elimination and a shrinkage factor of 1.2 generatewith only 77 linear inequalities. For comparison,

shrinkage factors of 1.4, 1.75, 2.0, and 10 resulted in with, re-spectively, 65, 51, 42, and 22 inequalities; the correspondingto 1.2 was selected as the best compromise between the numberof inequalities and conservatism of the RG operation.


Fig. 7. Compressor pressure and flow trajectory for different load governordesigns.

Fig. 8. Simulation during tip-in transient when oxygen starvation constraint isactive.

We examine the performance of the fast load governor whichuses (subsequently referred to as -governor), and fastload governor which uses (subsequently referred to as

-governor) during a series of step-ups and step-downs in thecurrent demand (which correspond to driver tip-in and tip-outcommands, respectively) with maximum current step of 100 A.6

Fig. 7 shows the compressor map during the entire load cycle.The current profile is shown at the top left corner.

Fig. 8 shows the oxygen excess ratio, current, and compressormotor command during the 100-A tip-in. The thin solid lineis the actual current demand. The small insert plot shows theoptimum values of the parameter . In the plots, three differentcases are shown: unconstrained, with -governor, and withthe -governor. In the unconstrained case, the surge constraint

6A simple kinetic energy calculation shows that accelerating a 1000-kg ve-hicle from 20 to 21.5 m/s (45 to 48 mph) in 1 s, requires almost 100 A on a300-V BUS that connects the fuel cell with a traction motor.

TABLE INUMBER OF ONLINE FLOATING POINT OPERATIONS AND

CPU TIME FOR DIFFERENT LOAD GOVERNOR DESIGNS

is violated during tip-outs as is shown in the compressor mapplot. Also, the oxygen starvation constraint is not met duringthe tip-in period and oxygen excess ratio almost reaches thecritical value of 1. The -governor and -governor enforceall the constraints. The governor negotiates the constraints bymoving along the constraint boundary. The -governor per-forms more conservatively and avoids constraint violation by acertain margin. This happens because . The advantageof the P-governor is in reduced online computation load as wewill illustrate later.

B. Computational Requirements

The computational requirements of each algorithm can bea deciding factor for which algorithm gets implemented. Thecomputational load of an algorithm may be characterized bythe number of floating point operations (flops) performed. Inreference [8], the number of flops (multiplication, additions,comparisons) for online calculation of FRG is estimated to be

, where , , and are the number of ref-erence commands, total number of states, and total number ofconstraints. In the simulations presented so far, 1 and10. For the -governor, 348 and, therefore, the estimatednumber of flops is 10 092 per sampling instant. For the -gov-ernor, 77 and the number of flops reduces to 2233, an al-most five-fold reduction. We also used the flops command inMATLAB7 to get an estimate for the actual number of flops fordifferent load governor designs.8 The average, maximum, andminimum number of recorded flops is summarized in Table I.MATLAB also provides an estimate of the CPU time spent onrunning a selected portion of the code. The last column of Table Ishows the total CPU time required for running a full simulationof the linear model and load governor on an 866-MHz Intel9

Pentium III processor. It is shown that the CPU time spent onthe calculations of the load governor are considerably reducedby moving from the -governor to the -governor.

The number of flops given by MATLAB has little variationfrom one step to another and appears to be relatively close tothe theoretically estimated value. We note that 2082 flops at anupdate rate of 10 ms for the -governor case is within the ca-pability of automotive microcontrollers although this is still arather large computational task. In fact, an optimized imple-mentation in a production automotive micro-controller simu-lator showed that the can be calculated within 1.3 ms; andthis calculation requires close to 4 kB of ROM (this represents

7MATLAB is a registered trademark of The MathWorks Inc., Natick, MA.8The command flops is not supported in newer releases of MATLAB. We

used the release 11.1 of MATLAB to get a flop count.9Intel is a registered trademark of Intel Corporation, Santa Clara, CA.


the total size of the code size and constants). The computationaleffort involved can be further mitigated by changing the updaterate. For example, it has been shown via simulations that the up-date rate of 20 ms still yields acceptable performance. Further,when 1 and the reference is constant or varies slowly,then can be kept equal to 1 without performing any calcu-lation, thereby further reducing the computational load for theload governor case.

V. APPLICATION OF THE FAST LOAD GOVERNOR

TO THE NONLINEAR MODEL

The fast load governor design described previously, is basedon the assumption that the plant is linear. In this section, we ex-plain how the fast load governor can be modified and applied tothe nonlinear model of the fuel cell system. Our focus will be onthe -governor, but the discussions apply to the -governoras well.

Recall that in the -based load governor, the parameterwas a function of the linear state of the system, current demand,current taken from the fuel cell, and . In the first trial ofthe load governor on the nonlinear plant we used to guardagainst constraint violation, while replacing the linear stateswith nonlinear states. While this approach worked well for smallto medium deviations from the operating point (50-A steps incurrent demand up or down) it failed to perform satisfactorily forlarger steps. Specifically, during a current step of 100 A, wasset to zero by the governor and the reference current trackingwas lost. This was due to differences between linear and non-linear systems which caused the nonlinear state to be outside of

in steady-state. Next, we attempted to remedy this situationby adjusting the nonlinear state by the difference between linearand nonlinear steady states. Using (3), the linear state equilib-rium for a current level is . The non-linear equilibrium corresponding to was calculated offlineby simulation under different loads and stored in a look-up table

. In particular, we simulated the nonlinear plant model for18 different values of in the range of and in in-crements of 10 A. After the plant reached steady-state for eachvalue of current, we recorded the steady states in the look-uptable. The adjusted state is then calculated according to

(11)

where denotes the augmented states of the nonlinear fuelcell model and the compressor controller. Since is con-structed using the linear model, the nonlinear state is ad-justed by subtracting the nonlinear steady-state , andadding the linear steady-state , then thisadjusted state along with is used to find the value of .10

Note that even with the adjustment given by (11), the dy-namics of the process are still predicted by the linearized model.Figs. 9 and 10 show the compressor trajectory and the time re-sponse of the nonlinear plant for a series of step-ups and downsin demand. Figs. 9 and 10 show that the load governor modifiedby (11) is able to reduce large excursion into the surge regionduring the step of 100 A and that it reduces (but fails to elimi-nate) oxygen starvation constraint violation of 1.9 during

10In other words, we are shifting the polytope ~O in the state-space by thedifference �(I (k))� (I � A ) B �I (k).

Fig. 9. Compressor pressure and flow trajectory with the load governor ad-justed through (11).

Fig. 10. Simulations for the case of the load governor adjusted by (11).

the first most aggressive tip-in. The load governor enforces theoxygen starvation constraint during other, less aggressive tip-ins. The system response, however, is jittery, since the load gov-ernor often and sporadically sets to zero. Moreover, it is notrealistic to assume all the states of the nonlinear plant can bemeasured.

Finally, to improve our results, another approach was pursuedwherein the difference between the nonlinear plant and its linearmodel was reflected in the construction of . Specifically, weredefined as follows:

(12)where is a constant output disturbance term. At each instant

during online calculations, the disturbance is the differencebetween the plant output and the output predicted by the linearmodel

(13)


Fig. 11. Schematics of the plant, linear model, and the RG and their interac-tions.

where denotes the value of the outputs at the operating point.The dimension of the admissible set is increased by the numberof outputs , i.e., . The governor parameteris determined as follows:

(14)where in (14) we use the state predicted by the linear modeland account for the mismatch between the plant and the modelvia the disturbance term . This approach is intended to reducethe governor sensitivity to model uncertainty which in our caseis the mismatch between the linear and nonlinear models. Thiscorrection can help in eliminating constraint violation and jitterin the response.

Fig. 11 shows the schematic of this implementation. The loadgovernor stores the system matrices that represent the linearizedmodel of the plant in addition to the inequalities describing theset (or ). The maximum value of is calculatedbased on the current demand , linear model states , andthe difference between the plant and model outputs . Conse-quently, the allowable fuel cell current is determined and ap-plied to both the nonlinear plant and the linearized model. Theprocess is repeated at each sampling time.11

Figs. 12 and 13 show the simulation results and confirmthat surge constraint violation is considerably reduced and thatoxygen starvation constraint violation and jitter are eliminated.Note also that for this implementation of the governor, only thestate predicted by the linear model and the measurement of theoutput are needed; the full plant state (i.e., state of thenonlinear model) does not need to be known.

Note that by including the -term the number of online com-putations increases by approximately , where is thedimension of the disturbance vector (number of measured out-puts) and is the total number of constraints. The approximatenumber of flops is then .

VI. REDUCED-ORDER LOAD GOVERNOR

One possible way of reducing the number of computations isto use a low-order linear fuel cell model for construction of theload governor.

11Note that adding an output injection term to the linear model is anotherpossibility which may further reduce the estimation error.

Fig. 12. Compressor flow trajectory when the generalized load governor isused. The surge excursion is noticeably reduced when the disturbance observeris used.

Fig. 13. Performance of the generalized fast load governor in nonlinear sim-ulations without state-feedback. Inclusion of the disturbance term eliminatesconstraint violation.

The nine-state fuel cell model discussed so far, representsthe anode, cathode, and return manifold dynamics in additionto the compressor and inlet manifold dynamics. While thesesubsystems interact, the compressor and oxygen flow dynamicsmight still be approximated well by a lower-order model. A re-duced model of the nine-state fuel cell system is obtained in[18] for simulation of the air supply side. The reduced modelhas only four dynamic states: partial oxygen pressure inside thecathode , partial nitrogen pressure inside the cathode ,compressor motor speed , and the supply manifold pressure

. Fig. 14 compares the response of this reduced four-statemodel to the original nine-state model. Only during the two in-tervals of step-down in current (between the third and fourthseconds and sixth and seventh seconds), the response of the two


Fig. 14. Comparison of the original model and the four-state reduced model.

models is different. The reason for this difference may be at-tributed to the fact that during these two intervals, the com-pressor enters the surge region where the models are highlynonlinear. Therefore, a small mismatch in the supply manifoldpressure can cause large deviations in the compressor flow andthe oxygen excess ratio response. The interesting question thatarises here is, if an RG designed based on the reduced modelcan enforce constraints of the full-order fuel cell system. If thisis in fact possible, we may reduce the computations further byusing a reduced-order load governor.

To address this question, we designed new and -gover-nors based on a linearized approximation of the reduced-model.We then applied the reduced-order governors for constraint en-forcement of the full-order nonlinear model. The difference be-tween the two models was compensated by the disturbance termdescribed by (13).

Figs. 15 and 16 show the performance of the reducedand governors when applied to the full-order nonlinearfuel cell model. The reduced-order -governor satisfies theconstraints of the full-order nonlinear model except duringrapid step-downs in current when the surge constraint is in-stantaneously violated.12 The -governor allows operationof the compressor near the surge line, the drawback is thata small disturbance can result in the compressor surge. The

-governor, on the other hand, is more conservative. As aresult, it was able to satisfy the constraints (practically at alltimes) during large disturbances even though it was designedfor a reduced-order model of the plant.

When using the reduced model, the closed-loop system has atotal of 5 states (4 model states and 1 compressor controller

12The excursion into the surge region lasts not more than a few sampling timesin our simulations. In practice, once the compressor starts to surge, the recoverymay take much longer.

Fig. 15. Compressor pressure and flow trajectory when reduced governors areapplied to the full-order nonlinear model.

Fig. 16. Time history of response when reduced governors are applied to thefull-order nonlinear model.

TABLE IICOMPUTATIONS FOR FULL- AND REDUCED-ORDER GOVERNORS

state). Therefore, the set which includes the disturbanceterm belongs to and is characterized by 404 linear con-straints. Constraint elimination and shrinkage factor of 1.2resulted in the approximation of with only 66 con-straints. Table II compares the number of flops (per samplingtime) calculated analytically for the full- and reduced-ordergovernors and shows that the required number of online com-putations for the reduced governors is considerably lower. In


Fig. 17. Response when oxygen excess ratio is not measured and is estimatedusing a static map.

the case of the limited computational resource (which is usuallythe case in real-time implementation), the governor with theleast computational effort is recommended.

VII. OBSERVER DESIGN

The FRG described above requires measurements of all theconstrained outputs. While the compressor flow and manifoldpressure are relatively easy to measure, oxygen excess ratio isnot a measured variable. Simulations show that there is a cor-relation between stack voltage and oxygen excess ratio. Hence,we used stack voltage, compressor flow, and manifold pressureto construct an observer for oxygen excess ratio. We first trieda Kalman filter based on the linearized reduced-order fuel cellmodel. During sudden transients the Kalman filter over esti-mated the drop in oxygen excess ratio, leading to conservativeRG action. In our next attempt, we used the measurements fromthe full-order fuel cell model and a least square curve fit to con-struct a static map which relates oxygen excess ratio to com-pressor air flow (kilograms per second), manifold pressure (bar),stack voltage (volt), and stack current (amp) as follows:

The static map could predict oxygen excess ratio in differenttest cycles closely; based on its performance and simplicity, wechose the linear static map over the more complicated observerdesigns. Fig. 17 shows the response of the reduced -governor(applied to the full-order fuel cell model) when the static mapwas used to estimate oxygen excess ratio. The oxygen excessratio is estimated with small error, as a result, the FRG is ableto enforce the lower bound on oxygen excess ratio.

VIII. REAL-TIME SIMULATIONS

Our analysis of the computational effort has so far been basedon an approximate flop count. While this approach provides

TABLE IIIMODEL VARIABLES AND PARAMETERS

useful insights into computational efficiency of different algo-rithms, it does not provide conclusive results on the CPU usageon a real-time platform. To obtain a more realistic estimate ofcomputational requirements of the FRG, we also carried out thesimulations on a real-time Opal-RT platform. The target pro-cessor was a 2.1-GHz processor running QNX real-time op-erating system. The simulations were executed at a fixed inte-gration step of 0.01 s. The processor was able to calculate themodified reference , in a fraction of a sampling time. For the

-governor implementation, for example, the combined simu-lation time for the plant and the governor was around 130 ms. Ofthis total time, only 14% were spent on running the linear modeland the governor, while the rest was used to run the plant model.In other words, the effective computational time for the gov-ernor was almost 1/500 of the sampling interval on the real-time2.1-GHz processor.


TABLE IVCALCULATION OF COMPRESSOR FLOW

IX. CONCLUSIONS

This paper presented the design of a load governor forpreventing oxygen starvation and compressor surge and chokein fuel cell systems. The main design objective was to enforcethese constraints without compromising performance whenconstraints are inactive and with minimal computational burdenon a microcontroller. To meet these requirements the max-imal constraint admissible set design procedure was used. Forlinear systems this approach allows carrying a large portionof computations offline, thus, reducing the online computationburden. We first employed the idea of such fast load governorson a linearized fuel cell model and provided some insight intocomputational requirements. When applied to the nonlinearplant model, the load governor required an adjustment tocompensate for differences between the linear model and thenonlinear system. Introducing a step disturbance observer inthe load governor design allowed to nearly eliminate constraintviolation and jitter in the response. The step disturbance ob-server eliminated the need for measurement or estimation ofthe plant states as well. We then showed that the computationscould be further reduced by using a lower-order model in de-signing the load governor. Overall, we showed that the numberof computational flops could be reduced 6–8 times withoutmuch sacrifice on the performance by moving to a reduced loadgovernor. It was shown that with the proposed design the loadgovernor computations can be easily handled in real-time ona rapid prototyping platform. Feasibility of computations onan automotive microcontroller was also established by runningthe governor in a production microcontroller simulator. Whilethe results were specific to the fuel cell problem, the insightsinto the computational requirements and the design procedurefor fast load governors applied to nonlinear plants may also beuseful in other applications.

APPENDIX

This appendix provides a summary of the compressor andthe fuel cell models governing equations and parameters. Themodels and parameters are explained in more detail in [9].Table III lists the parameters and variables of the model.

The Compressor Model: The compressor air flow andits temperature are determined using a nonlinear model for

the compressor which has been developed in [9] for an alliedsignals centrifugal compressor that has been used in a fuel cellvehicle [19].

The compressor outlet temperature and the torque required todrive the compressor are calculated using standard thermody-namic equations [20], [21]. The temperature of the air leavingthe compressor is calculated as follows:

(A-1)

where is the inlet atmospheric pressure, is the outletpressures, 1.4 is the ratio of the specific heats of air, isthe compressor efficiency, and is the atmospheric temper-ature. The compressor driving torque is

(A-2)

where is the compressor rotational speed, is the com-pressor flow, and is the specific heat capacity of air. Thecompressor rotational speed is determined as a function ofcompressor motor torque and the torque required to drivethe compressor

(A-3)

where is the compressor inertia. The compressor motortorque is calculated as a function of motor voltageusing a dc motor model

(A-4)

where , , and are the motor constants and is themotor mechanical efficiency.

The compressor air mass flow rate is determined asa function of pressure ratio across the compressor and bladespeed, using a compressor map shown in Fig. 2. In this map, thedashed lines represent boundaries beyond which compressorsurge and choke can occur. The equations used here to representcompressor dynamics are valid within these bounds. Enforce-ment of point-wise-in-time constraints as explained in the paperensured operation of the compressor inside the bounded region


TABLE VFUEL CELL MODEL GOVERNING EQUATIONS

and away from the surge and choke regions. In our simulations,this map is modeled using a nonlinear curve-fitting technique,which calculates compressor air flow as a function of inletpressure , outlet pressures , and compressor rotationalspeed

(A-5)

The details of compressor flow calculation are shown inTable IV. A summary of fuel cell governing equations is pro-vided in Table V.

ACKNOWLEDGMENT

The authors would like to thank Prof. J. Sun of the Universityof Michigan for his useful suggestions and for giving us accessto the Opal-RT platform. They would also like to acknowledgeC. Cox of Ford Motor Company who performed the assessment

of the load governor C-code implementation in the productionmicrocontroller simulator.

REFERENCES

[1] G. Boehm, D. Wilkinson, S. Khight, R. Schamm, and N. Fletcher,“Method and apparatus for operating a fuel cell,” U.S. 6,461,751, Oct.8, 2002.

[2] J. Pukrushpan, A. Stefanopoulou, and H. Peng, “Control of fuel cellbreathing,” IEEE Contr. Syst. Mag., vol. 24, no. 2, pp. 30–46, Apr.2004.

[3] A. Vahidi, A. Stefanopoulou, and H. Peng, “Model predictive controlfor starvation prevention in a hybrid fuel cell system,” in Proc. Amer.Contr. Conf., 2004, pp. 834–839.

[4] P. Moraal and I. Kolmanovsky, Turbocharger modeling for automotivecontrol applications, SAE, International, Warrendale, PA, Tech. Rep.1999-01-0908, 1999.

[5] B. de Jager, “Rotating stall and surge control: A survey,” in Proc. 34thConf. Dec. Contr., 1995, pp. 1857–1862.

[6] J. T. Gravdahl and O. Egeland, “Centrifugal compressor surge andspeed control,” IEEE Trans. Contr. Syst. Technol., vol. 7, no. 5, pp.567–579, Sep. 1999.


[7] J. Sun and I. Kolmanovsky, “A robust load governor for fuel celloxygen starvation protection,” in Proc. Amer. Contr. Conf., 2004, pp.828–833.

[8] E. Gilbert and I. Kolmanovsky, “Fast reference governors for systemswith state and control constraints and disturbance inputs,” Int. J. RobustNonlinear Contr., vol. 9, pp. 1117–1141, 1999.

[9] J. Pukrushpan, A. Stefanopoulou, and H. Peng, Control of FuelCell Power Systems: Principles, Modeling, Analysis, and FeedbackDesign. London, U.K.: Springer-Verlag, 2004.

[10] J. Pukrushpan, H. Peng, and A. Stefanopoulou, “Control-oriented mod-eling and analysis for automotive fuel cell systems,” ASME J. Dyn.Syst., Meas., Contr., vol. 126, pp. 14–25, 2004.

[11] A. Bemporad, A. Casavola, and E. Mosca, “A predictive referencegovernor for constrained control systems,” Comput. Ind., vol. 36, pp.55–64, 1998.

[12] A. Bemporad, “Reference governor for constrained nonlinear systems,”IEEE Trans. Autom. Contr., vol. 43, no. 3, pp. 415–419, Mar. 1998.

[13] A. Casavola and E. Mosca, “Reference governor for constrained un-certain linear systems subject to bounded input disturbances,” in Proc.35th Conf. Dec. Contr., 1996, pp. 3531–3536.

[14] E. Gilbert and K. Tan, “Linear systems with state and control con-straints:the theory and applications of maximal output admissible sets,”IEEE Trans. Autom. Contr., vol. 36, no. 9, pp. 1008–1020, Sep. 1991.

[15] E. Gilbert and I. Kolmanovsky, “Nonlinear tracking control in the pres-ence of state and control constraints: A generalized reference gov-ernor,” Automatica, vol. 38, pp. 2063–2073, 2002.

[16] I. Kolmanovsky and E. Gilbert, “Theory and computation of distur-bance invariant sets for discrete-time linear systems,” Math. ProblemsEng., vol. 4, pp. 317–367, 1998.

[17] J. Rossiter and B. Kouvaritakis, “Reference governors and predictivecontrol,” in Proc. Amer. Contr. Conf., 1998, pp. 3692–3693.

[18] K. -W. Suh and A. Stefanopoulou, “Coordination of converter and fuelcell controllers,” Int. J. Energy Res., vol. 29, pp. 1167–1189, 2005.

[19] J. Cunningham, M. Hoffman, R. Moore, and D. J. Friedman, Require-ments for flexible and realistic air supply model for incorporation intoa fuel cell vehicle (FCV) system simulation, SAE, International, War-rendale, PA, Tech. Rep. 1999-01-2912, 1999.

[20] M. Boyce, Gas Turbine Engineering Handbook. Houston, TX: Gulf,1982.

[21] J. Gravdahl and O. Egeland, Compressor Surge and Rotating Stall.London, U.K.: Springer-Verlag, 1999.

Ardalan Vahidi received the B.S. and M.S. degreesin civil engineering from Sharif University, Tehran,Iran, in 1996 and 1998, respectively, a second M.S.degree in transportation safety from George Wash-ington University, Washington, DC, in 2002, and thePh.D. degree in mechanical engineering from theUniversity of Michigan, Ann Arbor, in 2005.

Currently, he is an Assistant Professor of Mechan-ical Engineering at Clemson University, Clemson,SC. His current research interests include optimiza-tion-based control methods and control of vehicular

and energy systems.

Ilya V. Kolmanovsky (SM’04) received the M.S.and Ph.D. degrees in aerospace engineering and theM.A. degree in mathematics from the Universityof Michigan, Ann Arbor, in 1993,1995, and 1995,respectively.

Currently, he is a Technical Leader at Ford Re-search Advanced Engineering, Dearborn, MI. In ad-dition to expertise in the automotive powertrain con-trol, his research interests include constrained con-trol, optimization-based control, and control of non-linear systems. He has served as an Associate Editor

of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY (2002–2004)and the IEEE TRANSACTIONS ON AUTOMATIC CONTROL (2005–present).

Dr. Kolmanovsky was a recipient of the 2002 Donald P. Eckman Award ofAmerican Automatic Control Council and of the 2002 IEEE TRANSACTIONS ON

CONTROL SYSTEMS TECHNOLOGY Outstanding Paper Award.

Anna G. Stefanopoulou (SM’05) received theDiploma from the National Technological Universityof Athens, Greece, in 1991, the M.S. degree innaval architecture and marine engineering, a secondM.S. degree in electrical engineering and computerscience, and the Ph.D. degree from the Universityof Michigan, Ann Arbor, in 1992, 1994, and 1996,respectively.

Currently, she is a Professor in the Mechanical En-gineering Department at the University of Michigan.She was an Assistant Professor (1998–2000) at the

University of California, Santa Barbara, and a Technical Specialist (1996–1997)at the Scientific Research Laboratories at Ford Motor Company, Dearborn, MI.Her current research interests include control of advanced internal combustionengines and fuel cell power systems.

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