Improved Constrained Predictive Functional Control Using Extended Non-Minimal State Space Formulation for the Cement Production Process

Abstract

In this article, an improved constraint dealing method and an extended state space model-based constrained predictive functional control (PFC) approach is developed for the denitration process in cement production. To improve the control effect of the presented strategy, first, an enhanced form-based state space model in which the changes of the output predictions can be regulated to achieve smoother system dynamics is constructed. Then, a modified constraint dealing scheme, which integrates the relaxation factors to improve the success probability of solving the relevant optimization, is also introduced for the proposed method. On the basis of the two merits, better ensemble control performance is anticipated. Simulations on the regulation of the NOx concentration in the denitration process prove the validity of the proposed constrained PFC strategy.

1. Introduction

In order to guarantee the safety of the operation and the quality of the product, constraints on the key variables in industrial processes are common; hence, considering these constraints is necessary in the corresponding controller design [1,2]. To solve the constraints in industrial processes, model predictive control (MPC) has been presented as a promising control scheme in which the relevant optimization problem subject to constraints can be transformed into a standard quadratic programming (QP) issue [3].

In recent decades, there has been much progress in the research on QP problem-based constrained MPC approaches, and their corresponding applications in various industries have also been implemented. In [4], the NMPC scheme was proposed to exploit the prime-dual neural network for online optimization. Robustness was achieved by introducing an outer proximal-point iteration scheme that regularized the problem without altering the solution and by adaptively weighting the least squares problems encountered while solving the problem [5]. Based on the neural dynamic optimization, a nonlinear MPC method was put forward for the regulation of leader–follower mobile robots [6]. To solve the optimization problem in the constrained MPC scheme, Newton-Kantorovich type approaches were studied by Dontchev, et al. [7]. In [8], the strictly convex QP problems were solved by an active set method, which was proposed on the basis of nonnegative least squares. By employing the neural dynamic optimization, a constrained MPC strategy was designed for the nonholonomic mobile robots to track the desired trajectory [9]. To settle the convex QP problems in constrained MPC scheme, Lu, et al. [10] investigated the discrete-time neural network.

The constraints on critical variables may be very strict in industrial production; meanwhile, the actual working conditions are varying and unpredictable, so, there may be no feasible solutions for the relevant controller [11,12]. Under such situations, the value of the manipulated variables will not be updated; hence, the ideal system performance cannot be ensured.

Aside from the aforementioned conditions, the continuously improved control effect is also needed for MPC algorithms to adapt to the kinds of requirements [13]. Focusing on this point, many researchers have presented significant results [14,15]. Based on a mechanistic model, a small-scale MPC approach was implemented for pH regulation in the dechlorination process [16]. In [17], an enhanced PFC strategy in which an improved state space model and a linear quadratic form were employed was developed for linear processes. The control performance was improved by regulating both the process state and output error variables under partial actuator faults and unknown disturbances. Through introducing particle swarm optimization, a fast nonlinear MPC scheme was implemented on a field-programmable gate array by Xu et al. [18]. A two-tiered MPC method was designed for a chemical process under constraints and model/plant mismatch [19]. By combining this with a distributed moving horizon estimation, a distributed MPC strategy was studied for nonlinear process systems [20]. In [21], a novel state space MPC algorithm was investigated for linear systems with partial actuator failures. The idea of incorporating the measured process inputs, outputs, and their past measured values into a non-minimal state space model (NMSS)-based MPC has been intensively studied. However, there are still issues for an MPC to deal with the model/process mismatch for desired product quality.

Among the improvements in MPC strategies, the extended state space model based MPC schemes in [22,23,24] has shown its superiority. In the modified model, the increments of state variables are synthesized with the tracking error to form a novel state vector; then, an extended state space model is derived. By utilizing this enhanced model, the changes in output predictions can be adjusted; thus, the system dynamics can be tuned to be smoother, and the improved system performance is expected. To deal with unknown mathematical models, some controllers using fuzzy logic systems have been developed for this problem. A novel method based on interval type-3 fuzzy logic systems (IT3-FLSs) and an online learning approach was designed for power control and battery charge planning for photovoltaic (PV)/battery hybrid systems [25]. In [26], new fractional-order learning rules were derived to tune the T3-FLSs such that stability was ensured. In addition, new compensators were proposed by using fractional-order calculus.

For the traditional QP-based constrained MPC scheme, when the constraints are strict, feasible solutions may not be obtained, which will deteriorate the performance of the control system. In this article, an improved constraint dealing method is presented to ensure that the quadratic programming problem always has acceptable solutions under various conditions. The main contributions of this work include the following: (1) A modified state space model is derived to improve the control performance for the constrained PFC scheme. (2) An enhanced constraint dealing method is also employed; unlike conventional approaches, the integrated relaxation factors will take effect when the controller is overconstrained, and a relative optimal solution will be obtained. With such a solving mechanism, the desired system performance can be ensured to a larger extent.

The organization of this article is as follows. In Section 2, the conventional constrained PFC scheme, which adopts the traditional state space model, is described. The proposed constrained PFC strategy in which the extended state space model and the enhanced constraint dealing approach are introduced is presented in Section 3. In Section 4, case studies on the control of NOx concentration in the denitration process are presented, and Section 5 concludes this article.

2. Traditional Constrained PFC Method

For simplicity, we consider a general single-input single-output (SISO) discrete time model that can be constructed from the response data in industrial processes.

$$\begin{array}{l}y(k+1)+g_{1}y(k)+\cdots +g_{m}y(k+m-1)\\ =n_{1}u(k)+\cdots +n_{n}u(k+n-1)\end{array}$$

(1)

where $y(k),u(k)$ are the process output and process input at time instant $k$, respectively.

To obtain the set-point tracking under constraints, the relevant cost function is adopted as

$$\min J={\Vert Y(k)-Y_r(k)\Vert }_Q^2+{\Vert \Delta u(k)\Vert }_R^2$$

(2)

subject to

$$\begin{array}{l}\Delta u_{\min }\le \Delta u(k)\le \Delta u_{\max }\\ u_{\min }\le u(k)\le u_{\max }\\ y_{\min }\le y(k+i)\le y_{\max },i=1,2,\cdots ,P\end{array}$$

where

$$Y_r(k)=\left[\begin{array}{c}{y}_r(k+1)\\ y_r(k+2)\\ ⋮\\ y_r(k+P)\end{array}\right]$$

$y_r(k)$ is the reference trajectory, $Q,R$ are the corresponding weighting matrices, and $\Delta u_{\min },\Delta u_{\max }$, $u_{\min },u_{\max }$, and $y_{\min },y_{\max }$ are the relevant constraint limits.

As to the discrete time model in Equation (1), it can be formulated into the following through adding the difference operator $\Delta$.

$$\begin{array}{l}\Delta y(k+1)+g_1\Delta y(k)+\cdots +g_m\Delta y(k+m-1)\\ =n_1\Delta u(k)+\cdots +n_n\Delta u(k+n-1)\end{array}$$

(3)

Here, we construct a new state vector as

$$\begin{array}{l}{x}_m(k)=[\Delta y(k),\Delta y(k-1),\cdots ,\Delta y(k+m-1),\\ \begin{array}{ccc}& & \end{array}\Delta u(k-1),\cdots ,\Delta u(k+n-1){]}^{{\rm T}}\end{array}$$

(4)

and a corresponding non-minimal state space model is derived as

$$\begin{array}{l}{x}_m(k+1)=A_m{x}_m(k)+B_m\Delta u(k)\\ \Delta y(k+1)=C_m{x}_m(k+1)\end{array}$$

(5)

where

$$\begin{gathered} A_m=\left[\begin{array}{ccccccccc}-g_1& -g_2& \cdots & -g_{m-1}& -g_m& n_2& \cdots & n_{n-1}& n_n\\ 1& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ 0& 1& \cdots & 0& 0& 0& \cdots & 0& 0\\ ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 1& 0& 0& \cdots & 0& 0\\ 0& 0& \cdots & 0& 0& 0& \cdots & 0& 0\\ 0& 0& \cdots & 0& 0& 1& \cdots & 0& 0\\ ⋮& ⋮& \cdots & ⋮& ⋮& ⋮& \cdots & ⋮& ⋮\\ 0& 0& \cdots & 0& 0& 0& \cdots & 1& 0\end{array}\right] \\ {B}_m={[\begin{array}{cccccccc}{n}_1& 0& 0& \cdots & 1& 0& \cdots & 0\end{array}]}^{{\rm T}} \\ {C}_m=[\begin{array}{cccccccc}1& 0& 0& \cdots & 0& 0& 0& 0\end{array}] \end{gathered}$$

Further, we can introduce another state vector containing the process output.

$$x(k)={[x_m{(k)}^{{\rm T}},y(k)]}^{{\rm T}}$$

(6)

Then, the non-minimal state space model in Equation (5) can be rewritten as

$$\begin{array}{l}x(k+1)=Ax(k)+B\Delta u(k)\\ y(k+1)=Cx(k+1)\end{array}$$

(7)

where

$$A=\left[\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right];B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right];C=\left[\begin{array}{cc}0& 1\end{array}\right]$$

Here, $0$ in $A$ and $C$ are zero vectors of proper dimensions.

To make the controller design simple, here, we select the basis function of the PFC strategy as a step function. Then, the state prediction using Equation (7) is derived as

$$X(k)=Fx(k)+\theta \Delta u(k)$$

(8)

where

$$X(k)=\left[\begin{array}{c}x(k+1)\\ x(k+2)\\ ⋮\\ x(k+P)\end{array}\right];F=\left[\begin{array}{c}A\\ A^2\\ ⋮\\ A^P\end{array}\right];\theta =\left[\begin{array}{c}B\\ AB\\ ⋮\\ A^{P-1}B\end{array}\right]$$

$P$ is the prediction horizon.

Further, the output prediction can be acquired as

$$Y(k)=C_{T}X(k)$$

(9)

where

$$Y(k)=\left[\begin{array}{c}y(k+1)\\ y(k+2)\\ ⋮\\ y(k+P)\end{array}\right];C_T=\left[\begin{array}{cccc}C& 0& \cdots & 0\\ 0& C& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & C\end{array}\right]$$

We denote $c(k)$ as the set-point, and the reference trajectory $y_r(k)$ can be expressed as

$$y_r(k+i)={\beta }^{i}y(k)+(1-{\beta }^i)c(k),\begin{array}{cc}& i=1,2,\cdots ,P\end{array}$$

(10)

where $\beta$ is the smoothing factor.

For the constrained optimization problem in Equation (2), it can be transformed into a standard QP problem.

$$\underset{\Delta u(k)}{\min }q(\Delta u(k))=\frac{1}{2}\Delta u{(k)}^{{\rm T}}H\Delta u(k)+f^{{\rm T}}\Delta u(k)$$

(11)

subject to

$$C\Delta u(k)\le d$$

where

$$\begin{gathered} \begin{array}{l}H=2({\theta }^{{\rm T}}{C}_T{}^{{\rm T}}Q{C}_T\theta +R)\\ f=2{({\theta }^{{\rm T}}{C}_T^{{\rm T}}Q(C_{T}Fx(k)-Y_r(k)))}^{{\rm T}}\end{array} \\ C=\left[\begin{array}{c}1\\ -1\\ 1\\ -1\\ C_T\theta \\ -C_T\theta \end{array}\right];d=\left[\begin{array}{c}\Delta u_{\max }\\ -\Delta u_{\min }\\ u_{\max }-u(k-1)\\ u(k-1)-u_{\min }\\ y_{\max }-C_{T}Fx(k)\\ C_{T}Fx(k)-y_{\min }\end{array}\right]. \end{gathered}$$

By settling the QP problem in Equation (11), we can obtain the optimal solution of the traditional constrained PFC scheme.

Remark 1.

For the QP problem in Equation (11), there may be no feasible solutions if the constraints are too rigorous at some time instants. Under such conditions, the control law sent to the actuator will not be updated; thus, the ideal system performance may not be guaranteed.

3. Presented Constrained PFC Strategy

3.1. Extended Form Based State Space Model

In this part, the same process in Equation (1) is taken into account, and the non-minimal state space model in Equation (5) is also derived.

Similarly, the reference trajectory $y_r(k)$ is selected as:

$$y_r(k+i)={\beta }^{i}y(k)+(1-{\beta }^i)c(k),\begin{array}{cc}& i=1,2,\cdots ,P\end{array}$$

(12)

where $\beta$ is the smoothing factor, and $c(k)$ is the set-point at time instant $k$. $P$ is the optimization horizon.

We define the tracking error at time instant $k$ as

$$e(k)=y(k)-y_r(k)$$

(13)

Here, the basis function of the proposed PFC approach is also chosen as a step function, and based on Equation (5) we have

$$\begin{array}{ll}\Delta y(k+i)& =C_m{x}_m(k+i)\hfill \\ & =C_m{A}_m{}^i{x}_m(k)+C_m{A}_m{}^{i-1}{B}_m\Delta u(k)\hfill \end{array}$$

(14)

Further, at time instant $k+1$, we have

$$\begin{array}{l}e(k+1)=e(k)+\Delta y(k+1)-\Delta y_r(k+1)\\ \begin{array}{ccc}& & =e(k)+\end{array}{C}_m{A}_m{x}_m(k)+C_m{B}_m\Delta u(k)-\Delta y_r(k+1)\end{array}$$

(15)

By synthesizing the tracking error with $x_m$, the extended state vector is formed as

$$x(k)=\left[\begin{array}{c}{x}_m(k)\\ e(k)\end{array}\right]$$

(16)

Then, the new state space model is acquired as

$$x(k+1)=Ax(k)+B\Delta u(k)+C\Delta y_r(k+1)$$

(17)

where

$$A=\left[\begin{array}{cc}{A}_m& 0\\ C_m{A}_m& 1\end{array}\right];B=\left[\begin{array}{c}{B}_m\\ C_m{B}_m\end{array}\right];C=\left[\begin{array}{c}0\\ -1\end{array}\right]$$

$0$ in $A$ and $C$ are the appropriate zero vectors.

Using Equation (17), the corresponding state prediction is obtained as

$$X(k)=Fx(k)+\theta \Delta u(k)+S\Delta Y_r(k)$$

(18)

where

$$\begin{gathered} X(k)=\left[\begin{array}{c}x(k+1)\\ x(k+2)\\ ⋮\\ x(k+P)\end{array}\right];\Delta Y_r(k)=\left[\begin{array}{c}\Delta y_r(k+1)\\ \Delta y_r(k+2)\\ ⋮\\ \Delta y_r(k+P)\end{array}\right] \\ F=\left[\begin{array}{c}A\\ A^2\\ ⋮\\ A^P\end{array}\right];\theta =\left[\begin{array}{c}B\\ AB\\ ⋮\\ A^{P-1}B\end{array}\right] \\ S=\left[\begin{array}{ccccc}C& 0& 0& \cdots & 0\\ AC& C& 0& \cdots & 0\\ A^{2}C& AC& C& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ A^{P-1}C& A^{P-2}C& A^{P-3}C& \cdots & C\end{array}\right] \end{gathered}$$

3.2. Modified Constraint Dealing Method

Here, the following performance index is first introduced for the proposed PFC scheme to yield the desired control performance under constraints.

$$\min J={\Vert X(k)\Vert }_Q^2+{\Vert \Delta u(k)\Vert }_R^2$$

(19)

subject to

$$\begin{array}{l}\Delta u_{\min }\le \Delta u(k)\le \Delta u_{\max }\\ u_{\min }\le u(k)\le u_{\max }\\ y_{\min }\le y(k+i)\le y_{\max },i=1,2,\cdots ,P\end{array}$$

where $Q,R$ are the relevant weighting matrices. $\Delta u_{\min },\Delta u_{\max }$, $u_{\min },u_{\max }$, and $y_{\min },y_{\max }$ are the corresponding constraint bounds.

Remark 2.

From the objective function in Equation (19), it can be easily seen that $X(k)$is forced to be a zero vector, i.e., the changes in the output predictions also can be penalized; hence, smoother system dynamics are anticipated for the proposed PFC method.

To improve the success probability for solving the optimization in Equation (19), we utilize some relaxation factors to the output constraints, and its modified form is

$$\begin{array}{l}y(k+i)\le y_{\max }+L_{\max }\\ y_{\min }\le y(k+i)+L_{\min }\\ L_{\max }\ge 0;L_{\min }\ge 0\\ i=1,2,\cdots ,P\end{array}$$

(20)

where

$$L_{\max }=\left[\begin{array}{c}{L}_{\max 1}\\ ⋮\\ L_{\max P}\end{array}\right];L_{\min }=\left[\begin{array}{c}{L}_{\min 1}\\ ⋮\\ L_{\min P}\end{array}\right]$$

$L_{\max 1},\cdots L_{\max P},L_{\min 1},\cdots ,L_{\min P}$ are the corresponding relaxation factors.

Remark 3.

It is known that the constraints on control inputs are called hard constraints, and these constraints cannot be loosened because of the physical characteristics of the relevant actuators. Therefore, the output constraints are selected as the target to be loosened when the controller encounters overconstrained conditions.

Remark 4.

It is obvious that the modified output constraints in Equation (20) will be the same as the original output constraints in Equation (19) if the introduced relaxation factors are zeros.

Based on the enhanced output constraints, the corresponding performance index is rewritten as

$$\min J={\Vert X(k)\Vert }_Q^2+{\Vert \Delta u(k)\Vert }_R^2+{\Vert L_{\max }\Vert }_{Q_{\max }}^2+{\Vert L_{\min }\Vert }_{Q_{\min }}^2$$

(21)

subject to

$$\begin{array}{l}\Delta u_{\min }\le \Delta u(k)\le \Delta u_{\max }\\ u_{\min }\le u(k)\le u_{\max }\\ y(k+i)\le y_{\max }+L_{\max }\\ y_{\min }\le y(k+i)+L_{\min }\\ L_{\max }\ge 0\\ L_{\min }\ge 0\\ i=1,2,\cdots ,P\end{array}$$

where $Q_{\max }$ and $Q_{\min }$ are the relevant weighting matrices for the employed relaxation factors.

Remark 5.

From Equation (21), we can easily know that$L_{\max },L_{\min }$will be zero vectors when the output constraints are loosened. Under this situation, the modified optimization in Equation (21) is equivalent to the conventional optimization problem in Equation (19). Note that these relaxation factors will take effect when the corresponding controller is overconstrained, and a relative optimal solution can be obtained.

Remark 6.

Here$Q_{\max }$and$Q_{\min }$affect the degree of relaxation on output constraints under overconstrained conditions, and larger elements in $Q_{\max }$and$Q_{\min }$bring smaller looseness on output constraints.

Remark 7.

As to the multiple-input multiple-output (MIMO) process, the relevant extension of the presented constrained PFC approach is easy.

As to the optimization in Equation (21), it can be converted into the following QP problem.

$$\underset{\Delta u(k)}{\min }q(\Delta u(k))=\frac{1}{2}\Delta u{(k)}^{{\rm T}}H\Delta u(k)+f^{{\rm T}}\Delta u(k)$$

(22)

subject to

$$C\Delta u(k)\le d$$

where

$$\begin{gathered} H=\left[\begin{array}{ccc}{H}_1& 0& 0\\ 0& 2Q_{\max }& 0\\ 0& 0& 2Q_{\min }\end{array}\right] \\ {H}_1=2({\theta }^{{\rm T}}Q\theta +R) \\ f={[\begin{array}{ccc}2({\theta }^{{\rm T}}Q(Fx(k)+S\Delta Y_r(k)))& 0& 0\end{array}]}^{{\rm T}}. \\ {C}_T=\left[\begin{array}{cccc}{C}_y& 0& \cdots & 0\\ 0& C_y& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & C_y\end{array}\right];C_1=\left[\begin{array}{cccc}-1& 0& \cdots & 0\\ 0& -1& \cdots & 0\\ ⋮& ⋮& \ddots & ⋮\\ 0& 0& \cdots & -1\end{array}\right] \\ {C}_y=[\begin{array}{cccc}0& \cdots & 0& 1\end{array}] \\ C=\left[\begin{array}{ccc}1& 0& 0\\ -1& 0& 0\\ 1& 0& 0\\ -1& 0& 0\\ C_T\theta & C_1& 0\\ -C_T\theta & 0& C_1\\ 0& C_1& 0\\ 0& 0& C_1\end{array}\right];d=\left[\begin{array}{c}\Delta u_{\max }\\ -\Delta u_{\min }\\ u_{\max }-u(k-1)\\ u(k-1)-u_{\min }\\ y_{\max }-C_T(Fx(k)+S\Delta Y_r(k))-Y_r(k)\\ C_T(Fx(k)+S\Delta Y_r(k))+Y_r(k)-y_{\min }\\ 0\\ 0\end{array}\right] \end{gathered}$$

Finally, the optimal solution of the proposed constrained PFC strategy can be gained via solving the above QP problem.

For the two constrained PFC strategies, the main difference between the proposed PFC and the traditional PFC are listed in

Table 1. The main differences between the two strategies.

	Proposed	Conventional
Model	Extended state space model	Non-minimal state space mode
Constraints	Always has acceptable solutions	May have no feasible solutions
Performance	Better ensemble control performance	Performance may not be guaranteed.

.

4. Case Study

In this section, we introduce the regulation of the NOx concentration in the denitration process of the cement production to evaluate the validity of the presented constrained PFC scheme.

Figure 1 shows a part of the sketch for cement production. Firstly, the ground raw materials will be sent to a group of preheaters for preheating. Here, multistage preheaters are designed to preheat these raw materials adequately. Then the preheated materials will enter into the calcinator in which they will be heated and decomposed. After that, these materials will go to the rotary kiln for further decomposition and heating. During this process, the flow direction of the gases is inverse to that of raw materials. The high-temperature gases generated in the rotary kiln and calcinator will be pumped to flow through these preheaters. With these actions, the produced high-temperature gases are used reasonably, and the rate of energy utilization is also increased.

NOx will be produced under high temperature in cement production, and it will be harmful to the environment if these flue gases are untreated before emission. In order to process the NOx in the flue gases, ammonium hydroxide is ejected to react with the NOx in the calcinator. Under this operation, the NOx concentration in the flue gases will be reduced to a reasonable level to protect the environment, and this process is called denitration.

The NOx concentration is a key variable in the denitration process, and it should be controlled at an acceptable level. Figure 2 shows a general block diagram for the proposed method. In the following, a practical process model is considered.

$$y(k)=0.8119y(k-1)-48.9u(k-7)$$

(23)

Here, the conventional constrained PFC scheme was employed as the comparison. The set-point was chosen as 1 for both methods. To test the effectiveness of the two schemes further, we generated three model/plant mismatched cases via the Monte Carlo method, and the three cases are:

Case 1: $y(k)=0.8747y(k-1)-45.1u(k-7)$

Case 2: $y(k)=0.7361y(k-1)-55.8u(k-7)$

Case 3: $y(k)=0.9054y(k-1)-53.6u(k-7)$

Meanwhile, two groups of constraints (a loose group and a rigorous group) were also introduced.

The loose group:

$$\{\begin{array}{l}-0.001\le \Delta u(k)\le 0.001\\ -0.005\le u(k)\le 0\\ 0\le y(k+i)\le 1.1\end{array}$$

The rigorous group:

$$\{\begin{array}{l}-0.001\le \Delta u(k)\le 0.001\\ -0.004\le u(k)\le 0\\ 0\le y(k+i)\le 1.05\end{array}$$

For the two constrained PFC strategies, the corresponding control parameters under three mismatched cases are listed in the

Table 2. The parameters of the two strategies.

Parameters	Proposed	Conventional
$P$	10	10
$Q=diag(Q_1,\cdots ,Q_P)$	$\begin{array}{l}{Q}_1,\cdots ,Q_P=diag\\ (10,0,0,0,0,0,0,1)\end{array}$	$Q_1,\cdots ,Q_P=1$
$R$	0.1	0.1
$Q_{\max }$	$diag(1,1,\cdots ,1)$	\
$Q_{\min }$	$diag(1,1,\cdots ,1)$	\
$\beta$	0.3	0.3

.

Under the model/plant matched case, the conventional PFC approach adopted the control parameters listed in Table 2, and the proposed PFC scheme also employed the parameters listed in Table 1 except $Q$. Here, $Q_1,\cdots ,Q_P=diag(0,0,0,0,0,0,0,1)$ was selected for the presented PFC algorithm.

Figure 3a–c shows the corresponding responses for both strategies under model/plant matched cases and loose constraints. It can be easily seen that the responses of two methods were the same. Set-point tracking under constraints was achieved for both PFC approaches.

Remark 8.

Without regard to the constraints, the PFC scheme based on the extended state space model will be equal to the conventional PFC approach if only the tracking error in the extended state vector is weighted.

The relevant responses for two schemes under the model/plant mismatched cases are displayed in Figure 4a–c, Figure 5a–c and Figure 6a–c. In Figure 4a–c and Figure 5a–c, the chosen constraints were loose, and set-point tracking under constraints was reached for both methods. However, it is obvious that the responses of the proposed PFC strategy were smoother with smaller overshoots and oscillations, which implies that the developed constrained PFC scheme provided improved ensemble control performance. Under case 3 and rigorous constraints, there were no feasible solutions for the conventional PFC method after several time instants, but the control performance of the presented strategy was also acceptable, which verified the superiority of the proposed approach further. In a word, the ensemble control performance of the developed constrained PFC algorithm was enhanced.

5. Conclusions

In this article, an enhanced constrained PFC strategy was developed for the control of NOx concentration in the denitration process. By employing an extended state space model and an improved constraint dealing method, the system dynamics could be tuned in the proposed scheme, and the success of solving the relevant optimization was also increased; thus, a modified ensemble control performance was anticipated for the proposed PFC algorithm. Simulations on the regulation of denitration process in cement production demonstrated the validity of the developed constrained PFC method. For future work, the robustness and sensitivity analysis of the proposed PFC will be investigated as well as the potential for more rigorous stability and feasibility guarantees, while retaining simplicity.