Pattern-Moving-Modelling and Analysis Based on Clustered Generalized Cell Mapping for a Class of Complex Systems

Abstract

Considering a class of complex nonlinear systems whose dynamics are mostly governed by statistical regulations, the pattern-moving theory was developed to characterise such systems and successfully estimate the outputs or states. However, since the pattern class variable is not computable directly, this study establishes a clustered generalized cell mapping (C-GCM) to reveal system characteristics. C-GCM is a two-stage approach consisting of a pattern-moving-based description and analysis method. First, a density algorithm, named density-based spatial clustering of applications with noise (DBSCAN), is designed to obtain cell space $\Omega$ and the corresponding classification guidelines; this algorithm is initiated after the initial pre-image cells, and the total number of entity cells amounts to $N_s$. Then, the GCM provides several image cells based on a cell mapping function that refers to the multivariate ARMAX model. The global dynamic analysis employing both searching and storing algorithms depend on the attractor, domain of attraction, and periodic cell groups. At last, simulation results of two examples emphasise the practicality as well as efficacy of the technique suggested. The chief aim of this study was to offer a new perspective for a class of complex systems that could inspire research into nonmechanistic principles modelling and application to nonlinear systems.

1. Introduction

Research on system modelling and analysis has been hot for a long time in the control field in academia and industry. It is well known that there are, in industrial processes, a class of complex systems with a high energy consumption and substantial wastewater generation, e.g., Kiron blast furnaces, steel converters, sintering machines, cement rotary kilns, and so on. According to recent research, these systems have three distinguishing features: [1] (i) Certain physical or chemical processes of the system, like the transformation of the liquid phase to the solid phase during sintering, are largely governed by statistical rules rather than the traditional Newton principles mechanics. Specifically, particle size and sintering porosity are two actual variables that can be classified or measured by statistical rules. (ii) It is hard to construct a reliable system description using determined variables because of the complicated working mechanism and interaction effect between variables. Moreover, the relationship between the product quality and operating condition variables cannot be represented by deterministic equations, only existing in the statistical sense instead. (iii) The system involves an extremely intricate process that has nonlinear, distribution, and parameter perturbations. Since conventional modelling approaches are unfeasible, this investigation undertook a data-driven statistical scheme to uncover the system’s dynamic characteristics accurately.

A large body of evidence elucidated that pattern recognition (PR) was an appropriate tool to design and control the system in this case [2,3]. For instance, in application fields, PR has been applied in copper flash smelting and driving-cycle tests and is considered as a practical method for identifying the system structure [4,5]. Additional studies were carried out by Liu and Zhao with extended PR methods for different control systems [6,7]. In addition, PR was also applied to an air-conditioning system in building state estimation and fault diagnosis [8]. Over the past decades, PR has played a central role in the control system modelling domain in terms of data-driven approaches. Although PR has become commonly employed in industrial processes, the dynamical features of a system still rely on the state variables as opposed to pattern classes, which suggests there might be no statistic deviation among these methods. In fact, previous reports usually comply with the basic idea of designing different controllers for different systems, then judging the category of the current system or controller by a PR algorithm.

Taken together, Xu and colleagues claimed that this kind of production process conformed to the statistical motion formula in nature and proposed a new approach, namely, the pattern-moving theory (PMT) [9], to reveal system behaviours. Its fundamental idea was to extract the pattern class in the subspace of system operation and regard it as a variable (pattern class variable). The system motion principles were then explained based on how the pattern class variable varied over time. Because this variable is not measurable, it then needs to be converted to the calculable space. Recent research suggests that the solutions adopted for solving these computation challenges include generalized probability density evolution, interval numbers, and class centres [10,11,12]. Additionally, many studies have embraced the technique of integrating both implicit and explicit measures of the pattern class centre, where the modelling and control of the system are examined using the model permitted by the theory of control [13,14]. Of note, these methods are considered as linear for the modelling system. Further, the mapping process always results in the loss of a few statistical attributes in light of the mentioned methods.

Compared with the previous schemes, this paper develops a novel method that combines a clustering algorithm and generalized cell mapping (GCM) to address the computability of pattern class variables and maintain further statistical properties. Simple cell mapping (SCM), as a numerical method, was first introduced by professor Hsu to obverse the dynamics of nonlinear systems [15,16]. GCM is an extended version of SCM meant to uncover more of the global characteristics of the system, which accepts multiple image cells from a pre-image cell [17,18,19,20]. In general, the analysis of GCM leads to finding invariant sets, stable and unstable manifolds of saddlelike equilibrium states, and domains of attraction and their boundaries. The invariant sets are known as the persistent groups in the the Markov chain theory [21]. With regards to the nonlinear systems, GCM and its improved algorithms have been used in many fields such as fuzzy control, optimum trajectory planning in robotic systems, optimal control of populations, and semiactive control, among others [22,23,24]. Moreover, extensive efforts have been devoted to explore pursuit–evasion problems in the so-called differential games [25].

In this work, pattern class variables are obtained from the characteristic subspace of the system by using the density-based spatial clustering of applications with noise (DBSCAN) method [26], and the region of interest, $\Omega$, as well the cell space partition rules are acquired. The cell function can serve as a bridge between the computation of the category variables and the interpretation of the system’s dynamic properties and is constructed based on the multivariable autoregressive moving average with extra input (ARMAX) [27]. Whereafter, some pre-cells are randomly selected and mapped through the cell function to obtain the properties of the system.

Different from earlier proposals, the value and novelty of this research are summarized as follows.

• Concerning the computability of pattern class variables, G-GCM was designed as a two-stage method to describe and analyse the system peculiarity. First, the interesting space $\Omega$ (cell space) and initial number of cells were constructed by applying DBSCAN after data preprocessing. Then, the dynamic depiction and analysis of the system were described through GCM.
• Due to the system being governed by statistical law, the pre-image cells were selected randomly in a certain amount from uniform distribution $U(c_i,c_i+r_i)$, demonstrating the statistical behaviour of the system. Moreover, to greatly preserve the statistical properties between the state space and cell space, the study explored the multivariate ARMAX to estimate the initial maximum and minimum prediction outputs.
• A global analysis of system was examined according to attractors and domains of attraction, periodic cell groups, and transient cells, which made the stability and evolutionary process of the system clear. Searching algorithms for persistent group and domicile(s) of transient cells were proposed to discover more system dynamic characteristics in cell space.

An outline of the research is presented as follows. A brief summary of the cell concepts and problem formulation is given in Section 2. In the next section, a framework of evolution explained by the PMT on the basis of the clustered generalized cell mapping (G-GCM) is provided, and global analyses algorithms are developed for searching the attractor and its domain during cell mapping. Then, in Section 4, experimental simulations of the pattern-moving-based dynamic description with C-GCM are offered. Finally, Section 5 discusses a number of conclusions and future work.

2. Problem Statement and Preliminaries

2.1. Problem Formulation

Given a class of nonlinear systems governed by statistical laws, a system of dimension N is formulated in terms of N ordinary differential equations:

$$\dot{dx}=F(dx,t,u),x\in R^N,t\in R,\mu \in R^K,$$

(1)

where $dx$ is an N-dimensional pattern class vector, t denotes time, $\mu$ is a K-dimensional parameter vector, and F refers to a function vector with respect to $dx,t,\mu$. One of the motion properties of the system in N-dimensional space, namely $X^N$, is determined by the specified $\mu$. Then, the realization of the algorithm is implemented for a system structure and parameters through certain point and time period constants in the space. The main purpose of the kinetic analysis is to find out the general properties of the system, give analytical expressions using the vector flow, and obtain the general description of the system from the mapping information.

Remark 1.

Compared with the traditional description of a system, the PMT offers valuable insights into modelling ideas that depict the variations in pattern class variables to characterize a class of complex systems, which have two significant parts: the pattern class $dx$ and the moving space. Generally, during the operation of a real system, the working condition data (measurable and unmeasurable observable values) $S_n=\left[y_1,y_2,\cdots ,y_m\right]\in {\mathbb{R}}^m$ can be collected at any time $t_n$, and these data may change continuously over time. Let $P_i$ indicate the system working condition at time $t_i$, and each sampling time step $t_1,t_2,t_3,\cdots ,t_{n-1},t_n$ represents a realistic operation condition. With the variation in time, $t_1\to t_2\to t_3\to \cdots \to t_{n-1}\to t_n$, the production process conditions will also change accordingly from point $t_1$ to point $t_n$ in accordance with its inherent motion characteristics.

2.2. Preliminaries

A. 

Pattern class variables and its measurement

Definition 1.

Assume that sequences $\left\{S\right(k\left)\right\}$ and $\left\{Mx\right(k\left)\right\}$ represent the working condition data and the pattern sample data obtained by means of feature selection or extraction, respectively. Then, the pattern class variable $dx\left(k\right)$ can be calculated with the following two conversions:

$$\begin{array}{c}Mx\left(k\right)=T\left(S\right(k\left)\right)\hfill \\ dx\left(k\right)=M\left(Mx\right(k\left)\right)\hfill \end{array}$$

(2)

where $T(·)$ implies the process of feature extraction or selection, and $M(·)$ stands for the process of pattern classification.

From the above definition, it is clear that pattern class variables have two significant properties: (i) Pattern class variables $P_i$ are functions of time. As time changes, the value of the pattern class variable may vary accordingly. (ii) The attribute of a class, i.e., the pattern class variables have set and statistical properties. Hence, the pattern class is widely perceived as a set having the same or similar properties.

In this regard, if a variable is applied to represent all the pattern class data, which means it possesses coincident statistical characteristics in some manner, then applying the pattern moving of this variable can describe the system dynamics. However, the pattern class variable cannot be calculated straightforwardly, i.e., $P_1+P_2\ne P_3$, so it is not possible to implement the pattern class variables as scalars or vectors in a commutable space.

B. 

System operation space and Transformations

With respect to the patterns’ moving space, it can be exploited from actual production conditions on the basis of feature extraction and pattern classification, in order. Some basic concepts and logical relations of spatial transformations for different spaces are revealed in Figure 1 [9].

• System operating conditions’ subspace: If a subset of the system operating space referring to data on the operating conditions of a production process has been collected sufficiently long enough, then it contains enough information or features of the system. In general, all variables of a system operating space cannot be observed; moreover, it may be unnecessary. Thus, a sufficient database can be built over enough time from the production process, which provides enough support for pattern recognition or classification, and the corresponding system dynamics can then be basically characterized. For instance, 2–3 years of working condition data from a sintering process are considered sufficient to describe the overall operation of the system. Indeed, collecting data from the system operating subspace is the foundation of a data-driven approach.
• Condition feature subspace: By selecting or extracting features from the working-condition space, using techniques such as principal component analysis (PCA) or independent component analysis (ICA), the condition feature subspace can be structured, whose purpose is to minimize the computational space in terms of the PMT.
• Pattern-moving space: Pattern-moving space is defined as a virtual movement space because its structure and behaviour cannot be clearly expressed by a formula. We need to demonstrate how the pattern-moving space can be constructed successfully, after clustering or classifying from the condition feature subspace.
• Cell space: Since the pattern class variables are not computable in the pattern-moving space, they need to be mapped to another space, called the cell space, which consists of all entity cells in different dimensions, via discretizing the continuous state space, which is a computable space in terms of pattern class variables. It should be noted that there is a mutual mapping relationship between these two spaces.

As mentioned above, the basic ideas of depicting complex dynamic behaviours based on pattern class variables and the space transforms are stated and discussed. Furthermore, the GCM is introduced to solve the computability problem of pattern class variables.

2.3. The Generalized Cell Mapping

This part reviews a few essential concepts, terms, and properties associated with traditional GCM [15,16], which are used in the rest of the article. For this, the graph explaining cell mapping is rendered in following Figure 2 [28].

A. 

Definitions and abbreviations

• Cell: object with a unique index which implies a cell domain and different attributes (such an image or a pre-image).
• Cell index: cell property; a unique label.
• Pre-image cell: One or a variety of properties of a cell with respect to other cells. The cell domain that matches the cell is reached through the dynamics from the cell domain(s) determined by the pre-image(s).
• Image cell: One or a variety of properties of a cell with respect to other cells. The cell domain(s) indexed by the image(s) are reached with the dynamics from the cell domain that is related to the cell.
• Sink cell (SC): An extra cell, $z^{*}$ that indicates the area of the state space outside the CSS that is unbounded. Once a trajectory enters the sink, its evolution is terminated; to express this, the periods of the sink is equal to one.
• Cell domain: A contained area inside the state space, constituting a subset of the cell state space. A central point in the state space and the lengths along each dimension can serve as the most basic way to depict it.
• Cell sequence: a set of cells formed by subsequently tracking the image of cells (e.g., $\left\{3,5,z^{*}\right\}$, and $\left\{7,19,22,24,18,11,16\right\}$)
• Cell state space (CSS): The region of limited and discretized state space that arbitrary cell domains continually cover. In the basic situation, an n-dimensional state space may be separated using n-dimensional square rectangles of identical length.
• Periodic group (PG): A segment of a cell sequence that may make up a periodic motion is known as a periodic group (PG). A periodic group, with periodicity n, is formed by a periodic cycle of n cells (or succinctly an $n-P$ group). Every cell in the PG has period n, making it a periodic cell, or more simply, an $n-P$ cell.
• Group number (Gr): Every periodic group has a unique group number. All periodic cells within a PG and all transient cells leading to that PG have the same specific group number assigned.
• Step number(s): Property of a cell; the total number of steps recommended for reaching a PG. The step number of periodic cells is $St=0$, while the transient cells’ step number is $St>0$.
• Domain of attraction (DoA): The DoA of a PG with group number $Gr$ is a value sequence of positive integers that is the set of (transient) cells with the same group number $Gr$ and step number $St>0$. Thus, the DoA can be interpreted as the discretization of the basin of attraction.
• GCM solution: When the GCM process is successfully executed, each cell is assigned group number and step number properties together with the initial cell properties. We consider the cell state space and its properties the GCM solution, when every periodic group together with its DoA is discovered.

B. 

Generalized cell mapping

Definition 2.

A cell mapping indicates that a pre-image maps to an image through the cell function C; the evolution equation can be stated in the form:

$$z(n+1)=C\left(z\right(n),\mu )$$

(3)

where μ refers to the parameter vector, and C is cell function. Equation (3) indicates the $N-$array positive numbers map to $N-$array integers.

Definition 3.

An approach for describing the dynamic characteristics of the system is, from a pre-image cell containing a number of subcells, to obtain multiple image cells by transfer evolution with a certain probability of departure. The process can be expressed as:

$$\left\{\begin{array}{c}{z}^{\left(1\right)}(n+1)=p^{\left(1\right)}\left(n\right)z\left(n\right)\\ z^{\left(2\right)}(n+1)=p^{\left(2\right)}\left(n\right)z\left(n\right)\\ \cdots \\ \sum _{i=1}^{N_S}{p}^{\left(i\right)}=1,i\in \left[1,N_S\right]\end{array}\right.$$

(4)

where $z\left(n\right)$ is a pre-image cell at time n, $z^{\left(i\right)}(n+1)$ is an image cell at the next time, $n+1$, $p^{\left(i\right)}$ denotes $p^{\left(i\right)}$ in the ith pre-image cell evolution, and $N_s$ is the total number of cells.

Definition 4.

Cell probability vector: assuming $p_i\left(n\right)$ is the ith cell probability at time n, then $P\left(n\right)$ is a cell probability vector at the same time for all cells.

$$P\left(n\right)=\left[p_1\left(n\right),p_2\left(n\right),\cdots ,p_i\left(n\right)\right]i\in \left[1,N_S\right]$$

(5)

Definition 5.

Transitional probability matrix P: the transition probability $p_{ij}\left(n\right)$ is a conditional probability that cell j at time n transforms into the cell i at the next time:

$$p_j\left(n\right)=Prob\left\{i\mathrm{at}=n+1|j\mathrm{at}=n\right\}i,j\in \left\{N_S\right\}$$

(6)

P consists of all $p_{ij}\left(n\right)$ that are affiliated to time n. However, considering that the number of image cells is always limited in practical problems, it is approximately considered to be $p_{ij}\left(n\right)=p_{ij}$, introducing the notion of image cell set $A\left(j\right)$. It can be seen that

$$i\in A\left(j\right)\mathit{if}\mathit{and}\mathit{only}\mathit{if}{p}_{ij}>0$$

(7)

$p_{ij}\left(n\right)$ and $p_{ij}$ have the following properties:

$$\begin{array}{c}{p}_i\left(n\right)\ge 0,i\in S,\sum _{i\in S}{p}_i\left(n\right)=1\\ p_{ij}\left(n\right)\ge 0,\sum _{i\in S}{p}_{ij}=1,\sum _{i\in A\left(j\right)}{p}_{ij}=1\end{array}$$

(8)

Note: matrix P is without a null column.

Finally, the GCM can be shaped into:

$$p(n+1)=Pp\left(n\right)$$

(9)

By Equation (9), a finite, discrete, stationary Markov chain is defined (see, e.g., Isaacson and Madsen [29]). The dynamic behaviour of the system is entirely determined by the transition matrix P. Through employing the Markov chain theory for analysing the properties of P, the attractors and basins of attraction can be established. Next, some relevant properties of this theory are introduced.

Property 1.

For the transitional matrix of probabilities P, the total value of any column equals one.

$$\sum _{i=1}^{N_s}{p}_{ij}=1for1\le j\le N_s.$$

(10)

Property 2.

The probability of the system is preserved when the cell state space is closed without loss of probability, such that:

$$\sum _{i=1}^{N_i}{p}_i\left(k\right)=1fork\ge 0.$$

(11)

Property 3.

There exists at least one closed set of cells for a bounded Markov chain. This set is invariant and is called the persistent group.

Property 4.

Let us examine the sub-matrix $P_l$ representing the transition probability connected to the lth persistent group. If this group of cells represents the $period-K$ movement of the entire system, it can be separated into the following form:

$$P_l=\left[\begin{array}{cccc}& & & P_{l,K}\\ P_{l,1}& & & \\ & \ddots & & \\ & & P_{l,K-1}\end{array}\right].$$

(12)

where $P_{l,j}(1\le j\le K)$ acts as a sub-matrix of certain dimension. As soon as $P_l$ has more than one nonzero diagonal component, the period of the persistent group is one. It is also referred to as a periodic group. Later, we discuss a strategy for determining the duration of a persistent group. When it comes to the theory of Markov chains, two types of cells can be distinguished under the GCM:persistent cells and transient cells. A global analysis focuses on the spatial location of the individual attractors and their domains of attraction (global asymptotic stability), the stable and unstable manifolds of the unstable solutions, and the effect of parameter variations on the global structure of the phase space.

3. Design of Dynamic Description and Analysis Based on Clustered GCM

3.1. The Clustering Algorithm and Cell Function

A. 

The Clustering Algorithm

The density-based clustering algorithm known as density-based spatial clustering of applications with noise (DBSCAN) was proposed by Martin Ester et al. in 1996 [26]; it works on the assumption that clusters are dense regions in space separated with regions of lower density.

This method, which is believed to be nonparametric, accepts an array of points in space and clusters each of them that are closely spread together (i.e., the ones with a lot of neighbours nearby). It then identifies as outliers the points that are situated in low-density regions, for example, whose closest neighbours are too far away. Here, we give an overview of the DBSCAN algorithm procedure. (See Algorithm 1).

Algorithm 1 DBSCAN clustering algorithm process

Input:  The sample set $D=(x_1,x_2,\dots ,x_m),$ neighbourhood parameter $(ϵ,MinPts),$ sample distance measurement method; Output:  Cluster division C; 1: Initialize the core target set $\Omega =0$, the number of clusters $k=0$, the unvisited sample set $\Gamma =D,$ the cluster partition $C=0$; 2: For $j=1,2,\dots m,$ find all the core objects, follow the instructions below: 3:   (a) Finding the $ϵ$-neighbourhood subsample set $N_{ϵ}\left(x_j\right)$; 4:   (b) If the number of subsample set satisfies $\left|N_{ϵ}\left(x_j\right)\right| \ge MinPts,$ add sample $x_j$ to the core target set $\Omega =\Omega \cup \left\{x_j\right\}$; 5: If set $\Omega =0$, then the algorithm ends, otherwise go to step 6; 6: Select a core object o from set $\Omega$, and  initialize the current cluster core object queue ${\Omega }_{cur}=\left\{o\right\}$. Initialize class number $k=k+1,$ initialize the current set of cluster samples $C_k=\left\{0\right\}$, update the collection of unvisited samples $\Gamma =\Gamma -\left\{o\right\}$; 7: If the current cluster core object queue ${\Omega }_{cur}=0$, then the current cluster $C_k$ is generated; update cluster division $C=\{C_1,C_2,\cdots ,C_k\}$, update the core object collection $\Omega =\Omega -C_k,$ go directly to step 5. Otherwise update set $\Omega =\Omega -C_k.$ 8: Extract a core object $o^{\prime }$ from queueing ${\Omega }_{cur}$, find all $ϵ$-neighbourhood subsample set $N_{ϵ}\left(o^{\prime }\right),$ using neighbourhood distance threshold $ϵ$, let $\Delta =N_{ϵ}\left(o^{\prime }\right)\cap \Gamma$, update the current cluster sample collection $C_k=C_k\cup \Delta$, update unvisited sample set $T=T-\Delta$, update ${\Omega }_{cur}={\Omega }_{cur}\cup (\Delta \cap \Omega )-o^{\prime }$, go to step 7; 9: The output is cluster Division $C=\left\{C_1,C_2,\dots ,C_k\right\}$.

Remark 2.

In contrast to traditional clustering algorithms, the main difference of DBSCAN is that it does not use a presupposed number of k, which means it can be applied to arbitrarily shaped data clusters and find outliers. When dealing with complicated industrial processes, because the data set is typically dense and nonconvex, DBSCAN can be employed with outstanding applicability; $(ϵ,MinPts),$ is calculated by setting a threshold value in the moving space.

B. 

The Cell Function

After the construction of the cell space, there is a need for a mapping in a different space to analyse the system behaviours. Considering the high dimensionality and complexity of the moving space after extracting features from the original data, the multivariable ARMAX was seen as a feasible approach. Consequently, the identification algorithm for the multivariable ARMAX was derived and applied with the maximum likelihood estimation (MLE). In general, the multivariable time series function is often described as:

$$\left(z^{-1}\right)Y\left(z\right)-B\left(z^{-1}\right)U\left(z\right)+D\left(z^{-1}\right)V\left(z\right)$$

(13)

where

$$\left\{\begin{array}{cc}Y\left(z\right)={[y_1\left(z\right),y_2\left(z\right),\cdots ,y_n\left(z\right)]}^T\hfill & \\ U\left(z\right)={[u_1\left(z\right),u_2\left(z\right),\cdots ,u_r\left(z\right)]}^T\hfill & \\ V\left(z\right)={[v_1\left(z\right),v_2\left(z\right),\cdots ,v_n\left(z\right)]}^T\hfill \end{array}\right.$$

(14)

denote an ($n\times 1$)-dimension output vector, ($r\times 1$)-dimension control vector, and $n\times 1$ white noise vector with a normal distribution of the mean, respectively;

$$\left\{\begin{array}{c}A\left(z^{-1}\right)=A_0+A_1{z}^{-1}-\cdots -A_m{z}^{-m}\\ B\left(z^{-1}\right)=B_0+B_1{z}^{-1}+\cdots +B_m{z}^{-m}\\ D\left(z^{-1}\right)=D_0+D_1{z}^{-1}+\cdots +D_m{z}^{-m}\end{array}\right.$$

(15)

is the backward operator $z^{-1}$ of the matrix polynomial; here,

$$\begin{array}{cc}{A}_i=\left[\begin{array}{cccc}{a}_{11}^i& a_{12}^i& \cdots & a_{1n}^i\\ a_{21}^i& a_{22}^i& \cdots & a_{2n}^i\\ ⋮& ⋮& & ⋮& \\ a_{n1}^i& a_{n2}^i& \cdots & a_{nn}^i\end{array}\right]& B_i=\left[\begin{array}{cccc}{b}_{11}^i& b_{12}^i& \cdots & b_{1r}^i\\ b_{21}^i& b_{22}^i& \cdots & b_{2r}^i\\ ⋮& ⋮& & ⋮& \\ b_{r1}^i& b_{r2}^i& \cdots & b_{rr}^i\end{array}\right]\\ D_i=\left[\begin{array}{cccc}{d}_{11}^i& d_{12}^i& \cdots & d_{1n}^i\\ d_{21}^i& d_{22}^i& \cdots & d_{2n}^i\\ ⋮& ⋮& & ⋮& \\ d_{n1}^i& d_{n2}^i& \cdots & d_{nn}^i\end{array}\right]\end{array}$$

(16)

Equation (13) can be decomposed into n submodels as follows:

$$y_i\left(k\right)={\Psi }^T\left(k\right){\theta }_i^T+v_i\left(k\right),i=1,\cdots ,n$$

(17)

where

$$\begin{array}{c}\hfill {\theta }_i^T=[a_{i1}^1,\cdots ,a_{in}^1,\cdots ,a_{i1}^m,\cdots a_{in}^m;b_{i1}^0\cdots ,b_{ir}^0\cdots ,b_{i1}^m,\cdots b_{ir}^m;\\ \hfill d_{i1}^1,\cdots ,d_{in}^1,\cdots ,d_{i1}^m,\cdots d_{in}^m]=[{\theta }_i^1,{\theta }_i^2,\cdots ,{\theta }_i^{2mn+(m+1)r}]\end{array}$$

(18)

$$\begin{array}{c}\hfill \begin{array}{c}1{\Psi }^T\left(k\right)=[Y^T(k-1),\cdots ,Y^T(k-m);U^T\left(k\right),\cdots ,U^T(k-m)\\ V^T(k-1),\cdots ,V^T(k-m)]\end{array}\end{array}$$

(19)

This transforms a model with r inputs and n outputs into a model with n single outputs and multiple outputs.

To address the limitations of the classic MLE in SISO scenarios, we recursed this method to multiple variables. The equation of the single-variable ARMAX is described as:

$$\begin{array}{ccc}\hfill A\left(z^{-1}\right)y\left(z\right)& =& B\left(z^{-1}\right)u\left(z\right)+D\left(z^{-1}\right)v\left(z\right)\hfill \\ \hfill y\left(k\right)& =& {\psi }^T\left(k\right)\theta +v\left(k\right)\hfill \end{array}$$

(20)

where

$$\begin{array}{c}\theta =\left[a_1,\cdots a_m,b_0,b_1,\cdots b_m,d_1,\cdots d_m\right]\\ {\varphi }^T\left(k\right)=[y(k-1),\cdots y(k-m),u\left(k\right),\cdots u(k-m),\\ v(k-1),\cdots v(k-m)]\end{array}$$

(21)

The nonmeasurable noise terms of Equation (19) was replaced using the estimated value from the MLE. The formula is described as:

$$y\left(k\right)={\widehat{\Psi }}^T\left(k\right)\widehat{\theta }(k-1)+e\left(k\right)$$

(22)

The MLE recursive form of the above single ARMAX model is as follows:

$$\widehat{\theta }(k+1)=\widehat{\theta }\left(k\right)+K\left(k\right)e(k+1)$$

(23)

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

(24)

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

(25)

$$e(k+1)=y(k+1)-{\widehat{\Psi }}^{\tau }(k+1)\widehat{\theta }\left(k\right)$$

(26)

$$\widehat{v}(k+1)=e(k+1)$$

(27)

$$\begin{array}{cc}\hfill {\phi }^T(k+1)=-& [\frac{\partial e(k+1)}{\partial a_1}\cdots \frac{\partial e(k+1)}{\partial a_m},\frac{\partial e(k+1)}{\partial b_0}\cdots \frac{\partial e(k+1)}{\partial b_m},\hfill \\ & \frac{\partial e(k+1)}{\partial d_1}\cdots \frac{\partial e(k+1)}{\partial d_m}]\hfill \end{array}$$

(28)

${\phi }^T(k+1)$ can also be written as:

$$\begin{array}{cc}\hfill {\phi }^T(k+1)=& [y^{\prime }\left(k\right),\cdots ,y^{\prime }(k-m+1),u^{\prime }(k+1),\cdots ,u^{\prime }(k-m+1),\hfill \\ \hfill & {[(n-l)-(n_i-l_i)]}^{n_i-l_i}·\left[{(n-l)}^2-\sum _{j=1}^p{(n_i-l_i)}^2\right].\hfill \end{array}$$

(29)

Converting to a recurrence equation yields:

$$\left\{\begin{array}{c}{y}^{\prime }\left(k\right)=y\left(k\right)-{\widehat{d}}_1{y}^{\prime }(k-1)-\cdots -{\widehat{d}}_n{y}^{\prime }(k-m)\\ u^{\prime }(k+1)=u(k+1)-{\widehat{d}}_1{u}^{\prime }\left(k\right)-\cdots -{\widehat{d}}_m{u}^{\prime }(k-m+1)\\ e^{\prime }\left(k\right)=e\left(k\right)-d_1{e}^{\prime }(k-1)-\cdots -d_m{e}^{\prime }(k-m)\end{array}\right.$$

(30)

Comparing Equations (17) and (20), and noting how Equations (28)–(30) change through (5), it is straightforward to derive the MLE formula for Equation (17) as follows:

$${\widehat{\theta }}_i(k+1)={\widehat{\theta }}_i\left(k\right)+K_i\left(k\right)e_i(k+1)$$

(31)

$$K_i\left(k\right)=P_i(k+1){\phi }_i(k+1)=\frac{P_i\left(k\right){\phi }_i(k+1)}{\lambda +{\phi }_i^T(k+1)P_i\left(k\right){\phi }_i(k+1)}$$

(32)

$$P_i(k+1)=\left[1-K_i\left(k\right){\phi }_i^T(k+1)\right]P_i\left(k\right)/\lambda$$

(33)

$$e_i(k+1)=y_i(k+1)-{\widehat{\Psi }}^{\tau }(k+1){\widehat{\theta }}_i\left(k\right)$$

(34)

$${\widehat{v}}_i(k+1)=e_i(k+1)$$

(35)

$$\begin{array}{c}\hfill {\phi }_i^T(k+1)=-[\frac{\partial e(k+1)}{\partial a_i^1},\cdots ,\frac{\partial e(k+1)}{\partial a_{in}^1},\cdots ,\frac{\partial e(k+1)}{\partial a_{i1}^m},\cdots ,\frac{\partial e(k+1)}{\partial a_i^m};\\ \hfill \frac{\partial e(k+1)}{\partial b_i^0},\cdots ,\frac{\partial e(k+1)}{\partial b_{ir}^0},\cdots \frac{\partial e(k+1)}{\partial b_{i1}^m},\cdots \frac{\partial e(k+1)}{\partial b_{ir}^m};\\ \hfill \frac{\partial e(k+1)}{\partial d_{i1}^1},\cdots ,\frac{\partial e(k+1)}{\partial d_{in}^1},\cdots ,\frac{\partial e(k+1)}{\partial d_i^m},\cdots ,\frac{\partial e(k+1)}{\partial d_{in}^m}]\end{array}$$

(36)

When the partial derivatives are applied to the terms of the aforementioned equation, ${\phi }^T(k+1)$ refers to:

$$\begin{array}{c}\hfill {\phi }_i^T(k+1)=[y_1^{\prime }\left(k\right),\cdots ,y_n^{\prime }\left(k\right),\cdots ,y_1^{\prime }(k-m+1),\cdots ,y_n^{\prime }(k-m+1);\\ \hfill u_1^{\prime }(e+1),\cdots ,u^{\prime }(k+1),\cdots ,u^{\prime }(k-m+1),\cdots ,u^{\prime }(k-m+1);\\ \hfill e_1^{\prime }\left(k\right),\cdots ,e_r^{\prime }\left(k\right),\cdots ,e_1^{\prime }(k-m+1),\cdots ,e_i^{\prime }(k-m+1)]\end{array}$$

(37)

The elements in Equations (37) are determined by the following recurrence equation:

$$\left\{\begin{array}{c}{y}_1^{\prime }\left(k\right)=y_1\left(k\right)-\widehat{d_{i1}^1}{y}_1^{\prime }(k-1)-\cdots -\widehat{d_{i1}^m}{y}_1^{\prime }(k-m)\\ ⋮\\ y_n^{\prime }\left(k\right)=y_n\left(k\right)-\widehat{d_{in}^1}{y}_n^{\prime }(k-1)-\cdots -\widehat{d_{in}^m}{y}_n^{\prime }(k-m+1)\\ \\ u_1^{\prime }(k+1)=u_1\left(k\right)-\widehat{d_{i1}^1}{u}_1^{\prime }(k-1)-\cdots -\widehat{d_{i1}^m}{u}_1^{\prime }(k-m+1)\\ ⋮\\ u_r^{\prime }\left(k\right)=u_r(k+1)-\widehat{d_{ir}^1}{u}_r^{\prime }(k-1)-\cdots -\widehat{d_{ir}^m}{u}_r^{\prime }(k-m)\\ \\ e_1^{\prime }\left(k\right)=e_1\left(k\right)-\widehat{d_{i1}^1}{e}_1^{\prime }(k-1)-\cdots -\widehat{d_{i1}^m}{e}_1^{\prime }(k-m)\\ ⋮\\ e_n^{\prime }\left(k\right)=e_n\left(k\right)-\widehat{d_{in}^1}{e}_n^{\prime }(k-1)-\cdots -\widehat{d_{i1}^m}{e}_n^{\prime }(k-m)\end{array}\right.$$

(38)

Equations (31)–(38) are the MLE of Equation (16), where $i=1,\cdots ,n$; $\lambda$ is a forgetting factor, $0.9<\lambda \le 1$.

We initialized the settings as ${\widehat{\theta }}_i\left(0\right)=0,P_i\left(0\right)=\alpha ,I$, I denoting a unit matrix, $\alpha$ is a very large positive value. $e_i\left(j\right)=y_i\left(j\right)=u_i\left(j\right)=e_i^{\prime }\left(j\right)=u_i^{\prime }\left(j\right)=0$, $i=1,\cdots n;$ $j=0,-1,\cdots ,1-m$.

All in all, multiple inputs and outputs’ ARMAX parameters ought to be identified successfully in light of the inference of Equation (38).

3.2. System Described by Clustered GCM

GCM describes the transfer of numerous pre-image cells to many image cells that is capable of maintaining the statistical properties of a pattern-moving system. However, cell function C and the principle of selecting $\widetilde{dx}$ are not covered before GCM research. The transition between the pattern space and cell space is outlined in Figure 3.

Based on the correlation between the pattern-moving and cell spaces in Figure 3, a system description with improved GCM can be expressed as:

$$\left\{\begin{array}{c}dx(k+1)=M\left[\widetilde{dx}(k+1)\right]\hfill \\ =M\left\{C\left[dx\right(k),dx(k-1),\cdots ,dx(k-n)\right.\hfill \\ \left.u(k-\tau ),u(k-\tau -1),\cdots ,u(k-\tau -m)]\right\}\hfill \\ \widetilde{dx}(k+1)\in \{{\widetilde{dx}}_{max}(k+1),{\widetilde{dx}}_{min}(k+1),z^{*}\}\hfill \\ d{x}_i(k+1)\sim U(c_i,c_i+r_i).\hfill \end{array}\right.$$

(39)

where $M(·)$ denotes the classification of prediction pattern class variable $\widetilde{dx}(k+1)$, C is a generalized cell function that refers to the ARMAX, the value of $\widetilde{dx}(k+1)$ can be acquired in set $\{{\widetilde{dx}}_{max}(k+1),{\widetilde{dx}}_{min}(k+1),z^{*}\}$ at time $k+1$. It should be noted that $\widetilde{dx}(k+1)$ and $dx(k+1)$ were defined as a pre-image cell and an image cell with respect to the cell space, respectively. Here, the pre-image follows the uniform distribution $U(c_i,c_i+r_i)$, when the specific number of dimensions i is selected. Then, another k is set to $k=(0,1,2,3,\cdots ,k-n)$, and $z^{*}$ is the sink cell of the system, which satisfies $z^{*}=C\left(z^{*}\right)$ with $k⟶\infty$ meaning that we obtained the mean of ${\widetilde{dx}}_{max}(k+1)$ and ${\widetilde{dx}}_{min}(k+1)$.

We need to point out that the mapping calculation between two spaces is an iterative scheme by means of a generalised cell metric and classification mapping. In particular, as shown in the pattern-moving space in time k, the pattern variable $dx\left(k\right)$ and class radius $r_k$ are obtained, then they are mapped to the cell space by GCM, and the corresponding initial predictions can be gathered at the same time, such as ${\widetilde{dx}}_{max}\left(k\right)$ and ${\widetilde{dx}}_{min}\left(k\right)$. The classification mapping $M(·)$ to the pattern-moving space is designed to estimate $dx(k+1)$ at once. On the whole, the dynamic evolution of a complicated system can be described adequately by Equation (39) in the pattern-moving space.

If proper search and storing algorithms are developed, GCM may capture extra movement features of the system in addition to obtaining the system predictions using Equation (39)’s mapping. Therefore, a global analysis with clustered GCM is presented in the following.

3.3. System’s Global Analysis with Clustered GCM

GCM is an expansion of simple cell mapping, which can better reveal the statistical rules concerning the system built around a Markov chain. Additionally, the objective of invariant sets and the global analysis of nonlinear dynamical systems is to find instability solutions as well as invariant set zones of attractions. Thus, the cell mappings across the whole cell space are vital for the global analysis. GCM can be attempted for a global analysis of dynamical systems with moderately large dimensions, as is demonstrated in a later step, thanks to parallel computing.

GCM emanates from a directed graph: the closed, connecting cell groups in GCM make up the persistent groups, on which the depth-first search (DFS) technique is performed to figure out whether groups are persistent [29]. However, in graph theory, the strongly connected components (SCC) decomposing technique is an alternative to the DFS. SCC search visits a cell only once and is thus quite efficient. The computational cost of the SCC technique exceeds O(E + V) in which E and V are the number of edges and nodes in a graph. This approach enables the discovery of both stable and unstable manifold solutions that explore the GCM using inverse methodologies [30,31,32]. (See Algorithm 2).

Algorithm 2 Searching persistent group

Input: $GCM$, P Otput: $Pg$, $Gr$ 1: $SCC\leftarrow$ Tarjan(P) 2: $g\leftarrow 1,Gr\leftarrow 0,Pg\leftarrow 0$ 3: for $i=1$, $max\left(SCC\right)$ 4:    if $|SC{C}_i|=1$ AND $P\left(SC{C}_i\right)=SC{C}_i$ 5:       $Pg\left(SC{C}_i\right)\leftarrow 1$ 6:      $Gr\left(SC{C}_i\right)\leftarrow g$ 7:      $g\leftarrow g+1$ 8:   else 9:      $T\leftarrow DFS(P,SC{C}_i\left(1\right))$ 10:      $F\leftarrow DFS(P^{-1},SC{C}_i\left(1\right))$ 11:      if $|T\cap F|=|SC{C}_i|$ 12:         $Pg\left(SC{C}_i\right)\leftarrow 1$ 13:         $Gr\left(SC{C}_i\right)\leftarrow g$ 14:         $g\leftarrow g+1$ 15:      end 16:   end 17: end

Cells that are absorbed to more than one PG form the boundaries of the DoA. One basic manner for recognising transient cells constituting the boundaries of the DoA of different persistent groups is making use of the domicile matrix $Dm$, which is included in Algorithm 3. It should be noted that the state starting from the transient cell i ends up in the jth persistent group, as represented by the non-zero element of $Dm(i,j)$. Consequently, we argue that cell i will meet the boundary of at least two attraction domains if the ith row of $Dm$ has more than one non-zero element. Cell i may have more than one residence, particularly in high-dimensional dynamical systems. In a higher-dimensional cell state space, the basin boundaries usually become too difficult to visualize.

Algorithm 3 Storing domicile(s) of transient cells

Input: $Pg$, $Gr$, P Output:  $Dm\in R^{N\times g}$ 1: $Dm\leftarrow 0$ 2: for $i=1$, $max\left(SCC\right)$ 3:    $P{R}^1\leftarrow 0,TA{R}^1\leftarrow 0,$ 4:    $TA{R}^1\leftarrow Domi\_init(Gr,i)$, 5:      $n\leftarrow 1,$ 6:   while true 7:      $\left[P{R}^{n+1},TA{R}^{n+1},Dm\right.]\leftarrow$ DOmi_trans$(P{R}^n,TA{R}^n,P,Dm,i)$ 8:      if $|P{R}^{n+1}-P{R}^n|=0$ 9:         break 10:      end 11:      $P{R}^n\leftarrow P{R}^{n+1}$, $TA{R}^n\leftarrow TA{R}^{n+1}$ 12:      $n\leftarrow n+1$ 13:   end 14: end

Remark 3.

It is worthy to point out that this algorithm takes $TA{R}^n$ as the initial target set and $P{R}^n$ as the processing set. The label $TA{R}^{n+1}=1$ on the z image cells means that their processing starts at the next iteration. When the vectors $P{R}^{n+1}$ and $P{R}^n$ are identical, that is, once the number of processed cells stays unchanged in subsequent iterations, the expansion comes to an end. Here, each row of $Dm$ denotes the domiciles of a transient cell.

4. Simulation

To sum up, the proposed algorithm describes and analyses the global property of a system in accordance with the PMT. Establishing this procedure consists of the following steps:

• Construction of the pattern class variables and cell space

Step 1: Collect a large quantity of historical data to create a system’s operating subspace using data filtering and principal component analysis (PCA) for actual industrial processes. Whereafter, the DBSCAN algorithm is adopted for obtaining pattern class variables by applying it to different operating conditions of the system.

Step 2: The area of interest $\Omega$ (cell space), entity cell number $N_s$, and initial pre-image cells are determined based on categorical rules using Algorithm 1.

• Acquisition of the cell function and calculation of the image cells

Step 3: A multivariate ARMAX, the so-called cell function, is assessed by applying Equations (31)–(38) to search for the image cells and show the classification of different cells, such as the persistent cell group, transition cell group, and domains of attraction of invariant sets.

Step 4: Based on Algorithms 2 and 3, by creating a multidimensional array $Grps$ that includes the group number of the attractor in which each cell resides, the number of attractor cycles, and the convergence step, the image cells are searched and classified to a persistent group and a transition group ($Gr,P,St$).

• Prediction of pattern class variables and analysis of the dynamics of system

Step 5: By the classification mapping $F(·)$ in Equation (2), multiple pre-image cells are randomly selected with a uniform distribution $U(c_i,c_i+r_i)$. $c_i$ and $r_i$ are the class centre and radius, respectively. To maintain the statistical properties of the system, the maximum and minimum initial prediction outputs are reserved in the cell space.

Step 6: The final forecast outputs $\widehat{dx}\left(k\right)$ are exploited through classification $M(·)$ to model the complex system. Meanwhile, the reliability of the results is evaluated by the root-mean-square error (RMSE) among the real pattern class variables and its prediction outcomes with the goal of measuring the usefulness of the proposed approach.

• Case 1: Sintering Production Process

Consider Anyang Iron and Steel Co., Ltd.’s (Anyang, China) sintering procedure for manufacturing. The temperature of the three bellows (T20, T21, and T22) is a consequence of the sintering process and uses as its input the ignition temperature u. Both the original input and output data are shown in the Figure 4.

Figure 4 shows that the sintering process data have noise and are unequally distributed, thus data preprocessing like a filtering operation is usually necessary. After standardising the original data noted above, a feature extracting step was applied to capture significant information, such as applying PCA to produce the first two principal components, with the contribution rate reaching 95%. These data were then separated into nine pattern classes using the DBSCAN method, shown in

Table 1. The class centre and radius for two dimensions of the pattern-moving space for case 1.

Class No.	Class Centre ${\mathit{c}}_{\mathit{i}}$	Class Radius ${\mathit{r}}_{\mathit{i}}$
0	(33.69, −19.21)	40.39
1	(−45.46, −7.90)	31.92
2	(−7.72, −12.85)	57.88
3	(13.61, 34.04)	54.07
4	(−77.80, −4.0)	70.63
5	(78.60, −17.29)	181.06
6	(17.26, 6.07)	20.97
7	(−22.98, 14.43)	72.55
8	(48.77, 9.31)	57.96

, and the cell space $\Omega$ and the number of initial cells $N_s=9\times 9$ were calculated from the class centre and radius.

Table 1 presents the data provided by DBSCAN based on the sintering process; the number of pattern classes, together with their radius and centre $r_i$ and $c_i$, are found and explained. It should be noted that since the statistics of the system were unknown, the selection of crucial parameters $ϵ,MinPts$ with Algorithm 1 was randomized to some extent, and $ϵ=0.5$ and $MinPts=10$ were considered appropriate in this experiment. The distribution of different pattern classes is shown in Figure 5.

Based on the two-dimensional pattern-moving space, the cell function from the multivariate ARMAX model structure was obtained and its parameter identification was performed to analyse the complex system according to Equations (31)–(38). Here, the Akaike information criterion (AIC) was utilized to estimated the suitable structure appearing in Figure 6.

From the value in Figure 6, the structure of ARMAX(3,3,0) was regarded as reasonable for the sintering production process. According to the parameter estimation inference in Section 3, the cell function can be recognized and represented using a multivariable linear model as follows:

$$\begin{array}{c}y\left(t\right)=0.367537+0.9599641\left(t\right)+0.864634(t+1)-0.285809y(t-2)\\ +e\left(t\right)-0.543884e(t-1)-0.372571e(t-2)\end{array}$$

(40)

Assuming that state vector $x\left(t\right)={\left[y(t-1),y(t-2),e(t-1),e(t-2)\right]}^{\prime }$, input vector $u\left(t\right)={\left[x1\left(t\right),1\right]}^{\prime }$, and a noise vector $e\left(t\right)$, then the state-space expression can be written in the following form:

$$\begin{array}{c}x(t+1)=\left[\begin{array}{cccc}0.864634,& -0.285809,& 0,& 0\\ 1& 0& 0& 0\\ -0.543384,& -0.372571,& 0,& 0\\ 1& 0& 0& 0\end{array}\right]\ast x\left(t\right)+\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\\ 0& 1\end{array}\right]u\left(t\right)+\left[\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right]\ast e\left(t\right)\\ y\left(t\right)=[0,0,0,0]\ast x\left(t\right)+[0.959964,0.367537]\ast u\left(t\right)+e\left(t\right)\end{array}$$

(41)

Note that Equation (41) is only one of several representations using the selected state vector, but it does not imply any loss of generality for the cell mapping. Next, four pre-image cells were chosen from the uniform distribution $U(c_i,c_i+r_i)$ in each pattern class, and the corresponding image cells were obtained by ARMAX. The initial prediction outputs in the cell space were described in terms of the minimum and maximum prediction values, as shown in Figure 7.

In Figure 7, the red line denotes real values, the green and blue lines indicate the maximum and minimum initial predicted values, respectively, and it is clear the dynamic trajectory has been tracked properly and has maintained the inherent statistical characteristics. On the basis of the initial prediction outputs, we performed another classification step using Equation (2) of $M(·)$ on $\tilde{d}x\left(k\right)$ to convert the cell space variables to pattern category variables and compare them to the pattern class variables. Finally, the root-mean-square error (RMSE) of the system state prediction is 2.91 in Figure 8.

For the dynamic analysis of the two-dimensional cell state space, we utilized $z(i,j)$ to denote the cell on the state space and establish the mapping relationship $z(i^{\prime },j^{\prime })=C\left(z(i,j)\right)$ (C refers to the cell function). In this way, transforming the phase space into a cell space and the dynamical system into a cell-mapped dynamical system, we propose to use the three-dimensional array $Grps(Gr,P,St)$ to represent the group number of the attractor, the number of cycles of the attractor, and the number of convergence steps, respectively.

As shown in Figure 9, the phase space was replaced with the cell space, divided into $9\times 9$ cells, and the stable attractor and its domain were searched using Algorithms 2 and 3. The subplot on the right side of Figure 9 represents the system stabilizing the region of attraction, meaning the size of the region of attraction is different in the magnitude of the attraction compared to the surrounding transient cells. It is obvious that the system is globally stable from the attracting domain. Regarding the analysis of the system, the specific behaviours are summarized in

Table 2. The system’ global analysis with $Grps(Gr,P,St)$ from the first nine cells for case 1.

Pre-image cell	1	2	3	4	5	6	7	8	9
Image cell	2,6	4	4	5	3	3	3,8	3,5,7	2,7
St	2	1	0	0	0	1	1	1	2
P	0	0	3	3	3	0	0	0	0

, where the blue size of the text records the cell where the periodic cell resides.

The results of Table 2 clearly show that there is a three-period attractor ($P-3$) for the system, i.e., set $\left\{3,4,5\right\}$, whose one step of the DoA is the cell set $\left\{2,6,7,8\right\}$, two steps of the DoA is the cell set $\left\{1,9\right\}$.

• Case 2: Single-input Multiple-output Unknown Nonlinear System

Consider the following unknown nonlinear system with one input and several outputs:

$$\left\{\begin{array}{c}{x}_1(k+1)=\frac{(1+d_1\left(k\right))u\left(k\right)}{1+u^2\left(k\right)}+u^2\left(k\right)+0.9u(k-1)+d\left(k\right)\hfill \\ x_2(k+1)=0.3x_2\left(k\right)+0.1x_2(k-1)+u^2\left(k\right)+\frac{(1+d_2\left(k\right))u\left(k\right)}{1+u^2\left(k\right)}+d\left(k\right)\hfill \\ x_3(k+1)=0.5x_3\left(k\right)+\frac{(1+d_3\left(k\right))u\left(k\right)}{1+u^2\left(k\right)}+0.9u^2\left(k\right)+0.5u(k-1)+d\left(k\right)\hfill \end{array}\right.$$

(42)

where $d_i\left(k\right)$ is a parameter disturbance, $x_i\left(k\right)$ represents the multidimensional system output value, and $d_i\left(k\right)\sim N(0,0.1^2),i=1,2,3.$ The system input is $u\left(k\right)$, and it seems likely that $u\left(k\right)\in \left[-2,2\right]$. The measurement noise is given by $d\left(k\right)$, and $d\left(k\right)\sim N(0,0.{01}^2)$. The system’s operating subspace $\left\{\left[x_1\left(k\right),x_2\left(k\right),x_3\left(k\right)\right]\right\}$ was formed through collecting 2000 sets of historical output data under the control of input $u\left(k\right)$.

After data prepossessing and dimension reduction employing PCA, the two-dimensional data set $\left\{x_1\left(k\right)\right.\},\left\{x_2\left(k\right)\right\}$ was generated; the first two principal components were equal to 87%. Based on the DBSACN method, 12 pattern class variables were created as shown in Figure 10, where various colours indicate different pattern classes.

As in case 1, the parameters were set to $ϵ=0.5,MinPts=10$, then 12 pattern classes, class centres, and class radii were determined and are provided in

Table 3. The class centre and radius for a two-dimensional pattern-moving space for case 2.

Class No.	Class Centre ${\mathit{c}}_{\mathit{i}}$	Class Radius ${\mathit{r}}_{\mathit{i}}$
0	(−0.10, −0.92)	1.25
1	(3.96, −0.93)	0.98
2	(−1.51, −0.70)	0.90
3	(0.11, 1.40)	1.13
4	(1.28, −0.87)	1.28
5	(−2.88, −0.35)	0.86
6	(3.00, 2.41)	2.84
7	(−0.91, 0.67)	1.04
8	(1.29, 1.91)	1.16
9	(2.69, −1.00)	1.27
10	(−2.06, 0.32)	0.83
11	(5.79, −0.68)	1.69

. Meanwhile, the cell space $\Omega =[-2.88,5.79]\times [-1.00,2.41]$ was determined based on the DBSCAN algorithm. Subsequently, the structure and parameter identification of cell function ARMAX(2,2,0) were given by:

$$\begin{array}{c}y(k+3)-0.5362y(k+2)-0.2381y(k+1)-0.1420y\left(k\right)\hfill \\ =-0.0953u(k+2)+0.3476u(k+1)\hfill \end{array}$$

(43)

After the transformation of Equation (43), the state-space model after the transformation of the difference equations of a discrete-time state-space model was:

$$\begin{array}{c}x(k+1)=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 0.1420& 0.2381& 0.5362\end{array}\right]x\left(k\right)+\left[\begin{array}{c}-0.0953\\ 0.3476\\ 0\end{array}\right]u\left(k\right)\\ y\left(k\right)=x\left(k\right)\end{array}$$

(44)

As a result, in Figure 11, the minimum and maximum initial predictions were determined by applying Equation (44). On the side, Figure 12 illustrates the accuracy of the algorithm in estimating the pattern class variables versus the real values, which shows that the G-GCM accurately described the dynamic characteristics of the system. Finally, the root-mean-square error (RMSE) of the system predictions was 2.59. In addition, a global analysis of case 2 using Algorithms 2 and 3 is portrayed in Figure 13 and the

Table 4. The system’s global analysis with $Grps(Gr,P,St)$ from the first sixteen cells for case 2.

Pre-cell	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Image cell	2,5	4	5	5	6	4	3,8,12	5,7,11	5,13	4,13	14	13	12	11,13	13,16	15
St	1	1	1	0	0	0	1	0	1	1	1	1	2	1	1	2
P	0	0	0	3	3	3	0	0	0	0	0	2	2	0	0	0

.

By analysing Table 4, there exist two periodic attractors for the system: the cell set {4, 5, 6} constitutes the attractor of period $P-3$, and the cell set {12, 13} makes up the attractor of period $P-2$. The set {1, 2, 3, 8, 9} forms the DoA of the period $P-3$ attractor and the DoA of the period $P-2$ attractor is {9, 10, 11, 14, 15, 16}. The set of cells {7, 8, 9, 10} then forms the boundary between the DoAs. In a word, the stability and attractors behaviours of the system are successfully verified with the proposed algorithm.

5. Conclusions

In conclusion, this investigation developed the G-GCM, consisting of a clustering algorithm and generalized cell mapping, which brought a new insight into the system’s model and analysis for a class of complicated systems governed by statistical laws in industrial processes. On this basis, to address the computations for pattern class variables, first, the construction of a cell space and classification criteria were realized through applying DBSCAN. Secondly, the first two principal components’ information of the system was extracted by PCA, and the rate of change of the system state was considered constant in a sufficiently small time step. Given that, the global behaviour and maintained statistical characteristics were revealed with the improved GCM in accordance with the designed searching method. Finally, workshop-type and numerical examples were presented using the proposed method. Experimental results confirmed that the proposed technique was equipped to describe the dynamic behaviour of the system.

Despite favourable effects on the description and analysis of a class complex system using G-GCM, the study was mainly a theoretical analysis and was restricted in its applications. Therefore, further studies should consider the potential impact of clustering parameters and the search computational efficiency. Moreover, determining the different methods regarding the cell function (nonlinear model) and sampling of pre-image cells is important for analysing the dynamic system. Moreover, the design of a controller for this system is worth investigating due to the G-GCM descriptions.