Sensor Reduction in Diesel Engine Two-Cell Selective ... - IEEE Xplore

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IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 20, NO. 5, OCTOBER 2015

Sensor Reduction in Diesel Engine Two-Cell Selective Catalytic Reduction (SCR) Systems for Automotive Applications Hui Zhang, Junmin Wang, Senior Member, IEEE, and Yue-Yun Wang, Senior Member, IEEE

Abstract—In this paper, we have studied the sensor reduction problem for two-cell selective catalytic reduction (SCR) systems with applications to Diesel-engine-powered automotive systems. For the purpose of constructing feedback control loops, three NOx sensors and three ammonia sensors are necessary for the ideal case. However, a large set of physical sensors will not only increase the system integration cost, but also the burden of fault diagnosis. Therefore, we aim to reduce the number of physical sensors for the system. The proposed strategy consists of three physical sensors and two observers. The three physical sensors include two NOx sensors and one ammonia sensor among which one NOx sensor is placed at the tailpipe and the other two are located between the two SCR cells. For the first SCR catalyst cell, the available measurements are the NOx and ammonia concentrations. The function of the observer for the first cell is to estimate the engine-out NOx concentration, the ammonia injection input, and ammonia coverage ratio. Two observers are designed for the first cell to estimate the inputs and states simultaneously. One observer is used to estimate inlet ammonia concentration and the other one is based on the assumption that the urea solution is completely converted into gaseous ammonia before the cells. Both observers show good performances on the input and state simultaneous estimation in terms of experimental results. For the second SCR catalyst cell, the inputs are available. The output is the reading of the NOx sensor at the tailpipe. A Luenberger observer is designed for the second cell to estimate the ammonia coverage ratio and the NOx concentration. The designed observers are validated with experimental data. It infers from the results that the number of physical sensors has been successfully reduced. All the states and inputs of a two-cell SCR system can be made available by the three physical sensors and designed observers. Index Terms—Diesel engine, input/state observer design, selective catalytic reduction (SCR) system, sensor reduction.

NH∗3 θ Θ Cx Rx Kx

NOMENCLATURE Adsorbed ammonia in the catalyst cell. Ammonia surface coverage ratio. Ammonia storage capacity of SCR substrate(mole). Mole concentration of species x (mole/m3 ). Reaction rate. Preexponential factor.

Manuscript received June 17, 2014; revised September 10, 2014; accepted November 10, 2014. Date of publication December 4, 2014; date of current version August 24, 2015. Recommended by Technical Editor X. Chen. H. Zhang and J. Wang are with the Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]; [email protected]). Y.-Y. Wang is with the Propulsion Systems Research Laboratory, GM Global Research and Development, Warren, MI 48090 USA (e-mail: yue-yun. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2014.2370043

Ex T F V R

Activation energy. Temperature (K). Exhaust gas volume flow rate (m3 /s). SCR catalyst volume (m3 ). Ideal gas constant. I. INTRODUCTION

ESEARCH efforts on automotive powertrain systems have been consistently increasing to address the energy and environmental concerns; see [1]–[5] and the references therein. The emission issue of Diesel engines is one of the most challenging areas. It is well known that a Diesel engine has higher fuel efficiency than the gasoline counterpart. However, the Diesel engine combustion temperature is also higher and it leads to high engine-out NOx emissions. As the NOx emission regulations are becoming more and more stringent [6], much more efforts should be paid to deal with the Diesel engine emission issues. Generally, there are two main pathways to reduce the NOx emissions: combustion control and aftertreatment systems. It has been shown that only the combustion control such as new combustion mode homogeneous charge compression ignition [7], [8] is inadequate for meeting the stringent emission regulations. Therefore, aftertreatment systems are necessary for the NOx reduction. The commercial NOx -reduction techniques include the lean NOx traps [4] and selective catalytic reduction (SCR) systems [9]–[13]. Due to the fact that the lean NOx trap regenerations would deteriorate the vehicle fuel economy, the urea-based SCR technique has been widely regarded as the most promising and necessary one for the future Diesel enginepowered vehicles [14]–[20]. Roughly speaking, the urea-based SCR in automotive applications reduces the NOx emissions by the following steps [6], [9]: 1) inject 32.5% urea solution into the exhaust pipe; 2) the urea converts into gaseous ammonia in the environment with a high exhaust gas temperature; 3) the gaseous ammonia flows into the SCR catalyst cell and is adsorbed by the catalyst substrate; and 4) the NOx are catalytically deoxidized by the adsorbed ammonia into environment-friendly nitrogen (N2 ) and water (H2 O). Since the NOx reduction is a catalytic chemical reaction, the catalyst is critical for the SCR system operation. Unsurprisingly, research on the catalysts for NOx reduction by using ammonia has received substantial efforts [21]. The Fe–Mn based catalysts for low-temperature SCR was reported in [22]. A dual-bed of Pt/beta zeolite and RhOx/ceria monolith catalysts was used in the Diesel engine SCR systems by the authors in [23]. In addition, the urea dosing control is also critical for

R

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ZHANG et al.: SENSOR REDUCTION IN DIESEL ENGINE TWO-CELL SELECTIVE CATALYTIC REDUCTION (SCR) SYSTEMS

the NOx reduction performance of SCR systems. If the urea is overdosed, the ammonia would slip into the tailpipe, which is undesired and should be constrained. The ammonia slip and NOx reduction efficiency conflict with each other. In order to tradeoff between them, a model-based closed-loop control strategy is required; see [24]–[28] and the references therein. For the model-based closed-loop control, the two-cell SCR system intends to achieve better NOx reduction performance than the single-cell SCR system with the same total SCR catalyst volume [29], [30]. The main reasons arise from: 1) the two-cell SCR system has a better model approximation accuracy; and 2) the NOx and ammonia concentrations between the cells can be employed in the feedback control loop. However, more feedback signals means that there should be more physical sensors to measure the information, and consequently increase the system cost and maintenance expense. In order to enjoy the benefits of the two-cell SCR system with less physical sensors, we aim to develop observers to estimate the system input and/or the states simultaneously. Simultaneous estimation of unknown inputs and states is a typical control problem and has a great application to the fault diagnosis and detection; see [33]–[36] and the references therein. Recently, the proportional-integral (PI) observer design has attracted attention. The main idea is to augment the original system with the dynamics of the disturbance [37]–[40]. If the first derivative of the disturbance is nonzero, multiple integrators are employed. Then, an observer is designed for the augmented system. Since the designed observer has the dynamics of the estimated disturbance, it is called PI/MPI observer. We can see from the existing work that the PI observer has good performance on estimating the disturbance for linear systems, Takagi–Sugeno fuzzy systems, and singular systems. In this paper, we propose a scenario to obtain the states and inputs of the two-cell SCR systems with only three physical sensors. One NOx sensor and one ammonia sensor are placed between the SCR cells such that the inputs of the second cells are available and the outputs of the first cell consist of the NOx and ammonia concentrations. The other physical sensor is the NOx sensor located at the tailpipe. For the first cell, the unknown information contains the engine-out NOx concentration, the ammonia input, and the ammonia coverage ratio, that is, the estimation problem for the first SCR catalytic cell is an unknown input and state simultaneous estimation problem. Two observers are designed for the first SCR catalytic cell. The first observer is used to estimate the ammonia input concentration and the ammonia coverage ratio. The second observer is based on the availability of the inlet ammonia concentration to estimate the engine-out NOx and the ammonia coverage ratio. The design of the second observer is established under the framework of linear parameter varying (LPV) systems. For the second SCR cell, the main work is to estimate the SCR system state with the NOx sensor reading. The three observers are evaluated with the experimental dataset based on a medium-duty Diesel engine. It can be concluded from the comparisons that the proposed observers achieve the prescribed design objectives. The rest of this paper is organized as follows: in Section II, the SCR system operation principle is introduced, the system

Fig. 1.

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Schematic diagram of a urea-based SCR aftertreatment system.

is modeled, and the observer design problem is formulated; Section III provides the main results including observer design and the performance evaluation; Section IV concludes this paper. II. SYSTEM AND PROBLEM PRELIMINARY A. SCR System Operational Principle and Introduction It can be seen from Fig. 1 that the main principle of the SCR system in Diesel engines is to catalytically convert the Diesel-engine-out NOx to N2 and H2 O by using the injected urea. Theoretically speaking, if the urea dosing matches with the amount of engine-out NOx well, the NOx can be completely removed and the tailpipe emissions would only consist of environment-friendly gas species. The NOx conversion to N2 and H2 O in an SCR aftertreatment system is a complicated process, which can be roughly described as follows [6]. The liquid urea is first evaporated into gaseous ammonia. Then, the gaseous ammonia is adsorbed on the SCR substrate and catalytically reacts with the NOx . Specifically, the main chemical reactions started from the liquid urea to the final tailpipe emissions are composed of [31]: Urea evaporation: NH2 − CO − NH2 (liquid) → NH2 − CO − NH∗2 + nH2 O. (1) Urea decomposition: NH2 − CO − NH∗2 → NH3 + HNCO.

(2)

Isocyanic acid hydrolyzation: HNCO + H2 O → NH3 + CO2 .

(3)

Bidirectional ammonia absorption and desorption: NH3 + θfree NH∗3 .

(4)

4NH∗3 + 4NO + O2 → 4N2 + 6H2 O

(5)

2NH∗3

(6)

NOx reduction: + NO + NO2 → 2N2 + 3H2 O

4NH∗3 + 3NO2 → 3.5N2 + 6H2 O.

(7)

NH3 oxidation: 4NH∗3 + 3O2 → 2N2 + 6H2 O.

(8)

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In the previous reactions, we use θfree , NH∗3 , NH2 − CO − NH∗2 , and n to denote the free substrate site of the SCR catalyst cell, the adsorbed ammonia in the cell, the solid phase of the urea, and the mole number of H2 O, respectively. It is noted that the reactions in (5), (6), and (7) are well known as the “standard SCR,” the “fast SCR,” and the “slow SCR” in terms of reaction rate [32], respectively. Moreover, the ammonia oxidation only occurs when the temperature is high such as more than 450 °C. Also, the NO species dominate in the NOx emissions such that the fast and slow SCRs are not the main chemical reactions in the reduction. B. Problem Formulation and Research Objective Obviously, the NOx can be reduced in the SCR system if the urea is injected, that is, the urea dosing is critical for the NOx reduction efficiency. If the urea is underdosed, the NOx reduction efficiency cannot reach the maximal level. On the other hand, if the urea is overdosed, the extra gaseous ammonia may slip into the environment. Since the ammonia slip is undesired and needs to be constrained, the urea dosing control is one of the main challenges in the SCR system design and integration. In order to achieve a high NOx conversion efficiency and constrain the tailpipe ammonia slip, the open-loop control is insufficient and a model-based feedback control strategy is required. In order to capture the dynamics conveniently, the SCR modeling is based on two assumptions: the SCR catalyst cell is a continuous stirred tank reactor and the distribution of the SCR states is homogeneous. As pointed that the NO species dominate in the NOx emissions, the NO and NO2 are lumped together as NOx for simplicity. The kinetics of chemical reactions (4), (5), and (8) are considered. The reaction rates are represented by EF RF = KF exp − (9) CNH 3 (1 − θ) RT ER RR = KR exp − (10) θΘδ2 RT E2 (11) R2 = K2 exp − CNO x CO 2 θΘV 2 RT E3 (12) R3 = K3 exp − CO 2 θΘV δ1 . RT A three-state SCR model of an SCR catalyst cell with a volume of V is described by [24]: ⎤ ⎡ ⎡ ⎤ C˙ NO x f1 (CNO x , CNO x ,in , θ) ⎥ ⎢ ⎢ ⎢ θ˙ ⎥ = ⎣ f2 (CNH 3 , CNO x , θ) ⎥ (13) ⎦ ⎦ ⎣ f3 (CNH 3 , CNH 3 ,in , θ) C˙ NH 3

where F F CNO x ,in − CNO x V V − r2 CNO x CO 2 θΘV

f1 (CNO x , CNO x ,in , θ) =

f2 (CNH 3 , CNO x , θ) = rF CNH 3 (1 − θ)V − rR θδ1 − r2 CNO x CO 2 θV 2 −r3 CO 2 θV δ2

Fig. 2.

Original NOx sensors and ammonia sensors placement.

f3 (CNH 3 , CNH 3 ,in , θ) =

F F CNH 3 ,in − CNH 3 V V

rR θΘδ1 − rF CNH 3 (1 − θ)Θ + V Ei ri = Ki exp − , i = F, R, 2, 3 RT P1 P3 δ1 = exp − P , Θ = S1 exp (−S2 T) , δ2 = exp − P . θ 2 θ 4 It is necessary to mention that the first state of the system is the NOx concentration, the second state θ is called ammonia coverage ratio which is defined as the ratio of (adsorbed ammonia)/(maximal ammonia storage capacity Θ), and the third state is the NH3 concentration. In addition, F denotes the gas flow rate, Ki , Ei , P1 , P2 , P3 , and P4 are constant scalars identified with experimental data. During the modeling of the SCR system, it is assumed that the states are homogenous inside the catalyst. This assumption holds if the SCR catalyst cell is small. However, for medium-duty and heavy-duty Diesel engines, the SCR catalyst cell is relatively large. In these cases, the homogenous assumption would lead to a large modeling error and the corresponding model-based feedback control may not achieve the desired performance due to the modeling error. Therefore, recently, two-cell SCR systems have attracted increasing attention [29], [41], [51]. The main idea of the two-cell SCR system is to slice a SCR system into two parts. Compared to the one-cell SCR system with the same total volume, the main advantages include: 1) the model can be more accurate since the homogenous assumption becomes two piecewise-homogenous ones; and 2) not only the tailpipe information (NOx and ammonia concentration) but also the ones between two cells can be employed to construct the feedbackcontrol loop. In theory, if the total volume of the SCR cells is the same, the two-cell SCR system can achieve better performance on the NOx reduction. However, in order to obtain the species concentration information, more sensors are necessary in the two-cell SCR systems. Ideally, as shown in Fig. 2, three NOx sensors and three ammonia sensors are required to obtain the NOx and ammonia concentrations before the cells, between the cells, and after the cells, respectively. Note that one NOx sensor is placed before the urea injector. The main reason is that the NOx sensor is cross sensitive to ammonia. If the NOx sensor is placed before the urea injector, the sensor reading will not be contaminated by the ammonia concentration and we can trust on the reading of engine-out NOx concentration. Though more sensors involved in the system integration would help for the feedback control and system monitoring, the system cost will also increase. What is more,


Fig. 3.

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Proposed NOx sensors and ammonia sensor placement.

the system maintenance cost and sensor diagnosis will also be burdened. As the NOx and ammonia concentrations before, between, and after the cells are necessary for the system feedback control but six physical sensors are expensive for commercial production vehicles, the objective of this work is to develop soft sensors (observers) to estimate the gas concentrations, system inputs, and ammonia coverage ratios with fewer (three) physical sensors. III. MAIN RESULTS A. Proposed Sensor Arrangement The objective of this work is to reduce the number of physical sensors but keep the availability of the gaseous concentration before, between, and after the SCR cells. The proposed sensor arrangement can be seen in Fig. 3 in which there are two NOx sensors (located between and after the cells) and one ammonia sensor between the two cells. Our research is to propose estimator design methods to estimate unavailable NOx concentration, ammonia concentration, and ammonia coverage ratios. As pointed in [42], the NOx sensor is cross sensitive to ammonia. Therefore, the NOx sensor readings are inaccurate. The reading of a NOx sensor is a summation of the actual NOx and NH3 concentrations as follows: CNO x ,m = CNO x + K(T )CNH 3

(14)

where CNO x ,m stands for the measurement of NOx sensor, CNO x is the actual NOx concentration, CNH 3 denotes the ammonia concentration, and K(T ) is a cross-sensitivity factor depending on the gas temperature. Since the readings of the ammonia sensor are accurate and the cross-sensitivity factor K(T ) can be calibrated, the actual NOx concentration between the cells can be obtained via the expression (15). Therefore, the inputs of the second SCR cell are available and the outputs of the first SCR cell are NOx and ammonia concentrations. With the reading correction method, we assume that both the NOx and ammonia concentrations between the cells are accurate. For the NOx sensor after the cells, the reading is a combination of the tailpipe NOx and tailpipe ammonia slip CNO x = CNO x ,m − K(T )CNH 3 .

(15)

Fig. 4.

Inlet ammonia concentration estimation scheme for the first SCR cell.

temperature is low. The objective of this part is to design an observer to estimate the ammonia coverage ratio and the ammonia input of the first SCR cell with the NOx and ammonia measurements between the two cells. The scheme of the estimation is illustrated in Fig. 4. Suppose that the NOx concentration of the first cell is CNOx,1 , the ammonia concentration of the first cell is CNH 3 ,1 , the ammonia coverage ratio is denoted as θ1 , the inlet NOx concentration of the first cell is CNOx,in , and the measurement of NOx sensor between the cells is CNOx,1,m . The nonlinear model of the first SCR cell is represented by ⎡ ⎤ ⎡ ⎤ C˙ NO x ,1 f1 (CNO x ,1 , CNO x ,in , θ1 ) ⎢ ⎥ ⎥ ⎢ θ˙1 ⎥ = ⎢ (16) ⎣ ⎦ ⎣ f2 (CNH 3 ,1 CNO x ,1 θ1 ) ⎦ f3 (CNH 3 ,1 CNH 3 ,in , θ1 ) C˙ NH 3 ,1 After the reading correction, the available measurements of the first SCR cell are CNOx,1 and CNH3,1 . Before the observer design, we introduce the following lemma. Lemma 1 [43]: Consider the following nonlinear system

x˙ 1 = g(y, u) + Gd y = x1

(17)

where x1 is the available state vector, u is the known system input, d is the unknown external input, y is the measurement, G is a known matrix, and g(y, u) is a nonlinear mapping. If the matrix Ghas a full-rank and the second derivative of d is bounded, the input estimation error of the following observer ε˙1 = g(y, u) + Gdˆ ˙ dˆ = −K0 (ε1 − x1 ) − K1

t

(ε1 − x1 )dτ

(18)

0

B. Unknown Input and State Observer of the First Cell In this section, we study the unknown input and state observer design for the first SCR cell. The unknown input is the inlet ammonia concentration of the first cell. Though it is generally assumed that the urea–ammonia conversion is 100% completed before the SCR cell, the assumption may not hold when the gas

will asymptotically approach to zero if K0 > 0 and K1 → ∞. Note that the previous observer requires all the states available. However, for the SCR system, the ammonia coverage ratio cannot be measured by a physical sensor and it is obviously unavailable. Recalling the dynamics of the ammonia coverage ratio:

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θ˙1 = rF 1 CNH 3 ,1 (1 − θ1 ) V1 − rR 1 θ1 δ1,1 − r2,1 CNO x ,1 CO 2 θ1 V12 − r3 CO 2 θ1 V1 δ2,1 −rF 1 CNH 3 ,1 V1 − rR 1 δ1,1 = θ1 +rF 1 CNH 3 ,1 V1 −r2,1 CNO x , 1 CO 2 V12 − r3,1 CO 2 V1 δ2,1 (19) which is linear if CNH 3 ,1 and CNOx,1 are available. What is more, the eigenvalue of the linear system is negative since the terms rF 1 CNH 3 ,1 V , rR 1 θ1 δ1,1 , r2,1 CNOx,1 CO 2 V12 , and r3,1 CO 2 V1 δ2,1 are all positive. Then, an observer for the ammonia coverage ratio can be selected as follows: ˙ θˆ1 = −rF 1 CNH 3 ,1 V1 − rR 1 δ1,1

−r2,1 CNO x , 1 CO 2 V12 − r3,1 CO 2 V1 δ2,1

θˆ1 + rF 1 CNH 3 ,1 V1 .

(20) Defining the ammonia coverage ratio estimation error as eθ 1 = θ1 − θˆ1 , it is easy to prove that the estimation error will asymptotically converge to zero. According to Lemma 1, the input observer is selected as follows: ⎤ ⎤ ⎡ ⎡ ˙ h1 CNH 3 ,1 , CˆNH 3 ,in , θˆ1 Cˆ NH 3 ,1 ⎦=⎣ ⎣ ⎦ (21) ˙ h2 CNH 3 ,1 , CˆNH 3 ,1 Cˆ NH 3 ,in

Fig. 5.

Picture of medium-duty Diesel engine in the test.

Fig. 6.

Schematic diagram of the measurements in the test.

where

F ˆ F CNH 3 ,in − CˆNH 3 ,1 h1 CNH 3 ,1 , CˆNH 3 ,in , θˆ1 = V1 V1 rR − rF 1 CˆNH 3 ,1 (1 − θˆ1 )Θ1 + 1 θˆ1 Θ1 δ1,1 V1

+ λ1 sign CNH 3 ,1 − CˆNH 3 ,1 ,

h2 CNH 3 ,1 , CˆNH 3 ,1 = −K0 CˆNH 3 ,1 − CNH 3 ,1 − K1

t

CˆNH 3 ,1 − CNH 3 ,1 dτ,

0

K0 > 0, λ1 > |ΔM |, K1 → ∞, and |ΔM | is the maximal modeling error in the ammonia concentration dynamics. In order to evaluate the performance of designed input-state observer, an experimental test is carried out with a medium-duty Diesel engine. The exhaust line of the engine includes a DOC, a DPF, and two SCR catalyst cells, as shown in Fig. 5. The catalyst of the SCR is Fe–Zeolite and each catalyst cell with the volume of 0.0052 m3 . The measurement instruments are illustrated in Fig. 6. There are three K-type thermocouples which are used to measure the gas temperatures before the SCR cells, between the SCR cells, and after the SCR cells. Two Siemens VDO NOx sensors are placed between the SCR cells and after the SCR cells, respectively. A Delphi NH3 sensor is located between the SCR cells. In addition, two-channel Horiba MEXA 7500 is placed before the cells and after the cells for NOx concentration measurements. The urea injector is controlled via a dSPACE MicroAutoBox in real time, that is, the urea injection rate is available from the dSPACE MicroAutoBox. It is assumed that

Fig. 7. Oxygen concentration, flow rate, and temperature of the exhaust gas in the first cell.

the temperature measurements, the Horiba NOx concentration readings, and the ammonia concentration between the cells are trustable and reliable. It is noted that the flow rate, temperature, and oxygen concentrations are necessary to run the SCR model and the designed estimator. Fig. 7 depicts the oxygen concentration, flow rate, and temperature of the gas in the first SCR cell during the test which is based on a warmed-up short cycle. We can see that the flow rate varies a lot, which indicates that the Diesel engine is working under a transient condition. The three parameters of the input estimator are K0 = 100, K1 = 15, and λ1 = 0.1.


Fig. 8. Ammonia coverage ratio comparison between the SCR model and the estimator for the first SCR cell.

Fig. 9. Ammonia input comparison between the predictive model and the estimator for the first SCR cell.

Since the initial value of ammonia coverage ratio for the SCR cell is unknown for the observer, it is selected as 0. The ammonia coverage ratio cannot be measured by a physical sensor. Thus, we can only compare the ammonia coverage ratio values between the SCR model and the designed observer for the first SCR cell. The comparison is illustrated in Fig. 8, where we can see that the estimated ammonia coverage ratio approaches to the one of the SCR model swiftly. Since the initial value of the estimator is zero, the estimation error is large at the beginning. However, after 80 s, the estimated value tracks the model value quite well and there is almost no estimation error. In addition, the model value and the estimated value are both smaller than one which is the maximal value for the ammonia coverage ratio, that is, both values are reasonable. The performance of the input estimator is validated in Fig. 9. The solid curve is the predicted ammonia input to the first cell and the predictive model for the ammonia concentration is τ uAdBlue (22) C˙ NH 3 , i n = −αCNH 3 , i n + 2α Nurea Fin where α is the inverse of a time constant, τ is the mass fraction of urea in the solution (it is 32.5% in this application), uAdBlue denotes the mass injection rate of urea injected into the exhaust pipe, Nurea stands for the atomic number of the urea, and Fin

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Fig. 10. Ammonia input comparison between the predictive model and the estimator for the first SCR cell when the ammonia reading is multiplied by a factor of 0.8.

is for the flow rate of the engine-out exhaust gas. The mass injection rate of urea can be obtained with the control signal of the injector. When the temperature is relatively high, the urea– ammonia conversion is completed before the cells. We can take the predicted value as the actual ammonia input. The dash curve is the estimated ammonia input to the first cell. As the ammonia input cannot be too large or negative, a saturator with a lower bound of 0 and an upper bound of 0.1 is used. We can see from Fig. 9 that the observer performance is not good during the first 100 s. However, the estimated value can capture the shape of predicted value during the rest experimental test. The large estimation error during the first 100 s is induced by the big difference between the initial values of the SCR model and the estimator. In order to show the robustness of the proposed estimator to the measurement error or sensor fault, the measurements of the first cell’s ammonia concentration are multiplied by a factor of 0.8. The modified measurements are then applied to the proposed estimator. Fig. 10 depicts the ammonia input estimation. We can see that there is no significant difference between the predicted values and the estimated values after 100 s though the ammonia measurements are reduced by a factor of 0.2. Since the sensors may be subject to faults and the readings are different from the actual values, the modification studies explore that the designed observers are robust to the sensor faults. C. Unknown Input and State Observer of the First Cell With Available Ammonia Input In this section, we assume that the ammonia input is available for the aforementioned observer. In practice, the ammonia input for the first cell can be determined by using the urea injection rate, the flow rate, and the assumption that the urea is completely converted into gaseous ammonia before entering into the first cell. The urea–ammonia conversion efficiency depends on the gas temperature, the injector position, the injector angle, and the geometry of the SCR catalyst cell. If the SCR system is well designed and integrated, the urea–ammonia conversion can be completed at a reasonable temperature range. If the ammonia input is available, the scenario of the estimation work is shown

2228


16 vertices as [44]: 16 16 βi (Ai , Ci ) |βi ≥ 0, βi = 1 (A, C) = i=1

where

Fig. 11.

Estimation scheme for the first cell with available ammonia input.

in Fig. 11 in which the outputs of the observer consist of the ammonia coverage ratio and the engine-out NOx concentration. To estimate the unknown input and the state simultaneously, we develop a PI observer. It is assumed that the derivative of the engine-out NOx concentration is small such as zero, that is, the engine-out NOx concentration varies slowly with respect to the time. The nonlinear SCR model can be reformulated as follows: ξ˙ = Aξ + B + Cu where

(23)

⎛⎡

0 0

⎜⎢ (A1 , C1 ) = ⎝ ⎣ 0 0 ⎛⎡

CNOx,1 ⎥ ⎢ ξ = ⎣ CNH 3 ,1 ⎦ , θ1

⎡

0 ⎢ A = ⎣0

0 a1

⎤

0 0

a ¯1

0 0

⎥ 0 a2 ⎦ ,

0

0 a3

⎛⎡

0 0

⎜⎢ (A16 , C16 ) = ⎝ ⎣ 0 0

0 ⎤ ⎡

a4

⎤⎞

⎥ ⎢ ⎥⎟ a2 ⎦ ⎣ 0 ⎦ ⎠ a3

0

a ¯1

⎤ ⎡

a ¯4

⎤⎞

⎥ ⎢ ⎥⎟ a ¯2 ⎦ ⎣ 0 ⎦ ⎠ a ¯3

0

β1 =

b3

.. .

F −r2,1 CNO x ,1 CO 2 V1 δ2,1 , a4 = V1 F F b1 = − CNO x ,1 b2 = − − rF 1 Θ1 CNH 3 ,1 V1 V1

b3 = rF 1 V1 CNH 3 ,1 ,

⎤⎞

¯1 1 | |a2 − a ¯2 | |a3 − a ¯3 | |a4 − a ¯4 | |a1 − a |¯ a1 − a1 | |¯ a2 − a2 | |¯ a3 − a3 | |¯ a4 − a4 | a1 − a1 |a2 − a ¯2 | |a3 − a ¯3 | |a4 − a ¯4 | 1 β2 = |¯ a1 − a1 | |¯ a2 − a2 | |¯ a3 − a3 | |¯ a4 − a4 |

b1 ⎢ ⎥ B = ⎣ b2 ⎦

a3 = − rF 1 CNH 3 , 1 V1 − rR 1 δ1,1

F CNH 3 ,in V1

a4

.. .

a1 = − r2,1 CNOx,1 CO 2 Θ1 V1 rR a2 = rF 1 CNH 3 ,1 Θ1 + 1 Θ1 δ1,1 V1

+

⎤ ⎡

⎥ ⎢ ⎥⎟ a2 ⎦ ⎣ 0 ⎦ ⎠ a3

⎜⎢ (A2 , C2 ) = ⎝ ⎣ 0 0

⎤

⎡

a1

0 0

0 0 ⎤

⎡

β16 =

¯4 | |a1 − a1 | |a2 − a2 | |a3 − a3 | |a4 − a . |¯ a1 − a1 | |¯ a2 − a2 | |¯ a3 − a3 | |¯ a4 − a4 |

Since it is assumed that the derivative of the input is small, an augmented system with the dynamics of the input is described as follows: ¯ +B ¯ η˙ = Aη

⎡

a4

where

⎤

⎢ ⎥ C = ⎣ 0 ⎦,

(25)

i=1

u = CNOx,in .

0 Note that the matrices B and C are not constant but available online. The signal u is to be estimated. The system matrix A contains three nonzero elements which are varying but bounded. We further assume that ¯1 , a2 ∈ a2 a ¯2 a1 ∈ a1 a (24) ¯3 , a4 ∈ a4 a ¯4 a3 ∈ a3 a ¯1 , a2 , a ¯2 , a3 , a ¯3 , a4 , and a ¯4 are known where the bounds a1 , a scalars which are estimated with several experimental tests beforehand. In order to cover all the possible choices for ai , i = 1, 2, 3, 4, the interval ranges can be expanded. However, too large ranges would lead to conservative results. T Choose the scheduling variable vector ρ = a1 a2 a3 a4 . Then, the matrix pair (A, C) can be described by a polytope with

η=

ε u

, A¯ =

A C 0

0

(26)

¯= , B

B 0

.

¯ in (26) is available It is necessary to mention that the matrix B online and the system matrix A¯ is still determined by the matrix pair (A, C), that is, the system matrix A¯ can be described by the polytope with 16 vertices. The PI observer for the augmented system in (26) has the following form: ¯η + B ¯ + KPI (y − yˆ) , ηˆ˙ = Aˆ CˆNOx,1 CNOx,1 y= , yˆ = CNH 3 ,1 CˆNH 3 ,1 ⎡ ⎤ CˆNOx,1 ⎢ ⎥ εˆ ⎢ ˆ ⎥ ηˆ = = ⎢ CNH 3 ,1 ⎥ ⎣ θˆ1 ⎦ u ˆ u ˆ

(27)


2229

where KPI is the gain to be determined. Define the estimation error as eη = η − ηˆ. The dynamics of the error system is governed by e˙ η = A¯e eη A¯e = A¯ − KPI C¯ 1 0 0 0 ¯ C= . 0 1 0 0

(28)

In order to derive less conservative result [45]–[49], we design the gain-scheduling observer, that is, the gain KPI is dependent on the scheduling parameters. We can assume that the gain KPI is also linearly dependent on the scheduling parameter and KPI is described by KPI =

16

βi KPI,i .

(29)

Fig. 12. Ammonia coverage ratio comparison between the SCR model and the NOx -input estimator for the first SCR cell.

i=1

Here, there are 16 gains KPI,i , i = 1, . . . , 16 to be determined. According to the Lyapunov stability, the LPV error dynamic system in (28) is asymptotically stable if there exists a positive-definite matrix P such that the following condition is satisfied: P A¯e + A¯T e P < 0.

(30)

Substituting the expression A¯e into the previous condition, we have the following expression: T P A¯ − KPI C¯ + A¯ − KPI C¯ P T (31) = P A¯ + A¯T P − P KPI C¯ − P KPI C¯ < 0. In the condition (31), the matrices P and KPI are both unknown. If we define a new variable as M = P KPI , the condition in (31) becomes T (32) P A¯ + A¯T P − M C¯ − M C¯ < 0. Considering the polytopic form of the matrices A¯ and KPI , the condition in (32) can be evaluated as 16

¯ ¯ T