Predictive control for sector bounded nonlinear model and its application to solid oxide fuel cell systems

Applied Mathematics and Computation 218 (2012) 9296–9304

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Predictive control for sector bounded nonlinear model and its application to solid oxide fuel cell systems S.M. Lee a, O.M. Kwon b, Ju H. Park c,⇑ a

Department of Electronic Engineering, Daegu University, Gyungsan 712-714, Republic of Korea School of Electrical Engineering, Chungbuk National University, Republic of Korea c Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea b

a r t i c l e

i n f o

Keywords: Model predictive control Sector bounded nonlinear model Linear matrix inequality Solid oxide fuel cell

a b s t r a c t In this paper, a nonlinear model predictive control method is presented for solid oxide fuel cell systems. For realistic modeling, a sector bounded nonlinear model with input constraint is considered. As a performance index, we consider one horizon cost function that is represented by the weighted sum of state, control input and nonlinear function. By minimizing the upper bound of the cost function, the optimized control input sequences are obtained. For cost monotonicity, a terminal inequality condition is derived in terms of a finite number of linear matrix inequalities (LMIs) by using augmented vector feedback law which consists of state and sector bounded nonlinear function. In order to show the effectiveness of the proposed method, the sector bounded nonlinear model is derived for the solid oxide fuel cell systems and the system performance is verified by applying the proposed MPC algorithm to the systems. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Solid oxide fuel cell (SOFC) has attracted considerable interest as it offers wide application ranges, flexibility in the choice of fuel, high system efficiency and rapid load following capability [1–4]. In this regard, many researches have been investigated for simulating transient behavior as well as controller design for the SOFC dynamic model. In [1,12], a simple mathematical model and detailed model with temperature dynamics for SOFC stack have been developed to establish controller over the fuel cell voltage. However, these mathematical models have a difficulty on the controller design due to its heavily nonlinear behavior and dependence on disturbances such as load current and inlet temperatures. Thus, numerical modeling techniques based on the experimental input–output data such as support vector machine, T–S fuzzy, and neural network have been proposed in [5–7,13,14]. In [23], a model predictive control method for SOFC systems was presented by use of a simplified dynamic model which is obtained by subspace identification method. Recently, the MPC technique has been applied to the SOFC systems due to its slow dynamics and tight operating constraints [8–14,23]. It should be pointed out that model predictive control (MPC) scheme is very useful since it is possible to handle input constraints and to use the current measurement state at optimization algorithms. But if the model is not accurate, the control technique does not guarantee the stability and performance [21]. Thus, the mathematical model based predictive control method is required to overcome the disadvantage of input–output data based MPC method for SOFC systems. In this paper, we consider a sector bounded nonlinear model to handle SOFC systems and propose a one horizon robust MPC method for the systems. The sector bounded nonlinear systems, which have a feedback connection with a linear ⇑ Corresponding author. E-mail addresses: [email protected] (S.M. Lee), [email protected] (O.M. Kwon), [email protected] (J.H. Park). 0096-3003/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2012.03.008

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dynamical system and nonlinearity satisfying certain sector type constraints, have been extensively studied in control theory research area [15–19]. Since the pioneer work of Lur’e was presented in 1940’, the notion of absolute stability has played an important role in stability analysis and controller design problems [20]. But few results have been published concerning controller synthesis for systems with sector nonlinearities, see e.g. [22–25]. Especially, previous design methods did not consider nonlinear feedback for sector nonlinearity. To the best of authors’ knowledge, there are no approaches considering state as well as nonlinear function feedback. The proposed nonlinear MPC method minimizes an upper bound of one horizon cost function subject to the terminal inequality. The solution to this nonlinear MPC, which is solved repeatedly at each sampling times, minimizes an upper bound on the considered finite horizon cost function and the control input stabilizes the systems satisfying the sector condition. In order to handle the sector nonlinearity, we adopt convex representation of sector bounds of the nonlinear function. To guarantee closed-loop stability of the nonlinear MPC, a stabilization criterion is expressed in terms of LMIs which can be solved very efficiently by various convex optimization algorithms [26]. Finally, we demonstrate the effectiveness of the proposed approach via numerical simulations. 2. Problem statement Consider the following discrete-time sector bounded Lur’e systems

xðk þ 1Þ ¼ AxðkÞ þ Ff ðqðxðkÞÞÞ þ BuðkÞ;

ð1Þ

qðkÞ ¼ CxðkÞ with input constraints

 6 uðkÞ 6 u ; u

for all k 2 ½0; 1Þ;

n

m

ð2Þ nn

np

nm

pn

where x 2 R is the state vector, u 2 R is the control input, A 2 R ; F 2 R ; B 2 R ; C2R and f ðqðkÞÞ 2 Rp is memoryless time-invariant nonlinearity with sector restriction such as

ai 6

fi ðqi Þ 6 bi ; qi

are constant matrices

ð3Þ

in which ai ; bi are lower/upper sector bounds, respectively. For simplicity, let us define

di ðkÞ ¼

fi ðqi ðkÞÞ ; qi ðkÞ

ð4Þ

then

f ðqðkÞÞ ¼ DðkÞCxðkÞ; 1

ð5Þ p

where DðkÞ ¼ diagfd ðkÞ; . . . ; d ðkÞg. In order to describe the convexity of the nonlinearity, let us define

D1 , diagfa1 ; . . . ; ap g;

D2 , diagfb1 ; . . . ; bp g

ð6Þ

then function DðkÞ can be represented by:

DðkÞ 2 CofD1 ; D2 g;

ð7Þ

where Co denotes convex hull. Here, the goal of this paper is to design a stabilizing control uðkÞ for (1) by the model predictive control strategy. To find such a control, we consider the following performance index

Jðk; k þ 1Þ , xðkjkÞT QxðkjkÞ þ uðkjkÞT RuðkjkÞ þ Vðk þ 1jkÞ

ð8Þ

where Vðk þ 1jkÞ is a terminal cost and Q > 0; R > 0. In order to design a control law for the future sampling time, we consider one step predictive controller with the following structure at each time k,

8  > < uðkjkÞ ¼ u ðkjkÞ; uðk þ 1jkÞ ¼ K 1 ðkÞxðk þ 1jkÞ þ K 2 ðkÞf ðqðxðk þ 1jkÞÞÞ > : ¼ KðkÞxa ðk þ 1jkÞ;

ð9Þ

  xðkÞ where u ðkjkÞ is the control input variable that minimizes the performance index (8), xa ðkÞ ¼ , and f ðqðkÞÞ KðkÞ ¼ ½ K 1 ðkÞ K 2 ðkÞ . For cost monotonicity, the following inequality condition should be satisfied as

Vðk þ j þ 1jkÞ  Vðk þ jjkÞ < ½xðk þ jjkÞT Qxðk þ jjkÞ þ uðk þ jjkÞT Ruðk þ jjkÞ; for j P 1

ð10Þ

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where

VðkjkÞ ¼ xa ðkjkÞT PðkÞxa ðkjkÞ > 0:

ð11Þ

By summing (10) from j ¼ 1 to j ¼ 1, we obtain:

VðkjkÞ 6 Jðk; k þ 1Þ:

ð12Þ

For simplicity, we define

J  ðkÞ ¼ Minimize uðkÞ;PðkÞ

max Jðk; k þ 1Þ

DðkþjÞ;jP1

subject to ð10Þ:

ð13Þ

Thus, our goal is redefined to find a nonlinear MPC for minimization of (13) that minimizes an upper bound on the worst case of the cost function Jðk; k þ 1Þ. Before deriving our main results, let define the following matrices for the sake of simplicity on matrix representation

 A¼

A F 0



0

;

B¼

  B ; 0

E¼

  0 ; I

CðkÞ ¼ CAF þ BKðkÞ; " # Q þ K 1 ðkÞT RK1 ðkÞ 0 ; WðkÞ ¼ 0 K 2 ðkÞT RK 2 ðkÞ   Q 0 Q¼ ; AF ¼ ½ A F   Q ðkÞ ¼ P1 ðkÞ: 0 0

ð14Þ

3. Main result In this section, we introduce one horizon predictive control method for the sector bounded nonlinear model and derive a stability condition of the MPC with the feedback control law (9). Theorem 1. Consider the system (1) at sampling time k. Then the terminal inequality Eq. (10) with the feedback control law (9) is satisfied for j P 1, if there exists GðkÞ; LðkÞ; MðkÞ and Q ðkÞ > 0 subject to

2

GðkÞ  GðkÞT þ Q ðkÞ

6 6 Di CðAF GðkÞ þ BLðkÞÞ 6 6 AGðkÞ þ BLðkÞ 6 6 1 6 Q 2 GðkÞ 4 1 R2 LðkÞ "

 2l u







2MðkÞ





FMðkÞ

QðkÞ



0

0

I

0

0

0

Ll ðkÞ



3

7  7 7  7 7 < 0; 7  7 5 I

i ¼ 1; 2;

ð15Þ

#

LTl ðkÞ GðkÞ þ GT ðkÞ  Q ðkÞ

P 0;

l ¼ 1; . . . ; m;

ð16Þ

 l are l-th row of LðkÞ; u  , respectively. where LðkÞ ¼ KðkÞGðkÞ and Ll ðkÞ; u Proof. For j P 1, the inequality (10) with control input uðk þ jjkÞ ¼ KðkÞxa ðk þ jjkÞ is satisfied if and only if

xa ðk þ j þ 1jkÞT PðkÞxa ðk þ j þ 1jkÞ  xa ðk þ jjkÞT PðkÞxa ðk þ jjkÞ h i <  xðk þ jjkÞT Qxðk þ jjkÞ þ xa ðk þ jjkÞT KðkÞT RKðkÞxa ðk þ jjkÞ :

ð17Þ

Let us define

xf ðk þ jjkÞ ¼



xa ðk þ jjkÞ f ðqðk þ j þ 1jkÞÞ

 :

ð18Þ

Then, the inequality (17) can be written " xTf ðk þ jjkÞ

# AT þ KðkÞT BT PðkÞ½ A þ BKðkÞ E xf ðk þ jjkÞ  xTa ðk þ jjkÞPðkÞxa ðk þ jjkÞ < xTa ðk þ jjkÞðQ þ KðkÞT RKðkÞÞxa ðk þ jjkÞ ET

where A, B, and E are defined in Eq. (14).

ð19Þ

S.M. Lee et al. / Applied Mathematics and Computation 218 (2012) 9296–9304

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The following equality holds for all nonzero matrix NðkÞ

f T ðqðk þ j þ 1jkÞÞNðkÞf ðqðk þ j þ 1jkÞÞ ¼ f T ðqðk þ j þ 1jkÞÞNðkÞDðkÞðCAF xa ðk þ jjkÞ þ CKðkÞxa ðk þ jjkÞÞ:

ð20Þ

Combining the inequality (19) with equality constraints (20) by use of the augmented vector given in Eq. (18), the following inequality is obtained

xTf ðkÞPxf ðkÞ < 0; where

"

P¼

ð21Þ #

ðA þ BKðkÞÞT PðkÞðA þ BKðkÞÞ  PðkÞ þ WðkÞ



ET PðkÞðA þ BKðkÞÞ þ NðkÞDðkÞCðkÞ

ET PðkÞE  2NðkÞ

ð22Þ

:

By Schur complement [26], inequality (21) is satisfied if and only if

2

PðkÞ þ WðkÞ

6 4 NðkÞDðkÞCðkÞ A þ BKðkÞ





2NðkÞ



E

QðkÞ

3 7 5 < 0:

ð23Þ

By use of an auxiliary full block matrix GðkÞ and the fact that the matrix ðGðkÞT  Q ðkÞÞQ 1 ðkÞðGðkÞ  Q ðkÞÞ is nonnegative definite, we can get the following inequality

GðkÞT Q 1 ðkÞGðkÞ  GðkÞ  GðkÞT þ QðkÞ > 0:

ð24Þ

T

By multiplying the matrix diagfGðkÞ ; I; Ig and diagfGðkÞ; I; Ig in the left and right side of Eq. (23), respectively, the following inequality is obtained with Eq. (24),

2 6 4

3

GðkÞ  GðkÞT þ Q ðkÞ þ GðkÞT WðkÞGðkÞ





NðkÞDCðkÞGðkÞ

2NðkÞ



AGðkÞ þ BLðkÞ

E

Q ðkÞ

7 5 < 0:

ð25Þ

After performing the congruence transformation with

diagfI; MðkÞ; I; Ig;

MðkÞ ¼ NðkÞ1

ð26Þ

in the LMI (25), the equivalent condition (15) is obtained by Schur complement. For the invariant set

(  ) T   xðk þ jjkÞ xðk þ jjkÞ 1 < 1; j P 1 ; nðPðkÞÞ ¼ fxjxa ðk þ jjkÞ PðkÞxa ðk þ jjkÞ < 1; j P 1g ¼ xj Q ðkÞ f ðqðk þ jjkÞÞ f ðqðk þ jjkÞÞ T

ð27Þ

 ; j P 1; i ¼ 1; . . . ; m, hold that the input constraint jui ðk þ jjkÞj ¼ jK i ðkÞxa ðk þ jjkÞj < u

maxjui ðk þ jjkÞj2 ¼ maxjK i ðkÞxa ðk þ jjkÞj2 ¼ maxjðLðkÞGðkÞ1 xa ðk þ jjkÞÞi j2 jP1

jP1

jP1

   2   xðk þ jjkÞ  6 kðLðkÞG1 ðkÞQ 1 ðkÞGðkÞT LðkÞÞ k2 : ¼ max ½ L1 ðkÞ L2 ðkÞ GðkÞ1 i 2  f ðqðk þ jjkÞÞ jP1

ð28Þ

i

 i ; j P 1; i ¼ 1; 2; . . . ; m. Thus Eq. (28) is equivalent to the Eq. (16) by Schur complement, then it is satisfied that jui ðk þ jjkÞj 6 u This completes the proof. h For j ¼ 0, minimization of Jðk; k þ 1Þ is equivalent to

Minimize cðkÞ

ð29Þ

uðkÞ;PðkÞ

subject to

xðkjkÞT QxðkjkÞ þ uðkjkÞT RuðkjkÞ þ



xðk þ 1jkÞ f ðqðk þ 1jkÞÞÞ

T

PðkÞ



xðk þ 1jkÞ f ðqðk þ 1jkÞÞÞ



6 cðkÞ:

ð30Þ

By Schur complement, the optimization problem above is equivalent to

Minimize cðkÞ

ð31Þ

uðkÞ;Q ðkÞÞ

subject to

2

cðkÞ

6 RðkÞ 6 6 1=2 4 Q xðkÞ R1=2 uðkÞ

3







Q ðkÞ



0

I

7 7 7P0 5

0

0

I

ð32Þ

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where



AxðkjkÞ þ BuðkjkÞ þ Ff ðqðkÞÞ

RðkÞ ¼

Di CðAxðkjkÞ þ BuðkjkÞ þ Ff ðqðkÞÞÞ

 ;

i ¼ 1; 2:

Remark 1. Based on Theorem 1, the proposed nonlinear MPC method can be summarized in the following MPC algorithm.  Step 0 : Set k ¼ 0.  Step 1 : Solve the following optimization problem using the measurement xðkjkÞ and compute uðkÞ.

Minimize

uðkÞ;QðkÞ;GðkÞ;LðkÞ;MðkÞ

cðkÞ

ð33Þ

subject to (15), (16) and (32).  Step 2 : Apply uðkÞ to the plant.  Step 3 : k ¼ k þ 1 and go to Step 1. Remark 2. If there exist a feasible solution of the problem in Theorem 1 at time k ¼ 0, then the system (1) with the MPC is robustly asymptotically stable. Since J  ðkÞ is greater than or equal to zero and strictly decreases as time goes to infinity, it plays a role of a Lyapunov function. Therefore we conclude that the closed-loop system is asymptotically stable. Also, the optimal solution of the optimization problem at the time k can be the solution at the time k þ 1, the feasible solution of the optimization problem at time k is also feasible at the next time k þ 1. Remark 3. When we assume that the nonlinear function is not available in Theorem 1, the control gain K 2 ðkÞ becomes zero. Then, to find a feasible solution using the convex optimization technique, the auxiliary matrix GðkÞ should have non-full   XðkÞ 0 . Because of the proposed augmented Lyapunov function and nonlinear function block structure, like GðkÞ ¼ YðkÞ ZðkÞ feedback control law, the proposed method has a less conservative condition for cost monotonicity. Via comparison with the method without the nonlinear feedback, the effectiveness of the proposed method will be shown in a numerical example. 4. Application to SOFC system In order to show the effectiveness of the proposed method, let us consider the dynamics of an SOFC system [8] and assume that all states are available. As a widely adopted dynamic model of the SOFC system, the average voltage magnitude of the fuel cell stack is determined by the partial pressure of hydrogen, oxide and water. Table 1 contains the parameters of the SOFC model. Applying Nernst’s equation, the output of the SOFC can be modeled as follows:

" V 0 ¼ N0

R0 T 0 pH2 ðpO2 =101:325Þ E0 þ ln 2F 0 pH2 O

1=2

# ð34Þ

;

where pH2 ; pO2 ; pH2 O are partial pressures of hydrogen, oxygen and water. Also, the dynamics of fuel processor and the stack voltage are

Table 1 Parameters in the SOFC system [8]. Parameter

Value

Unit

Representation

T0 F0

1273 96; 485

K

Absolute temperature Faraday’s constant

R0

8:314

E0 N0 Kr

0:9378 384

Cmol

1

J mol V –

1

K 1

Universal gas constant

mols1 A1

Ideal standard potential Number of cells n series in the stack Constant, K r ¼ N 0 =4F 0

K H2

3

8:32  10

mols1 KPa1

Valve molar constant for hydrogen

K H2 O

2:77  103

mols1 KPa1

Valve molar constant for water

K O2

2:49  102 26:1 2:91 1.145 5 0.126 10

mols1 KPa1 s s – s

sH2 sO2 sHO sf r C

0:995  103

X F

Valve molar constant for oxygen Response time of hydrogen flow Response time of Oxygen flow Ratio of hydrogen to oxygen Time constant of the fuel processor Ohmic loss Parallel Capacitance

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q_ H ¼

1

sH

ðqF  qH Þ

ð35Þ

1 V_ s ¼ ðV 0  V s  rID Þ rC

ð36Þ

where qH is the hydrogen flow rate, qF is inlet fuel flow rate, V s is stack voltage and ID is external current load that is assumed to be not changed. The dynamic equation of the partial pressure inside the channel of hydrogen, oxygen and water are as follows:

p_ H2 ¼ p_ O2 ¼

1

sH2 K H2 1

ðqH  2K r Is  pH2 K H2 Þ;   qH  K r Is  pO2 K O2 ;

ð37Þ

sO2 K O2 sHO

p_ H2 O ¼

1

sH 2 O K H 2 O

ð2K r Is  pH2 O K H2 O Þ;

where Is ¼ ðV o  V s Þ=r is stack current. In order to transform states around its equilibrium point, let us define

x1 ¼ pH2 K H2  pH2 K H2 ; x2 ¼ pO2 K O2  pO2 K O2 ; x3 ¼ qH  qH

ð38Þ

x4 ¼ V s  V s u ¼ qF  qF ;

ID ¼ ID ;

pH2 O ¼ pH2 O

where pH2 ; pO2 ; qH ; V s ; qF ; ID ; pH2 O are operating points of each states. It should be noted that the pressure of water pH2 O has constant value because external current load ID is assumed to be not changed in this paper. Then, the variation of the stack current is expressed as follows

! !! " #  1 N 0 R0 T 0 pH pO 1 1  ðV s  V s Þ V o  V o  ðV s  V s Þ ¼ ln  2 þ ln  2 r r 2 2F 0 pH2 pO2 " ! !! #   x1 =K H2 þ pH2 x2 =K O2 þ pO2 1 N0 R0 T 0 1 ln ¼ þ ln  x4 : r 2 2F 0 pH2 pO2

Is  Is ¼

ð39Þ

Therefore, the transformed state space model is described as

_ xðtÞ ¼ Ac xðtÞ þ F c gðxðtÞÞ þ Bc uðtÞ;

ð40Þ

where xðtÞ ¼ ½x1 x2 x3 x4 T 2 Rn is the states, uðtÞ is the fuel flow rate,

2

 1 6 sH 2 6 6 0 Ac ¼ 6 6 6 0 4 0

2K r r sH

1

0

sH2

1

 sO

7 Kr 7 r sO 7 2 7; 7 0 7 5  rC1

1

sHO sO2

2

3

2

0

 s1

0

0

f

2 3 0 6 7 607 7 Bc ¼ 6 6 1 7; 4 sf 5 0

2

r  r2K sH

3

2 7 6 6  Kr 7 h 6 r sO 7 Fc ¼ 6 gðxðtÞÞ ¼ N02FR00T 0 2 7 7 6 4 0 5

1 rC

2



x1 þpH K H

2

3

7 i6 ln pH K H2 2 7 N 0 R0 T 0 6  7: 6  4F 0  x2 þpO K O 5 4 2 2 ln p K O 2

O2

ð41Þ

2

1

Using the state transformation around the nominal operation point with qF ¼ 0:7023mols ; ID ¼ 300A; V s ¼ 240:7075V; pH2 ¼ 12:6623; pO2 ¼ 12:6466 and Euler’s first order approximation with a sampling time 1 sec for the derivative, we obtain a discrete-time sector bounded system (1) with the following matrices.

2

0:9617

6 6 A¼6 4

0



0 0

0

1 0 0

0 0

1 0 0

 ;

0

0:2063  3 x1 ðkÞþpH K H 2 2 ln  7 6 pH K H 2 2 7 6  7: f ðxðkÞÞ ¼ 6  x2 ðkÞþpO K O 5 4 2 2 ln p K O 2

O2



DðkÞ 2

3 0 6 0 7 7 6 B¼6 7; 4 0:2000 5 2

3

0:6564 0:3001 0:0027 7 7 7; 0 0:8000 0 5

0

C¼

0:0383 0:0006

6:67 0

  

9:07 0 ; : 2:83 0 3:52 0

0

3 0:0128 0:0064 6 0:0572 0:0286 7 7 6 F¼6 7; 5 4 0 0 2

16:7151

8:3576

ð42Þ

2

ð43Þ

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State x4 16 With nonlinear feedback Without nonlinear feedback

14 12 10 8 6 4 2 0 −2

0

20

40

60

80

100

120

140

160

180

200

iteration Fig. 1. Simulation result (stack voltage x4 ðkÞ).

Control input U

0.04

With nonlinear feedback Without nonlinear feedback

0.02 0 −0.02 −0.04 −0.06 −0.08 −0.1 0

20

40

60

80

100

120

140

160

180

200

iteration Fig. 2. Simulation result (control input uðkÞ).

Stack voltage (V) 270

260

250

240

230

220

210

0

100

200

300

400

500

iteration Fig. 3. Set point tracking of stack voltage (V).

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9303

Hydrogen flow rate (mol/s) 0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0

100

200

300

400

500

600

iteration Fig. 4. Hydrogen flow rate (mol/s) for the reference tracking.

It should be noted that lower and upper sector bounds of the nonlinear function vector f ðxðkÞÞ are obtained as ½6:67 2:83T and ½9:07 3:52T in the range of x1 2 ½0 0:1. The initial condition is x0 ¼ ½0:0867 0:0777 0:0898 14:9727T , the weighting  ¼ 0:1. Figs. 1 and 2 show that the comparison of the proposed methmatrices are Q ¼ 2I; R ¼ I and the input constraint is u od with the case of only state feedback control law K 2 ðkÞ ¼ 0. It can be seen that the proposed method provides a fully utilized fuel flow rate within input constraint and better performance than those without considering the sector nonlinearities. In addition, Figs. 3 and 4 show the stack voltage and fuel flow rate for set-point tracking at three operating points with x ¼ f½0:1054; 0:3149; 0:7023; 240:7075; ½0:1538; 0:3573; 0:7508; 250:0042; ½0:0671; 0:2815; 0:6640; 230:0098g, respectively. This results show that the proposed method can be well applied to the set-point tracking in the presence of input constraint. 5. Conclusions In this paper, we proposed a nonlinear MPC algorithm for SOFC systems using a sector bounded nonlinear model. For realistic modeling of SOFC systems, we adopted a mathematical sector bounded model which was represented as the convex combination of each vertices. For the performance, one horizon MPC method was considered with an augmented terminal cost which consists of state and sector bounded nonlinear function. Due to the augmented vector feedback, a less conservative cost monotonicity condition was obtained in the terminal inequality. The proposed one step predictive controller design conditions were expressed in the form of a finite number of LMIs. Through numerical simulations, we showed that the proposed method had better performance than those without considering the sector nonlinearities and a good tracking performance. Acknowledgements This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0011460). Also, the work of J.H. Park was supported by 2010 Yeungnam University Research Grant. References [1] J. Padullés, G.W. Ault, J.R. Mcdonald, An integrated SOFC plant dynamic model for power systems simulation, Journal of Power Sources 86 (2000) 495–500. [2] F. Mueller, F. Jabbari, R. Gaynor, J. Brouwer, Novel solid oxide fuel cell system controller for rapid load following, Journal of Power Sources 172 (2007) 208–323. [3] X. Wang, B. Huang, T. Chen, Data-driven predictive control for solid oxide fuel cells, Journal of Process Control 17 (2007) 103–114. [4] A.M. Murshed, B. Huang, K. Nandakumar, Control relevant modelig of planer solid oxide fule cell system,, Journal of Power Sources 163 (2007) 830–845. [5] X.J. Wu, X.J. Zhu, G.Y. Cao, H.Y. Tu, Modeling a SOFC stack based on GA-RBF neural networks identification, Journal of Power Sources 161 (2007) 145–150. [6] X.J. Wu, X.J. Zhu, G.Y. Cao, H.Y. Tu, Nonlinear modeling of a SOFC stack based on ANFIS identification, Simulation Modelling Practice and Theory 16 (2008) 399–409.

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