An Adaptive-Noise-Bound-Based Set-Membership Method for Process Identification of Industrial Control Loops

Abstract

Modeling of key variable data needs to consider the complex characteristics of systems in the catalytic cracking unit (CCU) of petroleum refining process, such as slow time-varying behavior, complex dynamic properties, distributed traits, and unknown stochastic noise. To fully capture the dynamics of a linear ordinary dynamic process without introducing incremental components, an adaptive-noise-bound-based set-membership method (RSMI) is proposed in this paper. Under the set-membership framework, the output set is typically represented as an ellipsoid based on the assumed conditions. Firstly, a CARMA model is considered; longer-duration historical data are selected to capture the intricate dynamic characteristics of industrial control loops. Secondly, RSMI introduces am approach to determine allowance factor, optimizing the noise bound for better suitability in real-world noise environments. The adaptive noise bound is achieved by designing an optimization algorithm that seeks the optimal parameters within the optimization framework. The stability of the RSMI algorithm is demonstrated through the application of the Lyapunov method. Next, the RSMI algorithm has been applied in engineering practice and designed for offline and online training stages of control processes. Finally, simulation experiments are performed to model and predict real-time data of flow, pressure, and liquid-level control loops within a catalytic cracking unit. Furthermore, the effectiveness of the RSMI algorithm is validated through two general examples, and frequency domain analysis is performed.

1. Introduction

The petroleum refining process involves high temperatures, high pressures, and flammable and explosive factors. Equipment failures, improper operations, and accidents may lead to safety incidents.

Catalytic cracking unit (CCU) is one of the processing units in the petroleum refining industry, used to convert heavy petroleum fractions into high-value-added products such as gasoline, diesel, and light hydrocarbons [1,2,3]. In response to increasingly severe environmental challenges and market demands, optimization and control of catalytic cracking technology have become particularly important over the past few decades, with continuous advancement of computer technology and control theory. Model predictive control (MPC) [4], as an advanced control strategy, has been widely applied in petroleum refining processes.

Based on MPC technology and the process characteristics of CCU [5], it is essential to divide CCU according to the relationship between heat, material, and energy. Absorption stabilization (ABSS) system can be separated from CCU based on weak interaction of material and energy. The reaction–regeneration system can be separated from energy and catalyst cycle. The design and operation of a fractionation system depend on target purity and yield requirements of the different fractions to be separated. By reasonably controlling the operating variables such as temperature, pressure, flow, and liquid, effective separation and collection of different components can be achieved. CCU in fractionation systems is particularly important and deserves attention.

Set-membership identification algorithm is a common modeling approach used in MPC. Its fundamental principle involves establishing dynamic mathematical models from process data to predict system dynamic behavior and response. The analysis of operating variables, advanced control, and online optimization in CCU heavily relies on dynamic mathematical models that describe the process [6,7]. In recent years, an increasing number of studies have applied SMI methods in modeling, predictive modeling, and loop control of CCU to achieve accurate modeling and optimization control of this complex process [8,9,10]. However, petroleum refining processes often exhibit slow time-varying, dynamic characteristics, complex distribution properties, and unknown random noise, making it challenging to accurately describe real industrial processes using traditional mechanistic models.

Traditional system identification methods [11,12,13,14] neglect the uncertainty associated with system noise. To tackle this issue, scholars have put forward more rational hypotheses concerning statistical characteristics. One notable assumption of noise is the utilization of unknown but bounded (UBB) noise, which is preferred over assuming random noise due to its minimal requirement for prior information. In practical systems, SMI algorithm is commonly based on UBB noise. The primary objective of the SMI method is not to estimate a value, but to obtain a solution set (SS) ensuring the incorporation of real parameters [15,16]. Consequently, this approach provides stringent bounds on parameter estimates. The SMI method can be categorized into various algorithms according to obtained SS, including ellipsoid method [17,18,19], box and polyhedron algorithm [20,21], interval estimation algorithm [22,23,24], and centrosymmetric polycell body algorithm [25]. The ellipsoid algorithm, renowned for its simplicity and computational efficiency, is often favored by researchers when dealing with optimization problems characterized by high computational complexity and difficulties in finding the optimal solution set.

Member sets are defined differently for different practical application objects. The definition of the member set varies for different practical application objects. This paper adopts outer bound ellipsoid algorithm to find SS in a parameter space that intersects with measured data and known noise bounds. Unlike traditional identification methods, SMI does not result in a nominal point within the parameter space, but rather a collection of points within the parameter space, where each set member in the collection represents a feasible solution. Under specific conditions, as the sample size increases, the range covered by the member set gradually narrows. When sample size tends to infinity, the set member eventually converges to actual parameters. In practical applications, the selection of different parameters for optimization criteria varies; methods such as minimum ellipsoid trace algorithm, minimum ellipsoid volume algorithm, generalized radius algorithm for optimization ellipsoids, and upper bound algorithm for minimum performance indicators are generally used [26,27,28,29,30]. The set-membership identification (SMI) algorithm, in combination with other techniques, has witnessed broader applications due to continuous development and enhancements. Specifically, researchers have proposed several variants of the SMI algorithm for various purposes. Aronow et al. [31] introduced the least squares support vector machine algorithm, which addresses challenges such as nonlinearity data. Zhou et al. [32] proposed an improved SMI algorithm that incorporates an adaptive noise bound mechanism, effectively mitigating the performance degradation. Sophie et al. [33] presented a convex relaxation method based on the McCormick envelope, which reduces the complexity of identification steps by converting the problem into set-membership identification error. V. Cerone et al. [34] introduced a data-driven modeling approach that eliminates the need for actual identification steps, designed new controllers, and defined feasible equivalent set-membership problems. Professor Chairez proposed a new learning law with adaptive adjustment rate in [35] to represent the stability conditions of the observer’s free parameters, with smaller mean square error and faster convergence in the early stages of the estimation process.

Overestimating the noise bound can lead to performance degradation in terms of algorithm stability and tracking capabilities. To address the issue of overestimating the noise bound, this paper proposes an adaptive-noise-bound-based set-membership method (RSMI). This algorithm adaptively estimates noise bound, overcoming performance degradation and identification errors caused by traditional SMI algorithms’ overestimation of noise bound. The proposed approach is applied to key variable data modeling in CCU of petroleum refining processes, exhibiting excellent predictive capabilities. For example, temperature measurements at different positions in the fractionation tower can be used for monitoring and controlling the heat balance of the cracking process. Pressure measurement in the monitoring system ensures normal operation and assesses performance and phase states of liquid and gas in the fractionation tower. Flow measurements of different fractions are used for monitoring and controlling product yields and quality [36,37,38,39].

In this paper, the primary aim is to build dynamic mechanistic models for key variables in CCU, catering to the needs of operating variable analysis, advanced control, and online optimization research. By considering dynamic characteristics of different loops, the models are constructed and accuracy is analyzed. While it is a fact that the majority of current SMI methods rely on simplistic assumptions regarding noise, due to inaccurate determination under real working conditions, this paper proposes the RSMI algorithm to find the optimal parameters through optimization algorithms, which estimate noise bound using mean and standard deviation approximations of residual data.

2. RSMI Algorithm

In this section, this paper proposes an adaptive-noise-bound-based set-membership method (RSMI).

2.1. The Framework of SMI Algorithm

The difference CARMA model is suitable for describing non-incremental ordinary linear dynamic systems and processes. Actual linear dynamic systems and processes are inevitably affected by random bounded noise, and distribution characteristics of the random noise may be complex or unknown. Therefore, a CARMA model is considered in this section,

$$A\left(z^{-1}\right)y\left(k\right)=B\left(z^{-1}\right)u\left(k\right)+v\left(k\right)$$

(1)

where $u\left(k\right)$ represents observable input sequences, $y\left(k\right)$ is denoted as observable output sequences, $v\left(k\right)$ is defined as the unmeasurable noise, $\gamma$ represents noise bound, and $v\left(k\right)$ satisfies the condition of bounded noise error as follows,

$$v^2\left(k\right)\le {\gamma }^2,(k=0,1,2,\cdots )$$

(2)

where $A\left(z^{-1}\right)$ and $B\left(z^{-1}\right)$ can be defined by Equation (3)

$$\left\{\begin{array}{c}A\left(z^{-1}\right)=1+a_1{z}^{-1}+a_1{z}^{-2}+\cdots +a_1{z}^{-n}\\ B\left(z^{-1}\right)=b_0+b_1{z}^{-1}+\cdots +b_m{z}^{-m},(b_0\ne 0)\end{array}\right.$$

(3)

By substituting Equation (3) into Equation (1), it can be expressed as

$$y\left(k\right)=-a_{1}y(k-1)-\cdots -a_{n}y(k-n)+b_{0}u\left(k\right)+\cdots +b_{m}u(k-m)+v\left(k\right)$$

(4)

To transform Equation (4) into least squares format, it can be written as

$$y\left(k\right)={\phi }^T\left(k\right)\theta +v\left(k\right)$$

(5)

By obtaining the intersection of Equations (2) and (5), a subset region of the parameter space can be determined. When generating the k-th pair measurement sequences $(y\left(k\right),\phi \left(k\right))$, use $S\left(k\right)$ to represent a subset of $R^{n+m+1}$ written by

$$S\left(k\right)=\{\theta :{(y\left(k\right)-{\phi }^T\left(k\right)\theta )}^2\le {\gamma }^2,\theta \in R^{n+m+1}\}$$

(6)

$S\left(k\right)$ yields a solution set at time k; the essence of the ellipsoid method is to obtain a spatial solution set that is approximate to the real parameter set and parameter space.

It is generally assumed that the actual parameter $\theta (k-1)$ existed in ellipsoid $E(k-1)$, and $E(k-1)$ is given as follows

$$E(k-1)=\{\theta :{(\theta -\widehat{\theta }(k-1))}^{T}P{(k-1)}^{-1}(\theta -\widehat{\theta }(k-1))\le {\sigma }^2(k-1)\}$$

(7)

$P(k-1)$ is defined as a positive definite matrix, ${\sigma }^2(k-1)$ is defined as a non-zero scalar. At $k-1$ time, $\widehat{\theta }(k-1)$ is described as the center of ellipsoid parameter estimation. $\lambda \left(k\right)$ is a scalar representing weight factor Parameter. $\theta$ lies in the ellipsoid; the parameter set $\theta$ is related to $E(k-1)\bigcap S\left(k\right)$. $E(k-1)\bigcap S\left(k\right)$ is not a convex set, so the set must be reshown as $E\left(k\right)\supset (E(k-1)\bigcap S\left(k\right))$. In this situation, the ellipsoid $E(k-1)\bigcap S\left(k\right)$ is shown as follows:

$$\begin{array}{c}{(\theta -\widehat{\theta }(k-1))}^T{P}^{-1}(k-1)(\theta -\widehat{\theta }(k-1))+\lambda \left(k\right){(y\left(k\right)-{\phi }^T\left(k\right)\theta )}^2\\ \le \sigma {(k-1)}^2+\lambda \left(k\right){\gamma }^2\end{array}$$

(8)

Equation (8) is transformed into

$$E\left(k\right)=\left\{{(\theta -\widehat{\theta }\left(k\right))}^T{P}^{-1}\left(k\right)(\theta -\widehat{\theta }\left(k\right))\le {\sigma }^2\left(k\right)\right\}$$

(9)

with

$$P^{-1}\left(k\right)=P^{-1}(k-1)+\lambda \left(k\right)\phi \left(k\right){\phi }^T\left(k\right)$$

(10)

$$\widehat{\theta }\left(k\right)=\widehat{\theta }(k-1)+\lambda \left(k\right)P\left(k\right)\phi \left(k\right)\delta \left(k\right)$$

(11)

$$\delta \left(k\right)=y\left(k\right)-{\phi }^T\left(k\right)\widehat{\theta }(k-1)$$

(12)

$$G\left(k\right)={\phi }^T\left(k\right)P(k-1)\phi \left(k\right)$$

(13)

$${\sigma }^2\left(k\right)={\sigma }^2(k-1)+\lambda \left(k\right){\gamma }^2-\frac{\lambda \left(k\right){\delta }^2\left(k\right)}{1+\lambda \left(k\right)G\left(k\right)}$$

(14)

Algorithm initialization can be guided by ellipsoid parameter. In general, we assume the following initial values:

$$\theta \left(0\right)=\epsilon ,P\left(0\right)=\frac{1}{\epsilon }I$$

(15)

where I denotes a positive matrix, and $\epsilon$ is a small positive value, typically set to $\epsilon ={10}^{-3}$.

2.2. Optimization Criterion

For the set-membership identification algorithm, there is a special ability to select and update data. If the newly measured data do not contain information that can further reduce the feasible set, then data are discarded, which means stopping the update of the feasible set. The information refers to the optimal criterion that minimizes the feasible set. The differences among various SMI algorithms lie in the utilization of different optimization criteria to obtain distinct optimal weight factors $\lambda \left(k\right)$. The selection of these factors affects the size of the final feasible solution set. Therefore, an optimization criterion needs to be employed for solving, such as the minimum volume ellipsoid criterion, minimum trace ellipsoid criterion, and minimum upper bound of performance index criterion [26], among others. The detailed calculation process can be found in [28]. In paper [26], even if the new measurements do not contain new information to improve the quality of the estimation, the updates do not stop, leading to a large computational burden. However, the algorithm referenced in this paper [28] exhibits good characteristics for set-membership identification, gradually enhancing the reliability of parameter estimation. That is to say, the optimization criterion of minimizing generalized semi-axes of the ellipsoid ${\sigma }^2\left(k\right)$ exhibits excellent tracking capabilities for time-varying systems and is not constrained by ellipsoid assumptions. It demonstrates good performance in the presence of variations in the actual noise bound.The optimal weight factor, denoted as ${\lambda }^{*}\left(k\right)$, can be determined by taking the derivative of ${\sigma }^2\left(k\right)$ with respect to $\lambda \left(k\right)$, as illustrated in Equation (14).

$${\lambda }^{*}\left(k\right)=\left\{\begin{array}{c}\frac{1}{\gamma G\left(k\right)}\left(\right|\delta \left(k\right)|-\gamma ),({\gamma }^2\le {\delta }^2\left(k\right))\\ 0,({\gamma }^2>{\delta }^2\left(k\right))\end{array}\right.$$

(16)

Note:

(i)

It should be noted that by choosing the optimization criterion of minimizing the generalized semi-axes of the ellipsoid ${\sigma }^2\left(k\right)$, the SMI algorithm has been improved, as evidenced in Equations (9)–(14), where $\lambda \left(k\right)$ is replaced with ${\lambda }^{*}\left(k\right)$, and ${\lambda }^{*}\left(k\right)$ is used to represent the variable in the subsequent algorithms.

(ii)

The optimization objective of this paper is to minimize generalized semi-axes of the ellipsoid ${\sigma }^2\left(k\right)$, achieving minimum generalized semi-axes of the ellipsoid.

2.3. Noise Bound Adaptive Estimation

In engineering practice, the noise boundary estimation typically approximates the actual boundaries. Underestimating noise boundaries can complicate the convergence of set-membership identification algorithm, while overestimating them can lead to significant errors. Even if the SMI algorithm converges, errors can still be substantial, resulting in a loose ellipsoid. Therefore, this paper focuses on adaptively estimating noise boundaries, assuming their estimation depends on the residual sequence,

$$\left|v\left(k\right)\right|<\widehat{\gamma }$$

(17)

The residual sequence ${\delta }^{\prime }\left(i\right)$ is described as follows

$${\delta }^{\prime }\left(i\right)=y\left(i\right)-{\phi }^T\left(i\right)\widehat{\theta }\left(i\right),(i=1,2,\cdots ,k)$$

(18)

where $\widehat{\theta }\left(i\right)$ represents current parameter, and ${\delta }^{\prime }\left(i\right)$ is residual data at time of i.

The system noise in an identification model generally comes from input noise, measurement noise, and model errors of the system. In this study, residual sequences ${\delta }^{\prime }\left(i\right)$ are considered, which include measurement noise, modeling errors, environmental noise, and other sources of noise that cannot be ignored. This assumption better aligns with the actual situation. In this assumption, an allowance factor h is introduced, the determination of allowance factor is determined on method described in Section 2.4. The residual data from the current and past 199 time steps, calculated using Equation (18), are stored in the variable $can$. The mean and standard deviation of $can$ are computed to estimate noise bound $\widehat{\gamma }$.

$$\begin{array}{c}|mean\left(can\right)-h\ast std\left(can\right)|\le \widehat{\gamma }\le |mean\left(can\right)+h\ast std\left(can\right)|\end{array}$$

(19)

Estimate the noise bound as the maximum difference between the sum and the difference of the mean and variance of the residuals.

$$\begin{array}{c}\widehat{\gamma }=max\left(\right|mean\left(can\right)-h\ast std\left(can\right)|,|mean\left(can\right)+h\ast std\left(can\right)\left|\right)\end{array}$$

(20)

By adapting the current noise bound through Equation (20), the rationality of the assumption is validated, and overestimating the noise bound is resolved. The convergence and stability of algorithm can be further enhanced. As the parameter solution set converges to true parameter, noise bound converges to true value, ensuring the convergence of the adaptive-noise-bound algorithm. The convergence proof is provided in Section 3.

 Remark 1.

Adaptive noise bound determination primarily optimizes noise bound in Equation (14). By utilizing residual sequence ${\delta }^{\prime }\left(i\right)$, an estimated noise bound $\widehat{\gamma }$ is obtained, which approximates the actual noise bound. In Equation (14), γ is replaced by $\widehat{\gamma }$, ensuring that updated parameters obtained by RSMI better align with actual system dynamics and hysteresis.

2.4. Allowance Factor

To achieve a closer approximation of adaptive noise bound estimation to actual noise bound, adaptive noise bound estimation is determined by residual sequence; a search for optimal allowance factor h is conducted. Maximum value $h\left(up\right)$ and minimum value $h\left(down\right)$ for allowance factor h are determined, with update interval of $h\left(er\right)$. The number of optimization cycles is $h\left(L\right)$; the optimization range lies within interval [$h\left(up\right)$, $h\left(down\right)$].

$$\begin{array}{c}h\left(L\right)=\frac{h\left(up\right)-h\left(down\right)}{h\left(er\right)}+1(\mathrm{for}k=1,2,\cdots ,L)\end{array}$$

(21)

Within the framework of optimization loop, the process begins by determining the optimal weight factor ${\lambda }^{*}\left(k\right)$. Then, the RSMI algorithm is updated using Equations (10)–(14). Next, adaptive estimated noise bound is calculated, used as a prior sequence, and substituted into Equation (14). Last, minimum generalized semi-axes of ellipsoid within the optimization loop are determined. Corresponding sequence value associated with this minimum generalized semi-axes is indexed, and the allowance factor h is identified. Minimum generalized semi-axes and the allowance factor h are manually entered, and SMI algorithm is iteratively updated again. This iterative process is repeated until the final identified model parameters are determined.

 Remark 2.

By employing the optimization criterion proposed in Section 2.2, the algorithm seeks to find the minimum generalized semi-axes of ellipsoid within this optimization loop. Based on the index value of this minimum ellipsoid generalized semi-axes, the optimal noise bound widening coefficient is determined.

2.5. Algorithm Overall Flowchart

The algorithm flowchart for estimating non-incremental dynamic processes using an adaptive-noise-bound-based set-membership method (RSMI) is as depicted in Figure 1.

3. Convergence Proof

Proving the stability of the RSMI algorithm using Lyapunov method, employing a Lyapunov method analysis similar to the one used in [29].

 Lemma 1.

Assuming nonnegative sequences $\chi \left(k\right)$ and $\eta \left(k\right)$ that satisfy the following inequality: $\chi \left(k\right)\le \chi (k-1)-\eta \left(k\right)$. We can conclude that ${\sum }_{k=1}^{\infty }\eta \left(k\right)<\infty$, and $\chi \left(k\right)$ converges to a finite random variable $\chi \left(c\right)$, denoted as $\underset{k\to \infty }{lim}\chi \left(k\right)=\chi \left(c\right)$.

 Theorem 1.

For any sequence of bounded measurement vectors $\left\{\phi \left(k\right)\right\}$, the algorithm exhibits the following characteristics: (1) $\left|\right|\tilde{\theta }\left(k\right)\left|\right|$ remains bounded and non-increasing. (2) The noise bound $\widehat{\gamma }$ to be estimated remains bounded.

 Proof.

(1) ${\lambda }^{*}\left(k\right)\ge 0$, and $P\left(k\right)$ is a positive definite matrix, based on the matrix inversion theorem; it can be inferred that the formula for $P\left(k\right)$.

$$P\left(k\right)=P(k-1)-\frac{{\lambda }^{*}\left(k\right)P(k-1)\phi \left(k\right){\phi }^T\left(k\right)P(k-1)}{1+{\lambda }^{*}\left(k\right)G\left(k\right)}$$

(22)

Existence of the Lyapunov function $V\left(k\right)={\tilde{\theta }}^T\left(k\right)P^{-1}\left(k\right)\tilde{\theta }\left(k\right)$, where $\tilde{\theta }\left(k\right)={\theta }^{*}-\widehat{\theta }\left(k\right)$. $\tilde{\theta }\left(k\right)$ is given as follows.

$$\tilde{\theta }\left(k\right)=\tilde{\theta }(k-1)-{\lambda }^{*}\left(k\right)P\left(k\right)\phi \left(k\right)\delta \left(k\right)$$

(23)

$$\begin{array}{cc}\hfill V\left(k\right)& ={\tilde{\theta }}^T\left(k\right)P^{-1}\left(k\right)\tilde{\theta }\left(k\right)\hfill \\ \hfill & \hfill ={(\tilde{\theta }(k-1)-{\lambda }^{*}\left(k\right)P\left(k\right)\phi \left(k\right)\delta \left(k\right))}^T{P}^{-1}\left(k\right)(\tilde{\theta }(k-1)-{\lambda }^{*}\left(k\right)P\left(k\right)\phi \left(k\right)\delta \left(k\right))\hfill \\ \hfill & =V(k-1)+{\lambda }^{*}\left(k\right)(v^2\left(k\right)-\frac{{\delta }^2\left(k\right)}{1+{\lambda }^{*}\left(k\right)G\left(k\right)})\hfill \\ \hfill & \le V(k-1)+{\lambda }^{*}\left(k\right)({\widehat{\gamma }}^2-\frac{{\delta }^2\left(k\right)}{1+{\lambda }^{*}\left(k\right)G\left(k\right)})\hfill \end{array}$$

(24)

Define $f\left({\lambda }^{*}\left(k\right)\right)={\lambda }^{*}\left(k\right)({\widehat{\gamma }}^2-\frac{{\delta }^2\left(k\right)}{1+{\lambda }^{*}\left(k\right)G\left(k\right)})$,

$$V\left(k\right)\le V(k-1)+f\left({\lambda }^{*}\left(k\right)\right)$$

(25)

From Equation (16) that

$$0\le {\lambda }^{*}\left(k\right)\le \frac{1}{\widehat{\gamma }G\left(k\right)}\left[\right|\delta \left(k\right)|-\widehat{\gamma }]$$

It can be proved that $0\le {\lambda }^{*}\left(k\right)\le \frac{1}{{\widehat{\gamma }}^{2}G\left(k\right)}[{\delta }^2\left(k\right)-{\widehat{\gamma }}^2]$

$${\widehat{\gamma }}^2-\frac{{\delta }^2\left(k\right)}{1+{\lambda }^{*}\left(k\right)G\left(k\right)}<0$$

So

$$f\left({\lambda }^{*}\left(k\right)\right)<0$$

According to lemma 1, it can be concluded that $\eta \left(k\right)=-f\left({\lambda }^{*}\left(k\right)\right)$, $\chi \left(k\right)=V\left(k\right)$, $V\left(k\right)\le V(k-1)-(-f\left({\lambda }^{*}\left(k\right)\right))$, since $f\left({\lambda }^{*}\left(k\right)\right)<0$, there exists $-{\sum }_{k=1}^{\infty }f\left({\lambda }^{*}\left(k\right)\right)<\infty$, $V\left(k\right)$ converges to a variable on a finite field $V\left(c\right)$, $\underset{k\to \infty }{lim}V\left(k\right)=V\left(c\right)$.

$$V\left(k\right)\le V(k-1)$$

That is to say,

$${\tilde{\theta }}^T\left(k\right)P^{-1}\left(k\right)\tilde{\theta }\left(k\right)\le {\tilde{\theta }}^T(k-1)P^{-1}(k-1)\tilde{\theta }(k-1)$$

From Equation (11),

$$P^{-1}(k-1)\le P^{-1}\left(k\right)$$

It follows that

$$\left|\right|\tilde{\theta }\left(k\right)|{|}^2\le \left|\right|\tilde{\theta }(k-1)|{|}^2$$

Thus, $\left|\right|\tilde{\theta }\left(k\right)\left|\right|$ remains bounded and non-increasing.

(2) Based on (1), it follows that $V\left(k\right)$ is bounded and converges to $V\left(c\right)$.

$$V\left(k\right)={\tilde{\theta }}^T\left(k\right)P^{-1}\left(k\right)\tilde{\theta }\left(k\right)$$

we can deduce that $\widehat{\theta }\left(k\right)$ remains bounded.

$\phi \left(k\right)$ is within bounds, and $\widehat{\gamma }$ is related to residual data, so ${\delta }^{\prime }\left(i\right)$ is bounded. The expression for the residual sequence establishes the limit of the noise to be estimated.

$$\begin{array}{c}\widehat{\gamma }=max\left(\right|mean\left(can\right)-h\ast std\left(can\right)|,|mean\left(can\right)+h\ast std\left(can\right)\left|\right)\end{array}$$

Due to the fact that ${\delta }^{\prime }\left(i\right)$ is bounded, it implies that $\widehat{\gamma }$ is bounded. □

4. Engineering Application

For a single control loop with frequent changes in operating conditions, a long-period online slow-rate optimization tuning method for the controller is proposed.

4.1. Control-Oriented Identification

In the field of refining and petrochemical industries, the tuning of PID controllers is mostly performed offline and with long-term steady-state operation as the main focus. Online PID optimization tuning techniques are difficult to implement. Due to variations in feed composition, operating conditions, and equipment loads in industrial settings, control systems experience changes in operating conditions, leading to variations in the linear dynamic transfer function model of controlled processes. Additionally, during long-term operation of refining and chemical plants (over a year), it is challenging to maintain constant operating conditions, which can cause the PID controller to lose its original control effectiveness. Re-performing offline tuning under such circumstances is time-consuming and labor-intensive. Therefore, it is necessary to research online PID optimization tuning techniques specifically designed for underlying control loops of refining and chemical plants.

Step 1: Offline Identification Stage

(i) Collect a complete long-duration data segment while software runs continuously. Within a cyclic structure, perform recursive identification on this data segment to determine allowance for noise bound.

(ii) Repeat RSMI identification on this data segment to obtain identification model parameters.

Step 2: Online Identification and Model Application Stage

(i) Building upon offline identification stage, collect 3–5 recent days’ worth of data and divide these historical data into smaller segments. One piece of data is used as a training set, while another piece of data is used as a testing set. Perform RSMI identification on the training set to the determine allowance factor for noise bound. Divide the testing set into 10 pieces and perform recursive identification on each piece to identify final model. Continue recursively identifying model parameters for subsequent pieces.

(ii) Each small data segment will yield a model, resulting in a total of 10 models. Compare and analyze each model comprehensively, observing relevant control loops and their involvement in various operating conditions. Implement effective tuning based on observations and diagnose any anomalies in control loops.

 Remark 3.

Implementation Conditions for Engineering Applications: During the offline identification stage, collect a long-duration data segment consisting of several hundred thousand data points. The longer data segment should capture dynamic characteristics of different stages. By leveraging the historical dynamics of different stages, fully explore the dynamic characteristics of the model. Through dynamic data, stimulate internal characteristics of the system to ensure that the system is continuously excited for a significant period. This enables recursive identification to generate more accurate final models. It should be noted that in cases where system is less excited, models identified through recursive identification may be inaccurate, resulting in model mismatch.

4.2. Determination of Orders of System Model

Model order determination: With a fixed sampling time, the longer the response time of a control loop, the higher the order. Different types of control loops have different response times, such as temperature, liquid level, pressure, and flow. The order of the model is determined by the ratio of loop response time to sampling interval. In this paper, a sampling interval of 5 s is used in flow and pressure control loop, and a sampling interval of 7 s is used in liquid control loop. The step response settling time for flow control loop is approximately 45 s, for pressure control loop is approximately 1 min 15 s, and for liquid control loop is 3 min 17 s. The model orders for flow, pressure, and temperature control loops theoretically can be chosen as 9th order, 11th order, and 28th order.

5. Experiment

To verify the stability and robustness of the RSMI algorithm, the CARMA model and actual data of refining were simulated. Furthermore, the traditional SMI algorithm is introduced, which has fixed noise bound. By comparing the performance of these algorithms, a comprehensive evaluation of RSMI algorithm in real applications can be achieved.

5.1. Simulation

Considering the CARMA model

$$y_k=\frac{1z^{-1}+0.5z^{-2}}{1+1.5z^{-1}-0.7z^{-2}}{u}_k+\frac{1}{1+1.5z^{-1}-0.7z^{-2}}{v}_k$$

The equation above can be readily transformed into the forms of Equations (2) and (5). Real parameters are set to $\theta ={[-1.5,0.7,1,0.5]}^T$. In the simulation, we consider the presence of noise, where $vk$ represents a bounded random noise sequence following a uniform distribution with $\left|vk\right|<\gamma$, where $\gamma =1$. Furthermore, input data are a mixed univariate dynamic process with components of varying frequencies.

$$u\left(k\right)=\left\{\begin{array}{c}15+0.2sin\left(2k\right)cos\left(k\right),k\le L/6\\ 10+0.3sin\left(2k\right)cos(2k-\pi ),L/6<k\le L/3\\ 20+0.3sin\left(k\right)cos(k+\pi ),L/3<k\le L/2\\ 20+0.2sin\left(k\right)cos\left(3k\right),L/2<k\le 2L/3\\ 18+0.1sin\left(k\right)cos\left(2k\right),2L/3<k\le 5L/6\\ 8+0.2sin\left(k\right)cos(k+\pi ),k>5L/6\end{array}\right.$$

Determine output and input order based on model. The output order of model is $n=2$, the input order is $m=2$, and the initial value of the parameter is determined by Equation (17), $\theta ={[0.001,0.001,0.001,0.001]}^T$, $P={10}^3\ast I$. Maximum and minimum values of allowance factor are determined as $h_{up}=10$, $h_{down}=1$, and update interval is $h_{interval}=0.5$. The initial value of noise adaptive bound is set to $\gamma =1.0$, and initial value of ellipsoid generalized radius is set to ${\sigma }_k^2=1.0$; simulation steps L is 600. Determine the allowance factor through offline training. The determined parameters are shown in

Table 1. Optimal parameters.

h	$\mathit{\gamma }$	${\mathit{\sigma }}_{\mathit{k}}^2$
1.5	1.0	0.8

, model parameters were estimated through online training, and the convergence effect of the estimated parameters was observed.

Figure 2 shows the comparison results of two methods at optimal parameters, including comparison of parameter ${\theta }_1$, ${\theta }_2$, ${\theta }_3$ and ${\theta }_4$ estimates.

To study the effect of system fluctuation on performance of RSMI algorithm, root mean square error (RMSE) and mean absolute percentage error (MAPE) were introduced as evaluation indexes. RMSE is calculated by Equation (26), and MAPE is calculated by Equation (27). In order to compare the prediction effect of the model, the SMI algorithm with fixed noise bound is selected for comparison with the RSMI algorithm.For corresponding quantitative comparisons, the results are shown in

Table 2. Algorithm comparison.

Algorithm	RMSE	MAPE
SMI	0.1387	0.0837
RSMI	0.1303	0.0852

, representing average values obtained from 20 calculations.

$$RMSE=\sqrt{\frac{1}{L}\sum _{k=1}^L{(\widehat{y}\left(k\right)-y\left(k\right))}^2}$$

(26)

$$MAPE=\frac{1}{L}\sum _{k=1}^L|\frac{y\left(k\right)-\widehat{y}\left(k\right)}{y\left(k\right)}|$$

(27)

where, L denotes the length of training set, $y\left(k\right)$ represents actual output, and $\widehat{y}\left(k\right)$ signifies predicted output.

The comparison results demonstrate that the RSMI algorithm in parameter tracking compared to the SMI algorithm. However, it is important to note that the RSMI algorithm, which employs adaptive noise bounds, exhibits slightly higher time complexity. It is evident from the results that estimated noise bound progressively converges toward actual noise.

5.2. Industrial Equipment Experiment

The research data used in this study are from a distillation control loop involving detection instruments and actuating mechanisms in a domestic chemical plant.

Initially, dynamic historical data from a SISO control loop in refinery plant are acquired. Data segments are meticulously selected based on historical trends to establish a comprehensive training sample dataset. The historical data are systematically partitioned into time series, with a segment of 20,000 data points designated for the offline identification phase. During the offline identification phase, the optimal allowance factor is determined through rigorous analysis. Subsequently, a more recent subset of 10,000 data points is considered. Within this subset, first 5000 data points are utilized for identification, to determine the most suitable model parameters. The remaining 5000 data points are further divided into ten groups, organized according to time series, and subjected to multiple identification processes. These identifications enable a meticulous comparison between the tracking performance of predicted outputs and actual outputs. Finally, the degree of nonlinearity inherent in the identification model is thoroughly analyzed to provide a comprehensive understanding of the model’s performance.

5.2.1. Flow Control Loop

In flow control loop FIC1505, the step response settling time is measured to be 45 s. Following the Shannon sampling theorem, the sampling interval for flow loop is determined to be 5 s. Based on the finite impulse response (FIR) model order determination formula, the system model’s FIR model order is established to be 9. The output order of model is $n=3$, the input order is $m=3$, and the initial value of the parameter is determined by Equation (16), $\theta ={[0.001,0.001,0.001,0.001,0.001,0.001]}^T$, $P={10}^3\ast I$. The maximum and minimum values of the noise allowance factor are determined as $h_{up}=10$, $h_{down}=1$, and the update interval is $h_{interval}=0.5$. The initial value of the noise adaptive bound is set to $\gamma =2.0$; the initial value of the ellipsoid generalized radius is set to ${\sigma }_k^2=1.0$. Optimal parameters were determined through offline training, and the results of parameter determination are shown in

Table 3. Optimal parameters.

n	m	$\mathit{\gamma }$	${\mathit{\sigma }}_{\mathit{k}}^2$	h
3	3	2.0	0.84	1.5

. The final output model was determined in online training stage as shown in

Table 4. The final identification model.

${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_4$	${\mathit{\theta }}_5$	${\mathit{\theta }}_6$
−3.07	2.12	0.04	−0.28	0.2	−0.97

.

Using the recent 500 data sets as a test set, a comparison was made with the SMI algorithm with a fixed noise bound. The test results are illustrated in Figure 3 and Figure 4.

To assess the impact of system fluctuations on the performance of the RSMI algorithm, this study utilizes evaluation indices, namely, the Mean Absolute Percentage Error (MAPE) and the Root Mean Square Error (RMSE), calculated using Equations (26) and (27). The comparison between RSMI and SMI with a fixed noise bound is presented in

Table 5. Algorithm comparison.

Algorithm	RMSE	MAPE
SMI	0.192	0.223
RSMI	0.091	0.128

.

Based on the model identified by the RSMI algorithm, the final model is obtained, and the discrete open-loop transfer function is derived. The stability of the closed-loop transfer function is analyzed in the frequency domain, Nyquist and Bode plots of the open-loop transfer function are plotted to analyze the system’s stability, and test results are illustrated in Figure 5 and Figure 6.

According to the Nyquist criterion, it can be concluded that the closed-loop system is stable. The Nyquist curve does not intersect the negative real axis, indicating an infinite gain margin. There are no intersections with the phase of −180 degrees, indicating a good phase margin and good system stability.

5.2.2. Pressure Control Loop

In pressure control loop PIC1411, the step response settling time is measured to be 1 m 15 s. Following the Shannon sampling theorem, the sampling interval for pressure loop is determined to be 5 s. Based on FIR model order determination formula, the system model’s FIR model order is established to be 11. The output order of model is $n=6$, the input order is $m=5$, and initial value of parameter is determined by Equation (16), $\theta ={[0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001]}^T$, $P={10}^3\ast I$. The maximum and minimum values of allowance factor are determined as $h_{up}=10$, $h_{down}=1$, and the update interval is $h_{interval}=0.5$. The initial value of noise adaptive bound is set to $\gamma =2.0$, the initial value of ellipsoid generalized radius is set to ${\sigma }_k^2=1.0$. Optimal parameters were determined through offline training, and the results of parameter determination were shown in

Table 6. Optimal parameters.

n	m	$\mathit{\gamma }$	${\mathit{\sigma }}_{\mathit{k}}^2$	h
6	5	0.8	0.8	2.0

. The final output model was determined in the online training stage as shown in

Table 7. The final identification model.

${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_4$	${\mathit{\theta }}_5$	${\mathit{\theta }}_6$	${\mathit{\theta }}_7$	${\mathit{\theta }}_8$	${\mathit{\theta }}_9$	${\mathit{\theta }}_{10}$	${\mathit{\theta }}_{11}$
−1.378	0.425	−0.029	−0.044	−0.102	0.136	−0.001	−0.004	0.005	−0.002	0.002

.

Using the recent 500 data sets as a test set, a comparison was made with the SMI algorithm with a fixed noise bound. The test results are illustrated in Figure 7 and Figure 8.

To assess the impact of system fluctuations on the performance of the RSMI algorithm, this study utilizes evaluation indices, namely, the Mean Absolute Percentage Error (MAPE) and the Root Mean Square Error (RMSE), calculated using Equations (26) and (27). The comparison between RSMI and SMI with a fixed noise bound is presented in

Table 8. Algorithm comparison.

Algorithm	RMSE	MAPE
SMI	0.1077	0.0969
RSMI	0.1023	0.0959

.

Based on the model identified by RSMI algorithm, the final model is obtained, and the discrete open-loop transfer function is derived. The stability of the closed-loop transfer function is analyzed in frequency domain, Nyquist and Bode plots of open-loop transfer function are plotted to analyze system’s stability, and the test results are illustrated in Figure 9 and Figure 10.

According to the Nyquist criterion, it can be concluded that the closed-loop system is stable. The Nyquist curve does not intersect the negative real axis, indicating an infinite gain margin. There are no intersections with the phase of −180 degrees, indicating a good phase margin and good system stability.

5.2.3. Liquid Control Loop

In liquid control loop LIC801, the step response settling time is measured to be 3 min 17 s. Following the Shannon sampling theorem, the sampling interval for liquid loop is determined to be 7 s. Based on FIR model order determination formula, the system model’s FIR model order is established to be 28. The output order of model is $n=9$, the input order is $m=7$, and the initial value of the parameter is determined by Equation (16), $\theta ={[0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001,0.001]}^T$, $P={10}^3\ast I$. The maximum and minimum values of allowance factor are determined as $h_{up}=10$, $h_{down}=1$, and the update interval is $h_{interval}=0.5$. The initial value of noise adaptive bound is set to $\gamma =2.0$; the initial value of ellipsoid generalized radius is set to ${\sigma }_k^2=2.0$. Optimal parameters were determined through offline training, and the results of parameter determination are shown in

Table 9. Optimal parameters.

n	m	$\mathit{\gamma }$	${\mathit{\sigma }}_{\mathit{k}}^2$	h
9	7	1.0	1.2	3.0

. The final output model was determined in the online training stage as shown in

Table 10. The final identification model.

${\mathit{\theta }}_1$	${\mathit{\theta }}_2$	${\mathit{\theta }}_3$	${\mathit{\theta }}_4$	${\mathit{\theta }}_5$	${\mathit{\theta }}_6$	${\mathit{\theta }}_7$	${\mathit{\theta }}_8$	${\mathit{\theta }}_9$	${\mathit{\theta }}_{10}$	${\mathit{\theta }}_{11}$	${\mathit{\theta }}_{12}$	${\mathit{\theta }}_{13}$	${\mathit{\theta }}_{14}$	${\mathit{\theta }}_{15}$	${\mathit{\theta }}_{16}$
−1.328	0.111	−0.088	0.115	0.202	−0.045	0.0181	0.816	−0.686	0.303	−0.089	0.002	−0.086	−0.169	0.276	0.426

.

Using the recent 500 data sets as a test set, a comparison was made with the SMI algorithm with a fixed noise bound. The test results are illustrated in Figure 11 and Figure 12.

To assess the impact of system fluctuations on the performance of the RSMI algorithm, this study utilizes evaluation indices, namely, the Mean Absolute Percentage Error (MAPE) and the Root Mean Square Error (RMSE), calculated using Equations (26) and (27). The comparison between RSMI and SMI with a fixed noise bound is presented in

Table 11. Algorithm comparison.

Algorithm	RMSE	MAPE
SMI	0.167	0.027
SMIR	0.143	0.023

.

Based on the model identified by RSMI algorithm, the final model is obtained, and the discrete open-loop transfer function is derived. The stability of the closed-loop transfer function is analyzed in frequency domain, Nyquist and Bode plots of the open-loop transfer function are plotted to analyze the system’s stability, and the test results are illustrated in Figure 13 and Figure 14.

According to the Nyquist criterion, it can be concluded that the closed-loop system is stable. The Nyquist curve does not intersect the negative real axis, indicating an infinite gain margin. There are no intersections with the phase of −180 degrees, indicating a good phase margin and good system stability.

6. Conclusions

In view of the characteristics of slow time-varying, dynamic properties, complex distribution characteristics, and unknown stochastic noise in the petroleum refining process, a collective identification algorithm based on adaptive noise bounds is proposed. Compared to traditional collective methods, the collective identification algorithm based on adaptive noise bounds can make the system respond faster, be more robust, have more accurate modeling, and avoid unnecessary algorithm execution steps.

By introducing the residual sequence and adaptively estimating noise bounds, an improvement of the collective identification algorithm based on adaptive noise bounds is achieved, along with the introduction of optimization algorithms and tolerance factors to prevent local algorithm optima. This results in a more accurate model. The control algorithm is programmed in Python, and experimental results show that, compared to traditional collective identification algorithms, the collective identification algorithm based on adaptive noise bounds can further enhance modeling accuracy and exhibit better robustness.