1. 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
2O 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
2O. 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 a promising method is to control the NH
3 concentration and coverage of NH
3 in the SCR catalyst at the optimum [
5]. NH
3 coverage and the ammonia coverage ratio are defined in (1), where
is NH
3 storage capacity and
is the amount of NH
3 stored inside the SCR catalyst.
Much research has been conducted into SCR control systems [
4,
5,
6]. In general, the SCR control strategy can be divided into non-model-based [
7,
8] and model-based [
9,
10,
11,
12,
13,
14,
15,
16,
17]. The non-model-based methods include the pulse spectrum-based blue jet control method [
7] and the PID control method [
8]. Although the non-model-based method is relatively mature, its performance in transient and low temperature conditions makes it increasingly difficult to meet stricter emission regulations due to the problems of time delay, system inertia, and sensor measurement error and system uncertainty. As emission regulations become more and more stringent, model-based methods, such as predictive control [
9], are needed urgently. Simulation and test bench results show that, compared to the non-model-based method, the model-based control method has higher accuracy and better environmental adaptability.
It should be mentioned that the values of NH
3 concentration and NH
3 coverage of the catalyst is vital for the SCR control system. Unfortunately, it is inconvenient to measure NH
3 coverage directly through commercial sensors. To address the problem, observer based methods are prospective and widely used. Reference [
18] presents an observer for estimating the NH
3 concentration of catalysts in SCR. The observer can be used for NH
3 distribution control of the SCR catalyst and fault diagnosis of the diesel engine. Experiments show that the observer estimates converge to the sensor readings and can track the values well. However, the concentration cannot be estimated well in the first 1100 seconds. In [
4], an approach by utilizing two post-selective-catalytic-reduction nitrogen oxide sensors with different ammonia cross-sensitivity factors is proposed to estimate the nitrogen oxide concentration, the ammonia concentration, and the ammonia surface coverage ratio. Experimental results show that the proposed method can be useful in reducing the cost of SCR diagnosis, NH
3 coverage estimation, and advanced SCR controls. In addition, an extended Kalman filter [
19] can also be utilized to estimate the NOx sensor cross-sensitivity to ammonia. It is noteworthy that the performance of an urea-SCR system may be related to the NH
3/NO
X ratio and the NO/NO
2 ratio. Studies show that the ratio of NO to NO
2 varies with the reduction rate and conversion efficiency of NO
X [
1]. A sliding mode observer is widely used in system state estimation because of its strong robustness [
20,
21,
22,
23]. S. Hasan [
24] introduced the Luenberger term into the design of the sliding mode observer, which not only improved the robustness of the observer, but also improved the speed of parameter estimation. Based on the above analysis, a Luenberger sliding mode observer is designed to estimate the state of the two-cell SCR catalyst.
Backstepping control is mainly used to deal with robust control systems with nonlinear and parametric uncertainties [
25,
26]. SCR is a typical nonlinear system with uncertain parameters, which is very suitable for backstepping control. Thus, in [
27], the backstepping control is used successfully, but the unmeasurable problem of NH
3 concentration is not mentioned.
In this paper, considering the advantages of the Luenberger-sliding mode observer [
24] and backstepping control [
25,
26], a Luenberger-sliding mode observer based backstepping control method is applied to a nonlinear SCR system. The aims of this work are to simultaneously minimize the NOx emissions and limit the NH
3 slip under a certain input and output constraint. A Luenberger-sliding mode observer is designed to estimate NH
3 concentration and then, based on the Lyapunov stability analysis and the stepped distributed characteristic of the surface NH
3 coverage ratio along the SCR axial direction, a backstepping control method is designed for SCR system adblue dosing. After that, the stability analysis of the proposed control strategy is described. The proposed approach is validated through computer simulations that are compared with the traditional PID control. Simulation results show that the system controlled by the proposed method has promising performance in overshoot, setting time, and tracking error.
2. Selective Catalytic Reduction System
2.1. SCR System Operation Principles
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):
where
is the NOx sensor reading,
is the true value of the NOx concentration,
is the NH
3 concentration, and
denotes the cross-sensitivity factor. In this paper,
is considered to be a constant.
The reduction involves three processes. First, the urea solution sprayed into the upstream exhaust pipe is converted into NH3, 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:
Isocyanic acid (HNCO) hydrolysation:
Then, the converted NH3 is adsorbed on the surface of the catalyst matrix. Finally, the NH3 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
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):
2.2. 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]:
where,
,
,
, and
are the concentrations of NO, NH
3, inlet NO, and inlet NH
3, respectively.
,
,
, and
are standard reaction rate, adsorption rate, desorption rate, and oxidation rate, respectively.
is the exhaust flow rate and
is the SCR volume.
denotes the ammonia coverage ratio and
is the universal gas constant.
Let
where
,
,
.
Linearize the nonlinear model with respect to operating points and obtain the linear state space equation:
where
, , is the inlet ammonia concentration, is the inlet NO concentration.
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)
, ; the NH3 coverage ratio and the NOX concentration are uncontrollable. At that point, the NH3 coverage ratio reaches 100%. However, it will not happen in practice.
- (2)
, ; the NOX is uncontrollable. In the meantime, , the reasonable working temperature, is below 600 °C. Therefore, the loss of controllability due to this condition is not expected operationally.
4. 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
and control upstream NH
3 coverage at the desired target,
. 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
approach
under the condition
.
Stability of the backstepping is necessary for the controller design. For this system, two cases should be considered. One is ; at this time, the downstream ammonia coverage ratio is fairly high, and can converge to . Another is ; the constraint is satisfied, and can converge to .
4.1. Stability Analysis of Case 1
(1) For Equation (34), the Lyapunov function candidate can be defined as:
where
and
, and taking the time derivative of
gives
Combining (34) with (38) obtains:
Select the virtual control input
as
. Combined, (40) and (43), gives
Then
bacause
and
where,
is positive and definite. Therefore,
can converge to
.
(2) For Equation (35), in order to ensure that the real
can converge to the desired value,
, with the action of
, the Lyapunov function candidate can be defined as:
where
Taking the time derivative of
gives:
Because
, according to (48) and (64), we can get:
Letting
, as the virtual control signal, gets:
where,
is positive and definite. Therefore,
can converge to
.
(3) For Equation (36), in order to ensure that
can converge to the desired value,
, with the action of input signal
, the Lyapunov function candidate can be defined as:
where
. Analogously, according to (59), taking the time derivative of
gives:
Based on (40) and (61), it can be achieved by:
Because , , can converge to .
According to the above mentioned analysis, based on the Lyapunov functions (47), (55), (60), and the control law (39), , , and can converge to the desired value, respectively.
4.2. Stability Analysis of Case 2
In this case, the NH3 coverage ratio of the downstream SCR system should be lower than the value , therefore, the Lyapunov function is design to prove that can converge to with the action of .
(1) For Equation (34), select
as the virtual control input of
; the Lyapunov function candidate can be defined as:
where
.
Taking the time derivative of
, gives:
Let
, then:
At this moment, there are two different conditions needing consideration.
If
and
, then
,
, and
.
can be written as:
Since is negative and definite, can converge to .
If
and
, in order that
is negative and definite, according to (65), it can be achieved by:
Combined, (40), (41), and (43), obtain:
Since
and
, (68) can be achieved if the following condition is satisfied:
If , can converge to zero. If and and are large enough, can be very close to zero, which means that can converge to when .
(2) For Equation (35), the Lyapunov function candidate can be defined as:
According to (65) and (39), it can be achieved by:
Based on (45), is negative, can converge to , and can converge to .