Luenberger-Sliding Mode Observer Based Backstepping Control for the SCR System in a Diesel Engine

: In order to keep the ammonia (NH 3 ) slip of the downstream selective catalytic reduction (SCR) system at a low level and simultaneously achieve a high nitrogen oxide (NO X ) conversion rate, a Luenberger-sliding mode observer based backstepping control method is proposed. Considering that the internal working condition of the catalyst cannot be measured by commercial sensors directly, a Luenberger-sliding mode observer is designed to estimate the ammonia concentration at the middle of the catalyst. In addition, based on the stepped distributed characteristic of the surface ammonia coverage ratio along the SCR axial direction, a backstepping control method is utilized for the SCR system, in which the SCR system is decomposed into two subsystems. Firstly, the Lyapunov function is designed to ensure the convergence of the downstream subsystem, and then the virtual control law is obtained. After that, taking the virtual control law as the tracking target of the upstream subsystem, the Lyapunov function of virtual control law is given. Finally, the actual control law of the whole closed loop system is acquired. Simulations under di ﬀ erent conditions are conducted to investigate the e ﬀ ect of the proposed control method. In addition, comparisons with the traditional PID (Proportion Integration Di ﬀ erentiation) control are presented. Results show that the proposed method is much better than the PID control method in overshoot, setting time, and tracking error.


Introduction
Diesel engines have attracted more and more attention in recent years due to their high economy, high power, and low CO and HC emissions [1][2][3]. However, owing to the special combustion process, a diesel engine produces much more nitrogen oxide (NO X ) and particulate matter (PM), which is harmful to the environment and human health. Various regulations have been legislated against diesel engine NO X and PM emissions. In order to meet stringent regulations, devices such as selective catalytic reduction (SCR) systems and diesel particle filters (DPF) are installed in post-processing systems to reduce emissions. SCR refers to the use of reducing agents to selectively react with NOx in flue gas and generate non-toxic and pollution-free N 2 and H 2 O under the action of a catalyst. Generally, in SCR systems, 32.5% of aqueous urea solution is injected into the tail gas pipe of the engine; urea is then decomposed into ammonia, which reacts with NOx to generate N 2 and H 2 O. However, excessive urea can lead to NH 3 leakage in the tailpipe and increase the usage cost. On the other hand, insufficient NH 3 injection will lead to low NOx conversion and higher tail pipe NOx emissions [3,4]. A great deal of research has been done to minimize NOx emissions and limit NH 3 leaks at the same time, in which Figure 1 is a schematic diagram of an SCR system, in which temperature, NOx, and NH 3 sensors are located upstream and downstream of the SCR catalyst. In order to monitor the status of the intermediate catalyst, NOx and NH 3 sensors are installed between two SCR batteries. Note that the inlet NOx measurement will not be contaminated by NH 3 , while the intermediate and downstream NOx sensors will be affected by the cross sensitivity of NH 3 . According to [27], the concentration of NOx is a combination of the NOx and NH 3 concentrations, as shown in (2):

SCR System Operation Principles
where C NO x ,mea is the NOx sensor reading, C NO x is the true value of the NOx concentration, C NH 3 is the NH 3 concentration, and K denotes the cross-sensitivity factor. In this paper, K is considered to be a constant. The reduction involves three processes. First, the urea solution sprayed into the upstream exhaust pipe is converted into NH 3 , which generally consists of three chemical reactions: Urea solution evaporation, urea decomposition, and isocyanic acid hydrolysation. The main chemical reactions are summarized as: Aqueous urea solution evaporation: Urea decomposition: Isocyanic acid (HNCO) hydrolysation: Then, the converted NH 3 is adsorbed on the surface of the catalyst matrix. Finally, the NH 3 reacts with NOx to form nitrogen molecules.
It should be noted that that urea can be completely converted in the upstream tailpipe if the catalyst pool has a good geometric design and the exhaust has a suitable temperature [28]. Therefore, it is reasonable to assume that 100% of the urea aqueous solution is converted to gaseous NH 3 before the SCR catalyst unit.
The NH 3 adsorption and desorption reactions can be expressed as [29]: where Z is the active substrate site of the SCR catalyst cell and ZNH 3 represents NH 3 adsorbed on the SCR substrate. The adsorbed NH 3 is active enough to reduce the NOx in terms of the chemical reactions. The main NOx reduction process can be summarized as follows: In some cases, when the gas temperature is quite high, the adsorbed NH 3 can also be oxidized, as shown in (11):

SCR Dynamic Model and Analysis of Observability and Controllability
Assuming that the physical variables in the SCR catalyst unit are uniform, a SCR model is developed based on the above reaction. For convenience, the mass transfer and the surface phase concentration of species in the model are neglected. In this paper, the nonlinear model of the SCR model is expressed using the state-space form [29]: (12) where, C NO , C NH 3 , C NO,in , and C NH 3 ,in are the concentrations of NO, NH 3 , inlet NO, and inlet NH 3 , respectively. r red , r ads , r des , and r oxi are standard reaction rate, adsorption rate, desorption rate, and oxidation rate, respectively. F is the exhaust flow rate and V is the SCR volume. θ denotes the ammonia coverage ratio and R is the universal gas constant. Let where Linearize the nonlinear model with respect to operating points and obtain the linear state space equation: where The controllability grammian matrix takes the form: In most cases, the rank of the controllability grammian matrix is equal to 3. However, it may lose rank under certain operations: (1) A 23 = 0, rank(Q C ) = 1; the NH 3 coverage ratio and the NO X concentration are uncontrollable. At that point, the NH 3 coverage ratio reaches 100%. However, it will not happen in practice. (2) A 12 = 0, rank(Q C ) = 2; the NO X is uncontrollable. In the meantime, r oxi > r red × C NO , the reasonable working temperature, is below 600 • C. Therefore, the loss of controllability due to this condition is not expected operationally.

Two-Cell SCR Catalyst Ammonia Concentration Observer Design
According to [8], ammonia storage in SCR catalysts varies along the axis of the catalysts. Moreover, the ammonia storage in the upstream and downstream of the SCR catalysts has a direct impact on the conversion of NOx and the emission of NH 3 in the tail gas. In order to express the internal state of the SCR catalytic converter more accurately, a two-cell SCR catalytic converter system is designed, shown in Figure 2 [18]. Considering the NH 3 concentration and NH 3 coverage ratio, the dynamic model is presented as follows: . θ 1 = −θ 1 (r ads,1 C NH 3 ,1 + r des,1 + r red,1 C NO,1 + r oxi,1 ) + r ads,1 C NH 3 ,1 (16) . θ 2 = −θ 2 (r ads,2 C NH 3 ,2 + r des,2 + r red,2 C NO,2 + r oxi,2 ) + r ads,2 C NH 3 ,2 . .

Observer Stability Analysis
As mentioned above, the NH 3 coverage ratio is one of the important factors for the NH 3 storage distribution control, and its estimation requires the NH 3 concentration. The observability is demonstrated in the following [28]: First, select the Lyapunov function candidate as: Then, the differentiate (25), gives: .
It is apparent that . V θ 1 ≤ 0, which means thatθ 1 converges to θ 1 within a finite period of time.

Convergence Analysis ofĈ
can converges to C NH 3 ,2 in a finite period of time. Once the sliding mode is reached in the short term, there will be . C NH 3 ,1 = C NH 3 ,1 = 0. That means: Select the Lyapunov function candidate as: Then, differentiate (31) gives: .

Convergence Analysis ofθ 2
Let C NH 3 ,2 = 0 as a sliding surface; once the sliding mode is reached, there will be . C NH 3 ,2 = C NH 3 ,2 = 0. Select the Lyapunov function candidate as: Then, differentiate (32) gives: Let k 1,2 > C NH 3 ,2 r ads,2 max , then . V θ 2 is negative and definite, andθ 2 can converge to θ 2 in a finite period of time.

Backstepping Control Law Design
In order to keep NH 3 leakage of the downstream SCR system at a low level and achieve a high NOx conversion rate at the same time, the controller should keep downstream NH 3 coverage below constraint θ * 1 and control upstream NH 3 coverage at the desired target, θ * 2 . Based on the two-cell SCR system model, the dynamic equations are expressed as [18]: . . . where According to the backstepping theory, the control law is designed to let x 4 approach θ * 2 under the condition x 1 ≤ θ * 1 .
Stability of the backstepping is necessary for the controller design. For this system, two cases should be considered. One is x 1 > θ * 1 ; at this time, the downstream ammonia coverage ratio is fairly high, and x 1 can converge to θ * 1 . Another is x 1 ≤ θ * 1 ; the constraint is satisfied, and x 4 can converge to θ * 2 .

Stability Analysis of Case 1
(1) For Equation (34), the Lyapunov function candidate can be defined as: wherex = x 1 − θ * 1 and K 0 > 0, and taking the time derivative of V 1 gives .
(2) For Equation (35), in order to ensure that the real x 2 can converge to the desired value,x 2,tar , with the action of ξ 2 , the Lyapunov function candidate can be defined as: where Taking the time derivative of V 2 gives: Because x 2 = ξ 1 + z 2 , according to (48) and (64), we can get: .
where, Q 2 is positive and definite. Therefore, x 2 can converge to x 2,tar .
(3) For Equation (36), in order to ensure that x 3 can converge to the desired value, x 3,tar , with the action of input signal u, the Lyapunov function candidate can be defined as: where z 3 = x 3 − ξ 2 . Analogously, according to (59), taking the time derivative of V 3 gives: .
Based on (40) and (61), it can be achieved by: Because V 3 < 0, x 3 can converge to x 3,tar . According to the above mentioned analysis, based on the Lyapunov functions (47), (55), (60), and the control law (39), x 1 , x 2 , and x 3 can converge to the desired value, respectively.

Stability Analysis of Case 2
In this case, the NH 3 coverage ratio of the downstream SCR system should be lower than the value x 2,tar , therefore, the Lyapunov function is design to prove that x 4 can converge to θ * 2 with the action of ξ 2 .
(1) For Equation (34), select ξ 2 as the virtual control input of x 3 ; the Lyapunov function candidate can be defined as: Taking the time derivative of V 4 , gives: Let x 3,tar = ξ 2 , then: .
Since . V 4 is negative and definite, x 4 can converge to θ * 2 .
(2) For Equation (35), the Lyapunov function candidate can be defined as: According to (65) and (39), it can be achieved by: .

Experiment Results and Analysis
Several studies have reported that the combination of DOC (Diesel Oxidation Catalyst), DPF, and SCR has become one of the most common post-processing applications in heavy diesel engines, which can handle PM and NOx simultaneously [30][31][32]. Normally, DOC, installed upstream of the SCR catalysts, is utilized to convert part of NO into NO 2 . At the same time, DPF, installed between the DOC and the SCR, is used for reducing PM emissions. Figure 3 shows a schematic diagram of a SCR after-treatment system. The detail parameters of the parts are listed in Tables 1-4.   Table 2. Configuration parameters of SCR in GT power.

Item Quantity
Trap diameter (mm) 130 Filter wall thickness (inch) 0.014 Channel length (mm) 260 Inlet cell density (1/inch 2 ) 95 According to the proposed algorithm, the schematic diagram of the control system is designed as shown in Figure 4. In the system, the NH 3 concentration is estimated by the Luenberger-sliding mode observer and used as the input of the backstepping control. After that, the SCR is controlled by the controller.

Experiment Validation of Luenberger-Sliding Mode Observer
In this section, the effectiveness of the observer will be validated first. Because the main reactions on the catalyst are standard reactions and fast reactions, as shown in (8) and (9), the simple model-based controller is targeting a molar ratio of NH 3 /NO X of 1/1 in order to suppress NH 3 leakage. The observer result of the mid-catalyst NH 3 concentration at three different NO 2 /NO ratios and different temperatures are shown in Figures 5-13.        In order to show their performance more intuitively, the mean absolute error is given in Table 5.  As can be seen, the proposed Luenberger-sliding mode observer estimation can converge to sensor measurements very well at different working conditions. The experimental results show that the observation accuracy of mid-catalyst NH 3 concentration can be achieved by using the proposed observer.

Simulation Validation of the Luenberger-Sliding Mode Observer Based Backstepping Control for SCR System
To illustrate the validity of the Luenberger-sliding mode observer based backstepping control for the after-treatment process, NO X conversion efficiency and NH 3 leakage are taken as the output, and the injection of urea (concentration of the inlet ammonia) is taken as the input. To show the effectiveness of the proposed control strategy, traditional PID control is used for comparison. The control performance of the proposed Luenberger-sliding mode observer based backstepping control strategy is shown in Figures 14-22. The control performance of the two control methods is compared using integrated absolute error (IAE) criteria: where e(t) is the error between the reference value and the actual process output. The value of IAE is enumerated in Table 6.          As can be seen from Figures 14-22 and Table 6, traditional PID control and Luenberger-sliding mode observer based backstepping control can basically meet the control requirements, and both can achieve high NOx conversion rate when the NH 3 leakage in the tail gas exceeds the standard, or when a small amount exceeds the standard. Nevertheless, traditional PID control has a large overshoot, which is when it injects excessive adblue into the engine exhaust in a short time. As can be seen from Figure 14, Figures 16 and 19, NH 3 emission from the SCR catalytic converter outlet fluctuates for a period of time, which does not meet the requirements of emission regulations. The proposed controller reaches better operating points in which about 96.2% of NO X is reduced while allowing about 24 ppm NH 3 slip past the catalyst. Although the backstepping control method also has a small amount of overshoot, the downstream emission of the SCR catalysts does not exceed the limit, which is in line with the requirements of emission regulations. Moreover, the backstepping control method has a shorter adjustment time. Even in the case of overshoot, NH 3 emissions downstream of the SCR catalyst can quickly return to normal levels, which is conducive to achieving a higher NO X conversion rate. Furthermore, the control response obtained using the Luenberger-sliding mode observer based backstepping controller has smaller overshoot and relatively shorter settling time. The control responses indicate the efficiency of the proposed controller with excellent set-point tracking properties.

Conclusions
In this paper, a Luenberger-sliding mode observer based backstepping control strategy was proposed to estimate the mid-catalyst ammonia concentration and calculate the input of adblue. The dynamics of a SCR system was modeled to represent the actual process in the design study of the Luenberger-sliding mode observer based backstepping control strategy. The Lyapunov technique was used for demonstrating the stability of the observer and the backstepping SCR control method. Through the simulation test, the performance of the Luenberger-sliding mode observer and the proposed approach was verified under the conditions of different intake components and different intake temperatures. The results show that the observer has high estimation accuracy under different conditions, with a maximum average error of less than 4.2 × 10 −6 . Furthermore, the Luenberger-sliding mode observer based backstepping control strategy can keep the ammonia slip of the downstream SCR system at a low level and simultaneously achieve a high NO X conversion rate, which is much better than the popular PID control method in setting time, overshoot, and tracking error.

Conflicts of Interest:
The authors declare no conflict of interest.