Passivity and robust PI control of the air supply system of a PEM fuel cell model

Automatica 47 (2011) 2554–2561

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Passivity and robust PI control of the air supply system of a PEM fuel cell model✩ Reine Talj a,1 , Romeo Ortega b , Alessandro Astolfi c,d a

Heudiasyc, CNRS UMR 6599, Université de Technologie de Compiègne, Centre de Recherches de Royallieu, 60205 Compiègne, France

b

Laboratoire des Signaux et Systèmes, L2S, CNRS UMR 8506, Supélec, Univ Paris Sud-P11, 91192 Gif sur Yvette, France

c

Department of Electrical and Electronic Engineering, Imperial College, London, SW7 2AZ, UK

d

DISP, University of Roma, ‘‘Tor Vergata’’, Via del Politecnico 1, 00133 Rome, Italy

article

info

Article history: Received 6 April 2010 Received in revised form 8 March 2011 Accepted 29 April 2011 Available online 7 October 2011 Keywords: Fuel cells Robust control Passivity Monotonicity

abstract Fuel cells are electrochemical devices that convert the chemical energy of a gaseous fuel directly into electricity. They are widely regarded as potential future stationary and mobile power sources. The response of a fuel cell system depends on the air and hydrogen feed, flow and pressure regulation, and heat and water management. In this paper, the study is concentrated on the control of the air subsystem that feeds the fuel cell cathode with oxygen—whose dynamics is described with a widely accepted nonlinear model. Due to the complexity of this model, the model-based controllers that have been proposed for this application are designed using its linear approximation at a given equilibrium point, which might lead to conservative stability margin estimates for the usually wide operating ranges of the system. On the other hand, practitioners propose the use of simple proportional or proportional–integral controllers around the compressor flow, which ensures good performance in most applications. In this paper we provide the theoretical justification to this scheme, proving that this output variable has the remarkable property that the linearization (around any admissible equilibrium) of the input–output map is strictly passive. Hence, the controllers used in applications yield (locally) asymptotically stable loops—for any desired equilibrium point and all values of the controller gains. Ensuring stability for all tuning gains overcomes the inherent conservativeness of linearized dynamics analysis, and assures the designer on the current use of robust, high performance loops. Instrumental to prove the passivity property is the exploitation of some monotonicity characteristics of the system that stem from physical laws. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction A fuel cell is an electrochemical device in which the energy of a reaction between a fuel (e.g., hydrogen) and an oxidant (e.g., oxygen), is converted directly and continuously into electrical energy. Fuel cell systems offer a clean alternative to energy production and are currently under intensive development. This new device is known per its low operating temperature, few moving parts (mechanically robust construction), and low-to-zero emissions during its operation. In particular, it is considered one of the most promising alternatives to fossil-based fuel engines for automotive applications, where the fuel cell is combined with ultra-capacitors or batteries to feed the load. In such hybrid systems energy management is assured by the control of DC/DC

✩ The material in this paper was partially presented at the 2011 Fundamentals and Developments of Fuel Cells conference (FDFC 2011), January 19–21, 2011, Grenoble, France. This paper was recommended for publication in revised form by Associate Editor Yong-Yan Cao under the direction of Editor Toshiharu Sugie. E-mail addresses: [email protected], [email protected] (R. Talj), [email protected] (R. Ortega), [email protected] (A. Astolfi). 1 Tel.: +33 344234631; fax: +33 344234477.

0005-1098/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2011.08.028

converters. To predict, monitor and efficiently control the fuel cell system under a variety of environmental conditions and a wide operating range, it is essential to understand the highly nonlinear behavior of the fuel cell itself. We consider here a proton exchange membrane (PEM) fuel cell system composed of four main subsystems: the hydrogen subsystem that feeds the anode with hydrogen, the air subsystem that feeds the cathode by oxygen, the humidifier and the cooler that maintain the humidity degree and the temperature of the fuel cell, respectively. See Fig. 1. For several physical reasons (Suh, 2006; Talj, Hissel, Ortega, Becherif, & Hilairet, 2010), the control problem is concentrated on the air supply system equipped with an electrical motor and a compressor. The fuel cell presents the problem of oxygen starvation when the load changes rapidly (Pukrushpan, Stefanopoulou, & Peng, 2004a; Suh, 2006). There are two performance variables in compromise. Namely, the net power delivered to the load and the oxygen stoichiometry (ratio of the input oxygen flow over the reacted oxygen flow in the cathode). Previous work presented in Pukrushpan et al. (2004a) proves that, for all operating points, the maximum net power delivered to the load is reached approximately for the same

R. Talj et al. / Automatica 47 (2011) 2554–2561

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Fig. 1. Basic schematic of the fuel cell system.

constant value of stoichiometry, equal to two. Based on this result, we fixed our control objective: we need to avoid starvation by controlling the oxygen stoichiometry around a constant desired value, which corresponds approximately to the maximum net power delivered to the load in steady state. The dynamic behavior of the air supply system is highly nonlinear and uncertain, which render the analysis and design of suitable control laws very complicated (Pukrushpan, Stefanopoulou, & Peng, 2004b). In previous works (Talj, Hilairet, & Ortega, 2009; Talj et al., 2010) we have studied and compared different control laws to improve the transient performance and avoid unacceptable undershoot of the oxygen excess ratio during transients of current demand. In this study we aim to get a better understanding of the characteristics of the system. Once the stability limits are known and the system is well understood, it will be possible to extend the previous studies and even to propose new control strategies and control methods. In Pukrushpan et al. (2004a) and Suh (2006) a 9-th order nonlinear model – derived from physical principles – to describe the dynamics of the fuel cell was proposed. Reduced, 4-th and 3-rd order models of the air supply system were proposed in Suh (2006) and Talj et al. (2010) and Talj, Ortega, and Hilairet (2009), respectively. In spite of its widespread acceptance by the scientific community, the complexity of these models stymies its application for controller design, which is almost invariably carried out using its linear approximation at a given equilibrium point (Grujicic, Chittajallu, Law, & Pukrushpan, 2004; Pukrushpan, Peng, & Stefanopoulou, 2004; Pukrushpan et al., 2004b; Serra, Husar, Feroldi, & Riera, 2006). The validity of this analysis is restricted to a neighborhood of that particular point, even though the system has a wide operating region. Consequently, the resulting numerical designs may be conservative, and may fail to provide tuning rules for high performance applications. This limitation can be overcome incorporating uncertainty models in the linear design, for which several powerful techniques are available in the literature, see e.g. Green and Limebeer (1995). However, to the best of our knowledge, these techniques have not yet been used in fuel cell applications. On the other hand, practitioners propose the use of simple proportional or proportional–integral controllers around the compressor flow, which ensures good performance in most applications. The latter fact suggests that there is an underlying property of the fuel cell that makes it ‘‘easy’’ to control. Our objective in this paper is twofold: to provide the theoretical justification to the scheme used in practice – revealing its stabilization mechanism – and to give simple tuning rules for high performance applications. These objectives are attained invoking the property of passivity. It is well known, that passive systems can be controlled with simple PI loops (Castanõs, Jayawardhana, Ortega, & García-Canseco, 2009; Hernandez-Gomez, Ortega, Escobar,

& Lamnabhi-Lagarrigue, 2010; Van der Schaft, 2000) whose gains, moreover, can take arbitrary positive values. This last feature is essential to design ‘‘tight’’, high performance and robust, controllers. Robustness is further enhanced if the passivity property is established from the analytic, instead of the numerical, model. In other words, if it holds for all admissible equilibrium points. Our main contribution is to prove that the map from the compressor voltage to the flow is such that, the linearization, around any admissible equilibrium, of the 3-rd order nonlinear model is strictly passive (Van der Schaft, 2000). As a consequence, closing the loop with a PI controller yields – for all tuning gains and all equilibrium points – a robust (locally) asymptotically stable system. Instrumental to establish this result is the use of the monotonicity characteristics of the system, which stem from the basic physical laws. Passivity properties have also been exploited in Talj, Ortega et al. (2009), where the fuel cell system is decomposed into subsystems, one of them being output strictly passive. Talj, Ortega et al. (2009) yields a general methodology to tune all type of controllers – eventually nonlinear – with guaranteed local stability, which is established using linearization. The result presented in this paper is different, and in some sense, stronger than the aforementioned one. The remaining of the paper is organized as follows. The mathematical model of the air supply system proposed in Talj et al. (2010) and Talj, Ortega et al. (2009), and the control problem formulation are presented in Section 2. Some remarks on the limitations of numerical controller designs and the role of passivity are presented in Section 3. The linearization of the model and some useful structural properties of it are then given in Section 4. Section 5 contains the main result of the paper, namely, the proof of passivity of the map. The stability analysis of a PI controller and some simulation results are presented in Sections 6 and 7, respectively. We wrap up the paper with some concluding remarks and future work in Section 8. 2. Mathematical model and control problem formulation The dynamical equations of the reduced 3-rd order model, which suitably captures the behavior of the system (Talj et al., 2010), are2 x˙ 1 = −b1 x1 + b2 x2 + b3 − b4 ξ , x˙ 2 = ψ(x2 )



h(x2 , x3 )

x˙ 3 = −c9 x3 −

c16 c10 x3

 + x1 − x2 ,

ϕ(x2 )h(x2 , x3 ) + c13 u

(1)

2 The interested reader is referred to Talj et al. (2010) and Talj, Ortega et al. (2009) for further details on the model.

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R. Talj et al. / Automatica 47 (2011) 2554–2561

with

ϕ(x2 ) =



x2

c12

c11

− 1,

ψ(x2 ) = c14 c16 [1 + c15 ϕ(x2 )], where x1 , x2 and x3 are, respectively, the air pressure inside the cathode, the air pressure in the supply manifold (between the compressor and the fuel cell cathode input), and the motocompressor angular speed. Furthermore, ξ is the fuel cell current, which is a measurable disturbance input, and u is the voltage applied on the compressor motor that is the control input. In view of the difference in time scales between the electrical and the mechanical dynamics, ξ is assumed constant. The dynamical equation of x3 is calculated, as suggested in Suh (2006), using the mechanical equation of a simplified DC motor. All the constants bi and ci – that are functions of the physical parameters of the system – are positive (see details in Talj, Ortega et al. (2009)). The static function of the compressor, h(x2 , x3 ), has the shape shown in Fig. 2. We draw the reader’s attention to the monotonic behavior of the graph, that is essential for the development of our results. The vector of measurable outputs is y1 y2 y3

  y=

V (x1 , x2 ) x2 h(x2 , x3 )

 =

 (2)

where y1 is the fuel cell voltage (also known as polarization characteristic) and y3 is the compressor flow. See Grujicic et al. (2004) and Pukrushpan et al. (2004a) for the analytic expressions of the static functions V (x1 , x2 ) and h(x2 , x3 ), which are omitted here because they are very cumbersome and, for the purposes of our study, only their monotonicity properties are exploited. Note that the moto-compressor angular speed x3 is not measurable. In fact, the driving motor and the compressor are both integrated in a closed volume that, due to mechanical constraints, does not include a speed sensor. Speed can, in principle, be recovered from the measurements y2 and y3 , inverting the function h(x2 , x3 ) = y3 . But this function is highly nonlinear in x3 , complicating its inversion. The performance variables for the fuel–cell system are z z= 1 z2

y1 ξ − c21 u(u − c22 x3 ) c23 , (x2 − x1 ) c24 ξ



[ ] =



(3)

where z1 is the net power delivered to the load and z2 is the oxygen excess ratio, known also as the stoichiometry coefficient. As already mentioned, the control objective is to design a controller that will regulate z2 around a desired constant value, typically taken to be z2⋆ = 2. The pressures x1 and x2 are positive. Furthermore, the motocompressor is unidirectional, hence, the speed x3 , the flow y3 and the feeding voltage u are also positive. Finally, the fuel cell is an irreversible device, hence, its current ξ , voltage y1 and the net power z1 delivered by the fuel cell are always positive. Note that the stoichiometry z2 represents a flow ratio and is also positive. Consequently, all the variables of the system belong to the positive orthant. As indicated above, the control objective is to drive z2 to a constant desired value. The proposition below shows that this objective can be recast in terms of stabilization of an equilibrium point, that we denote x⋆ ∈ R3+ . Furthermore, it also proves that regulating x3 to its desired value drives x1 and x2 to their equilibrium value. Proposition 1. Consider the system (1), (3) with ξ fixed to a constant value.

Fig. 2. Compressor map: the curves are parameterized by x3 , and grow when it increases.

(i) The equilibrium points x⋆1 , x⋆2 are uniquely defined by x⋆3 . (ii) If x = x⋆ then z2 = z2⋆ . Furthermore, for all z2∗ there exists an x∗ , uniquely defined. Proof. To streamline the proof of the proposition the following observations are in order. First, note that, given an equilibrium x⋆ , the (constant) control that assigns this equilibrium, say u⋆ , is univocally defined by the third equation in (1) as ⋆

u :=

[

1

⋆

c9 x3 +

c13

c10 x⋆3

⋆

⋆

⋆

]

ϕ(x2 )h3 (x2 , x3 ) .

(4)

Hence, our attention is concentrated on the first two of Eqs. (1). Define the parameterized mapping r ξ : R3+ → R3 as

 −b1 x1 + b2 x2 + b3 − b4 ξ  h(x2 , x3 ) + c16 (x1 − x2 )  . r ξ (x) :=    c24 ⋆ x2 − x1 − ξ z2 

(5)

c23

It is clear that the set {x ∈ R3+ | r ξ (x) = 0} identifies the equilibrium points x⋆ such that z2 = z2⋆ . Now, r ξ (x) = 0 is a set of three nonlinear algebraic equations in three unknowns. We use the first two equations to prove the first claim. Then, the third equation is solved to find a suitable x⋆3 . (Proof of (i)). The first claim is established showing that the equilibria of the system admit the following parametrization x⋆1 =

 1  b2 ζ (x⋆3 ) + b3 − b4 ξ , b1

x⋆2 = ζ (x⋆3 ),

(6)

where ζ : R+ → R+ is a strictly increasing function. Now, fix ξ ξ x⋆3 ∈ R+ . Setting r1 (x⋆ ) = r2 (x⋆ ) = 0 get x⋆1 =

1 b1

[b2 x⋆2 + b3 − b4 ξ ]

x⋆2 − x⋆1 =

1 c16

h(x⋆2 , x⋆3 ).

Then h(x⋆2 , x⋆3 ) =

c16 b1

[(b1 − b2 )x⋆2 − (b3 − b4 ξ )].

In Talj, Ortega et al. (2009) it is shown that the term b1 − b2 is positive—see Section 4. Hence, the equilibrium point corresponds to the intersection between the curve h(x2 , x⋆3 ) and a straight line with positive slope. Since h(x2 , x3 ) is strictly increasing in x3 , see

R. Talj et al. / Automatica 47 (2011) 2554–2561

Fig. 2, it is clear that there exists a strictly increasing function ζ : R+ → R+ satisfying h(ζ (x⋆3 ), x⋆3 ) = c16

 1−

b2 b1



ζ (x⋆3 ) −

c16 b1

(b3 − b4 ξ ),

(7)

completing the proof. (Proof of (ii)). In view of the parametrization (6), to prove this claim it is enough to show that there exists x⋆3 such that x3 = x⋆3 implies ξ z2 = z2⋆ . Setting r3 (x⋆ ) = 0 and using (6) we get

ζ (x⋆3 ) =

1 b1 − b2

 b1

c24 c23

 ξ z2⋆ + b3 − b4 ξ .

Now, ζ (x3 ) is a strictly increasing function therefore it is oneto-one and admits a left inverse, say ζL : R+ → R+ , such that ζL (ζ (x3 )) = x3 . The proof is completed selecting x⋆3 = ζL

[

1 b1 − b2

 b1

c24 c23

ξ z2⋆ + b3 − b4 ξ

]

,

that, together with (6), defines the desired equilibrium.



3. Limitations of numerical controller designs and the role of passivity In view of the complexity of the nonlinear model – even the reduced 3-rd order system (1) – most of the controllers for this application are designed based on its linear approximation (Grujicic et al., 2004; Pukrushpan et al., 2004b; Romani, Godoy, Beauvois, & Le Lay, 2006; Serra et al., 2006).3 Obviously, the analytic model of the linearized dynamics still preserves some of the structural properties of the physical system. Unfortunately, this is lost when numerical values are inserted to obtain the model for a given equilibrium point. This limits the validity of the analysis – in particular, the predicted stability margins of the controller – to a neighborhood of that particular point. Using numerical ranges and designing controllers that are robust to parameter uncertainty, as done in Talj, Ortega et al. (2009), partially alleviates this problem. As a consequence of losing the physical information, the resulting numerical designs may be conservative, and fail to provide tuning rules for high performance applications. This significantly complicates the commissioning of the controller, which becomes a tedious and expensive trial-and-error procedure. As indicated in the Introduction several powerful linear controller design techniques are available in the literature to overcome these limitations, see e.g. Green and Limebeer (1995). In particular, incorporating uncertainty in the linear design and carrying out linearizations at different operating points can improve the quality of the approximation. However, to the best of our knowledge, these techniques have not yet been used in fuel cell applications. As explained in Section 1, to overcome the conservativeness problem we invoke the property of passivity. Our objective is to look for a passive output that should, additionally, be measurable and detectable. In this respect, it is interesting to recall the main result of Talj, Ortega et al. (2009), which proves that the nonlinear system (1) defines an output strictly passive map u → x3 .4 Unfortunately, the coordinate x3 is not measurable nor does it define a detectable output. Moreover, this result does not even

3 Piece-wise approximations, used for model predictive controllers, are an alternative that carry the heavy prize of a high computational burden and lack of physical insight. See Garcia-Canseco, Jeltsema, Ortega, and Scherpen (2010) for an illuminating example. 4 In Talj, Ortega et al. (2009) the statement is made only for the dynamics of x , 3

looking at x2 as an arbitrary, external signal. From the proof it is clear that this is also true for the whole dynamics.

2557

prove that the linearization of the system is passive. Indeed, it has been shown in Nijmeijer, Ortega, Ruiz, and Van der Schaft (1992) – see also Section 8.3 of Van der Schaft (2000) – that passivity of the nonlinear system implies (local) passivity of its linearization under some conditions on the storage function, in particular, that it has a minimum at the equilibrium. Unfortunately, the storage function in Talj, Ortega et al. (2009) is simply x23 , which clearly does not satisfy this condition. The task of defining an output signal that satisfies all the requirements for the nonlinear model (1) seems daunting. Instead, we will content ourselves with the establishment of such a result for the linearized dynamics in Section 5. Interestingly, it will be shown that the output that is most often used in applications satisfies these properties. The most common procedure to control the air subsystem (Pukrushpan et al., 2004b; Suh & Stefanopoulou, 2005; Talj, Ortega et al., 2009) is to close the loop with a PI controller around the compressor air flow error y3 − h⋆ —recall that y3 , defined in (2), is measurable. Although this configuration ensures good performance in applications, to the best of our knowledge, no rigorous theoretical analysis has been carried out to prove it. A notable exception is Talj, Ortega et al. (2009) where, linearizing only part of the dynamics, a procedure to tune the gains of a cascaded controller configuration – ensuring stability of the system – is proposed. Unfortunately, consistent with the discussion of Section 1, the admissible gain ranges predicted by the theory turned out to be extremely conservative, yielding below par performance (Talj et al., 2010). In the following sections the remarkable property of passivity of the linearization of the map u → (y3 − h⋆ ) is established. 4. Model linearization and some useful properties Proposition 2. The linearization of system (1) around any equilibrium point x⋆ is given by

η˙ = Aη + bu˜ , where u˜ = u − u⋆ ,

 −b 1  ⋆ A= ψ

b2



−ψ ⋆ 1 −

0



b= 0

0 h¯ 2



c16

ψ ⋆ h¯ 3   c16 

q1 0

c13

⊤



q2

,

(8)

where

 c10  q1 = − ⋆ ϕ ⋆ h¯ 2 +ϕ1 h⋆ , x3 [

ϕ⋆



q2 = − c9 + c10 ⋆ x3

h⋆ h¯ 3 − ⋆ x3

]

,

and we have defined the constants

ϕ ⋆ := ϕ(x⋆2 ), ψ ⋆ := ψ(x⋆2 ), h⋆ := h(x⋆2 , x⋆3 ), ∂ϕ ⋆ ∂h ⋆ ⋆ ∂h ⋆ ⋆ ϕ1 := (x ), h¯ 2 := (x , x ), h¯ 3 := (x , x ). ∂ x2 2 ∂ x2 2 3 ∂ x3 2 3 Proof. The proof is easily established writing the dynamics of the error system as

η˙ 1 = −b1 η1 + b2 η2 ,   1 ⋆ η˙ 2 = ψ(x2 ) [h(x2 , x3 ) − h ] + η1 − η2 , c16 [ ] 1 ϕ ⋆ h⋆ η˙ 3 = −c9 η3 − c10 ϕ(x2 )h(x2 , x3 ) − ⋆ + c13 u˜ x3

x3

computing the Taylor series expansion and retaining the first terms. 

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R. Talj et al. / Automatica 47 (2011) 2554–2561

Although the system (1) is a highly complicated set of nonlinear equations, there are several useful properties, which stem from the physical laws, that are instrumental to solving our task. It is shown in Talj, Ortega et al. (2009), that b1 > b2 ,

∀x2 ∈ R+ ,

(10)

ψ(x2 ) > 0,

∀x2 ∈ R+ .

(11)

Using these properties, and the monotonicity of some of the functions describing the dynamics, it is possible to identify some (sign) properties of the elements of the matrix A. Proposition 3. The constants appearing in the matrix A verify the following inequalities. (C1)

ϕ⋆

[

h⋆



]

for all t ≥ 0, all u˜ (t ) (such that the integrals are well-defined) and for some α ∈ R+ , β ∈ R, where | · | is the Euclidean norm. Proof. (P1) First, note that y˜ 3 is zero at the equilibrium. Thus, the linearization of the output, called y˜ ℓ3 , is

∂h ⋆ ⋆ ∂h ⋆ ⋆ (x , x )η2 + (x , x )η3 ∂ x2 2 3 ∂ x3 2 3 = h¯ 2 η2 + h¯ 3 η3 .

y˜ ℓ3 =

(15)

(16)

is strictly positive real (SPR). As is well known, see e.g. Lemma 6.1 of Khalil (2000), this is tantamount to verifying

h¯ 2 < 0,

(R1) ℜ[H (jω)] > 0, ∀ω ∈ R, (R2) limω→∞ {ω2 ℜ[H (jω)]} > 0.

h¯ 3 > 0.

Proof. (C1) The function f (x3 ) := x1 h(x⋆2 , x3 ) is strictly increasing 3 for all x3 in the operating domain. So

( x3 ) =

1 x23

[

] ∂ h(x⋆2 , x3 ) x3 − h(x⋆2 , x3 ) > 0, ∂ x3

⋆

( x3 ) =

1 x⋆3



h

⋆

h¯ 3 − ⋆ x3

s 2 + n1 s + n0 s3

+ d2 s2 + d1 s + d0

> 0,

∂ f2 ∂ϕ ∂h (x2 , x3 ) = (x2 )h(x2 , x3 ) + (x2 , x3 )ϕ(x2 ) > 0, ∂ x2 ∂ x2 ∂ x2

n0 = (b1 − b2 )ψ ⋆ ,

∂h (x2 , x3 ) > 0, ∂ x3

which evaluated at the equilibrium completes the proof.



d 2 = b1 + ψ ⋆ d 1 = b1 ψ ⋆

1−



h¯ 2

Setting s = jω, the frequency response function becomes RN + jIN RD + jID

R N = n0 − ω 2 , ⋆

,

IN = n1 ω,

RD = d0 − d2 ω ,

ID = ω(d1 − ω2 ).

(18)

Hence, R N R D + IN ID

(12)

ℜ[H (jω)] = c13 h¯ 3 .

(13)

Since c13 h¯ 3 > 0 and R2D + ID2 > 0, ℜ[H (jω)] is positive if and only if RN RD + IN ID > 0. Now,

(P1) The linearization of (1), (12) is

where (A, b) are given in (8) and c := col(0, h¯ 2 , h¯ 3 ).

c16

where RN , IN , RD and ID are the real and imaginary parts of the numerator and the denominator, given by



Proposition 4. Consider the system (1) and an equilibrium point x with the output

y˜ 3 = c η,

− q2 ,



c16 

2

η˙ = Aη + bu˜ ,

c16

c16

H (jω) = c13 h¯ 3 .

5. Main result: the compressor flow error is a passive output

⊤



h¯ 2

− b2 ψ ⋆ ] h¯ 2 ψ ⋆ h¯ 3 ⋆ − q2 b1 + ψ 1 − − q1 , c16 c16 [   ] h¯ 2 ψ ⋆ h¯ 3 d0 = −q2 b1 ψ ⋆ 1 − − b2 ψ ⋆ − q1 b1 . 1−

[

which evaluated at the equilibrium proves the claim. (C3) From Fig. 2 it is clear that the function h(x2 , x3 ) is strictly decreasing with respect to x2 and strictly increasing with respect to x3 —see also the analytic expression in Talj, Ortega et al. (2009). So

ℓ

(17)

and

which yields the desired result, given that c9 , c10 , ϕ > 0. (C2) The function f2 (x2 , x3 ) := ϕ(x2 )h(x2 , x3 ) is strictly increasing with respect to x2 in the operating domain of the fuel cell. So

y˜ 3 := h(x2 , x3 ) − h(x⋆2 , x⋆3 ).

,

where n1 = b 1 + ψ ⋆ ,

⋆

∂h (x2 , x3 ) < 0, ∂ x2

Given the matrices A, b and c in the Eqs. (8) and (14), and after some calculation, one can prove that H (s) = c13 h¯ 3 .

in particular, evaluated at the equilibrium, the derivative becomes

dx3

|η(τ )|2 dτ + β 0

0

ϕ ⋆ h¯ 2 +ϕ1 h⋆ > 0. (C3)

df

t

∫

H (s) = c ⊤ (sI3 − A)−1 b,

(C2)

dx3

u˜ (τ )˜yℓ3 (τ )dτ ≥ α

(P2) The proof of strict passivity is established showing that the transfer function

> 0.

h¯ 3 − ⋆ x3

c9 + c10 ⋆ x3

t

∫

(9)

ϕ(x2 ) > 0,

df

(P2) The map u˜ → y˜ ℓ3 is strictly passive. More precisely, along the trajectories of the linearized system

R2D + ID2

.

(19)

RN RD + IN ID (14)

= ω4 (d2 − n1 ) + ω2 (n1 d1 − d0 − n0 d2 ) + n0 d0 .

(20)

R. Talj et al. / Automatica 47 (2011) 2554–2561

From the monotonicity characteristics of the system and the expressions of ni and di mentioned above, it is clear that d2 − n1 = −ψ

h¯ 2

⋆

c16

[ + c9 + c10

ϕ⋆

h⋆



x⋆3

]

h¯ 3 − ⋆ x3

(21)

Furthermore,

[

− q1 ψ ∗2

h¯ 3 c16



h¯ 2

1−

− ψ⋆

h¯ 2 c16



c16

+ 2b2 ψ ⋆

We show that A is Hurwitz proving that it satisfies an algebraic Lyapunov equation of the form

P A + A⊤ P = −C ⊤ C ,

> 0,

n0 d0 > 0.

n1 d1 − d0 − n0 d2 = −q2 b21 + ψ ∗2

2559

]

where P > 0 and the pair (A, C ) is observable. Since the transfer function H (s) is strictly positive real, Kalman–Yakubovich–Popov’s Lemma yields the existence of positive definite matrices P ∈ R3×3 and Q ∈ R3×3 such that PA + A⊤ P = −Q ,

(b⋆1 + b2 ψ ⋆ ) > 0. (22)

The inequalities (21) and (22) show that the term (RN RD + IN ID ) in Eq. (20) is positive for all ω ∈ R, proving (R1). Furthermore, given that R2D + ID2 = (d0 − d2 ω2 )2 + [ω(d1 − ω2 )]2

= ω6 + (d22 − 2d1 )ω4 + (d21 − 2d0 d2 )ω2 + d20 ,

(23)

and considering (20), we find that

(26)

Pb = c .

(27)

Consider the positive definite matrix P =

P A + A⊤ P = −

Q + 2kp Pbb⊤ P

0

0

0

[

]



P 0

0 ki



, which yields

,

where (27) has been used. The matrix Q + 2kp Pbb⊤ P is positive definite, therefore it admits a factorization of the form Q + 2kp Pbb⊤ P = K ⊤ K , where K ∈ R3×3 is nonsingular. It is clear that (26) is satisfied with

lim {ω2 ℜ[H (jω)]} = c13 h¯ 3 (d2 − n1 ) > 0

ω→∞

.. .



where the inequality follows from c13 h¯ 3 > 0 and d2 − n1 > 0.



6. An asymptotically stable PI controller In this section we prove that, in view of strict passivity of the linearized model, the nonlinear system in closed-loop with a PI controller has an asymptotically stable equilibrium at x⋆ . Although the same result can be established for a simple proportional controller,5 u˜ = −kp y˜ 3 , the addition of the integral action – besides the well-known constant disturbance rejection feature – has the advantage that the controller can be implemented without the knowledge of u⋆ , which depends on uncertain model parameters. Proposition 5. Given a fuel cell current ξ and a desired value for the stoichiometry z2 = z2⋆ , let x⋆ be the corresponding equilibrium. Consider the system (1) in closed-loop with the PI controller x˙ c = y˜ 3 ,

C= K



3×4 . 0 ∈R

The observability claim follows from the Popov–Belevitch–Hautus test, e.g., showing that there is no eigenvector of A in the kernel of C . Indeed, for any vector v ∈ C4 ,

C v = 0 ⇔ K col(v1 , v2 , v3 ) = 0 ⇔ v = col(0, 0, 0, v4 ), for some v4 ∈ C. Hence,

Av =

[ ] −ki v4 b 0

,

and, clearly, there is no (eigenvalue) λ ∈ C such that Av = λv . Completing the proof.  7. Simulation results To compute the output for the PI controller (24) we write

u = −kp y˜ 3 − ki xc ,

(24)



⋆

where y˜ 3 is given in (12). The equilibrium x , −

 ⋆

u ki

⋆

z2⋆ c24 b2 − b4 c

x⋆1 =

, with u given

23

b1 − b2 z2⋆ c24 b1 − b4

ξ+

c

b3 b1 − b2 b3

in (4), is asymptotically stable, for any ki , kp ∈ R+ . Moreover, limt →∞ z2 (t ) = z2⋆ .

x⋆2 =

Proof. The proof is established invoking Lyapunov’s indirect method. That is, we prove that the linearization of the closed-loop system, around the equilibrium x⋆ , is asymptotically stable, which implies that x⋆ is a (locally) asymptotically stable equilibrium of the nonlinear system. The proof that limt →∞ z2 (t ) = z2⋆ then follows from claim (ii) in Proposition 1. The linearization of the closed-loop is given by (13) together with

which are obtained setting r1 (x⋆ ) = r3 (x⋆ ) = 0 in (5). The com-

ℓ

x˙˜ c = y˜ 3 , u˜ = −kp y˜ ℓ3 − ki x˜ c ,

(25) ⋆

⋆

where we have defined x˜ c := xc − xc , and used xc = − linearized closed-loop system then takes the form

[ ] [ ] η˙ η = A , x˜ c x˙˜ c

 A :=

A − kp bc ⊤

−ki b

c⊤

0

u⋆ . ki

The

 .

5 Actually, for any linear controller that ensures asymptotic stability of the closed-loop.

23

b1 − b2

ξ+

b1 − b2

,

ξ

ξ

ξ

putation of h(x⋆2 , x⋆3 ) is carried out from r2 (x⋆ ) = 0, with x⋆1 and x⋆2 being replaced by the expressions above, yielding h(x⋆2 , x⋆3 ) =

1 c16

(x⋆2 − x⋆1 ) = z2⋆

c24 c23

ξ.

Hence, y˜ 3 = h(x2 , x3 ) − h(x⋆2 , x⋆3 ) is known, given that the compressor airflow y3 = h(x2 , x3 ) is measured. Figs. 3 and 4 present the simulation results of the system (1) in a closed-loop with the PI controller (24). The gains of the PI are chosen to be kp = 3000, ki = 18 000, so that the poles of the closed-loop system, which are the roots of the characteristic polynomial Dcl (s) = s4 + (c13 h¯ 3 kp + d2 )s3 + [c13 h¯ 3 (ki + n1 kp ) + d1 ]s2

+ [c13 h¯ 3 (n1 ki + n0 kp ) + d0 ]s + c13 h¯ 3 n0 ki , are given by p1 = −1.209, p2 = −5.663, p3 = −27.822, p4 = −54.338, when the fuel cell current is at the nominal value of 191A (Suh, 2006). The simulation is performed with the current profile shown in Fig. 4(a). Fig. 3 shows that the three states x1 , x2

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R. Talj et al. / Automatica 47 (2011) 2554–2561

Fig. 3. Convergence of the three states to the equilibrium point.

Fig. 4. Simulation response of the closed-loop system with the PI controller.

and x3 , representing respectively the air pressure in the cathode, the air pressure in the supply manifold and the rotational speed of the compressor, converge to the equilibrium point corresponding to the desired output z2 = z2⋆ . Fig. 4(b) shows the control input, that is the voltage feeding the motor of the compressor. The convergence of the compressor airflow to its equilibrium point is shown in Fig. 4(c). Finally, Fig. 4(d) shows the convergence of the controlled output z2 to its desired reference z2⋆ , chosen equal to two. 8. Concluding remarks and future work In this paper a theoretical justification has been given to the common practice of regulating the air supply system of a PEM fuel cell with a simple PI controller around the compressor flow. Exploiting the monotonic characteristics of some static functions of the fuel cell, it has been verified that the linearized incremental model of the system – at any equilibrium point – is strictly passive. Based on the latter property the stability of the closed-loop system, with a simple PI controller, is proved around any equilibrium point in the operating domain for all kp > 0 and ki ≥ 0. Unfortunately, as shown in Nijmeijer et al. (1992), passivity of the linearized system does not imply (local) passivity of the original nonlinear map. Therefore, no conclusions can be drawn regarding

the behavior of the system in closed-loop with a nonlinear controller. However, as proven in Van der Schaft (2000), L2 stability of the linearized system implies, local, L2 -stability of the nonlinear system. Since the passivity property that we established is strict, the linearized system is L2 -stable and small gain theorem arguments can be used to analyze nonlinear controllers. Current research is under way in this direction. It should be pointed out that the passivity property has been established for the linearization at the equilibrium points and not at any point in the state space. Some preliminary calculations suggest that the latter, stronger, property is unfortunately not true. A limitation of the local asymptotic stability result of Proposition 5 is that we are unable to estimate the domain of attraction of the stable equilibrium. This requires a Lyapunov analysis of the nonlinear system. Another interesting open question is the potential advantage of using other passive outputs. For instance, it can be shown that the output h(x2 , x3 ) − h(x2 , x⋆3 ) = y3 − h(y2 , x⋆3 ), which is measurable and detectable, is also a passive output. This output is measurable since the compressor air flow y3 = h(x2 , x3 ) and the air pressure in the supply manifold y2 = x2 are measurable. Moreover, the analytic function h(x2 , x3 ) is known, allowing then the calculation of the term h(x2 , x⋆3 ). Simulation results have shown that using the PI with this output, instead of y˜ 3 , yields some nice transient performance.

R. Talj et al. / Automatica 47 (2011) 2554–2561

Another interesting and important point to be studied is the effect of saturation of the input voltage in fast transients, and its influence on the stability analysis. Saturation is a first–third quadrant static nonlinearity that defines a passive operator. This leads us to believe that, as done for mechanical systems, it is possible to carry out this study within our passivity framework. The result presented in this paper provides a first, modest, step towards the development of physically-based controller design methodologies, where the mass and energy balance properties of the system are explicitly exploited. This is the essence, and final objective, of the research area generically known as passivitybased control.

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Reine Talj was born in Saïda, Lebanon. She obtained her B.Sc. in Electrical Engineering from the Lebanese University—faculty of engineering II, Roumieh, Lebanon and her Master in Automatics from the University of ParisSud XI and Supelec, Gif-sur-Yvette, France in 2005 and 2006 respectively. Then she obtained her Ph.D. degree in Modeling and nonlinear control of fuel cells, from the University of Paris-Sud XI, at Laboratoire des Signaux et Systèmes (L2S) and Laboratoire de Génie Electrique de Paris, Gif-sur-Yvette, France in 2009. She was a teacher in IUT d’Orsay from 2006 to 2009, in Electricity, Electrotechnics and Automatics. In 2009/2010 she was a researcher–teacher in the University of Paris-Sud XI. Since September 2010, she is a CNRS researcher at the HEUristics and DIAgnosis of Complex SYstems (HEUDIASYC) laboratory in Compiègne (France). Her main research activities concern fuel cell systems, modeling, nonlinear control, electric machines and intelligent vehicles.

References Castanõs, F., Jayawardhana, B., Ortega, R., & García-Canseco, E. (2009). Proportional plus integral control for set point regulation of a class of nonlinear RLC circuits. Circuits, Systems, and Signal Processing, 28, 609–615. Garcia-Canseco, E., Jeltsema, D., Ortega, R., & Scherpen, J. M. A. (2010). Power-based control of physical systems: four case studies. Automatica, 46, 127–132. Green, M., & Limebeer, D. J. N. (1995). Linear robust control. Englewood Cliffs: Prentice Hall. Grujicic, M., Chittajallu, K. M., Law, E. H., & Pukrushpan, J. T. (2004). Modelbased control strategies in the dynamic interaction of air supply and fuel cell. Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy, 218(7), 487–499. Hernandez-Gomez, M., Ortega, R., Escobar, G., & Lamnabhi-Lagarrigue, F. (2010). Adaptive PI stabilization of switched power converters. IEEE Transactions on Control Systems Technology, 18(3), 688–698. Khalil, H. K. (2000). Nonlinear systems (third ed.) Upper Saddle River, New Jersey: Prentice Hall Pearson Education. Nijmeijer, H., Ortega, R., Ruiz, A. C., & Van der Schaft, A. (1992). On passive systems: from linearity to nonlinearity. In Proceedings of 2nd IFAC symposium on nonlinear control systems. Bordeaux. France (pp. 214–219). Pukrushpan, J. T., Peng, H., & Stefanopoulou, A. G. (2004). Control-oriented modeling and analysis for automotive fuel cell systems. Journal of Dynamic Systems, Measurement, and Control, 126, 14. Pukrushpan, J. T., Stefanopoulou, A. G., & Peng, H. (2004a). Control of fuel cell power systems: principles, modeling, analysis and feedback design. Springer. Pukrushpan, J. T., Stefanopoulou, A. G., & Peng, H. (2004b). Control of fuel cell breathing. IEEE Control Systems Magazine, 24(2), 30–46. Romani, N., Godoy, E., Beauvois, D., & Le Lay, V. (2006). Modélisation et commande du système d’alimentation en air d’un véhicule pile à combustible avec reformage embarqué. In Conférence internationale francophone d’automatique. IEEE CIFA’06. Serra, M., Husar, A., Feroldi, D., & Riera, J. (2006). Performance of diagonal control structures at different operating conditions for polymer electrolyte membrane fuel cells. Journal of Power Sources, 158(2), 1317–1323. Suh, K. W. (2006). Modeling, analysis and control of fuel cell hybrid power systems. Ph.D. thesis. University of Michigan. Suh, K. W., & Stefanopoulou, A. G. (2005). Coordination of converter and fuel cell controllers. International Journal of Energy Research, 29(12), 1167–1189. Talj, R., Hilairet, M., & Ortega, R. (2009). Second order sliding mode control of the moto compressor of a PEM fuel cell air feeding system, with experimental validation. In IEEE IECON. Porto, Portugal. 3–5 November. Talj, R., Hissel, D., Ortega, R., Becherif, M., & Hilairet, M. (2010). Experimental validation of a PEM fuel cell reduced order model and a moto-compressor higher order sliding mode control. IEEE Transactions on Industrial Electronics, 57(6), 1906–1913. Talj, R., Ortega, R., & Hilairet, M. (2009). A controller tuning methodology for the air supply system of a PEM fuel cell system with guaranteed stability properties. International Journal of Control, 82(9), 1706–1719. Van der Schaft, A. (2000). L2 -gain and passivity techniques in nonlinear control (2nd ed.) Springer.

Romeo Ortega was born in Mexico. He obtained his B.Sc. in Electrical and Mechanical Engineering from the National University of Mexico, Master of Engineering from the Polytechnical Institute of Leningrad, USSR, and the Docteur D‘Etat from the Polytechnical Institute of Grenoble, France in 1974, 1978 and 1984, respectively. He then joined the National University of Mexico, where he worked until 1989. He was a Visiting Professor at the University of Illinois in 1987–88 and at the McGill University in 1991–1992, and a Fellow of the Japan Society for Promotion of Science in 1990–1991. He has been a member of the French National Researcher Council (CNRS) since June 1992. Currently he is in the Laboratoire de Signaux et Systemes (SUPELEC) in Paris. His research interests are in the fields of nonlinear and adaptive control, with special emphasis on applications. He has published two books and more than 200 scientific papers in international journals. Dr. Ortega is a Fellow Member of the IEEE since 1999. He has served as chairman in several IFAC and IEEE committees and editorial boards. Currently he chairs the Automatica Paper Prize Award Committee (2009–2012), is an Editor at Large of IEEE Transactions on Automatic Control, and an Associate Editor of Systems and Control Letters and International Journal of Adaptive Control and Signal Processing.

Alessandro Astolfi was born in Rome, Italy, in 1967. He graduated in electrical engineering from the University of Rome in 1991. In 1992 he joined ETH—Zurich where he obtained an M.Sc. in Information Theory in 1995 and a Ph.D. degree with a Medal of Honour in 1995 with a thesis on discontinuous stabilization of nonholonomic systems. In 1996 he was awarded a Ph.D. from the University of Rome ‘‘La Sapienza’’ for his work on nonlinear robust control. Since 1996 he has been with the Electrical and Electronic Engineering Department of Imperial College, London (UK), where he is currently Professor in Non-linear Control Theory. From 1998 to 2003 he was also an Associate Professor at the Dept. of Electronics and Information of the Politecnico of Milano. Since 2005 he is also Professor at Dipartimento di Informatica, Sistemi e Produzione, University of Rome Tor Vergata. He has been visiting lecturer in several universities. His research interests are focused on mathematical control theory and control applications, with special emphasis for the problems of discontinuous stabilization, robust stabilization, robust and adaptive control and model reduction. He has authored more than 100 journal papers, 20 book chapters and over 200 papers in refereed conference proceedings. He is the author (with D. Karagiannis and R. Ortega) of the monograph ‘‘Nonlinear and Adaptive Control with Applications’’ (Springer Verlag). He is an Associate Editor of the IEEE Trans. on Automatic Control, Systems and Control Letters, Automatica, the International Journal of Control, the European Journal of Control, the Journal of the Franklin Institute, and the International Journal of Adaptive Control and Signal Processing. He has also served in the IPC of various international conferences. He is the Chair of the IEEE CSS Conference Editorial Board.