Abstract
This paper presents a new method to study the system stability of a kind of complex production processes. The system dynamics is described by multi-dimensional pattern class variables instead of state variables or output variables. The system asymptotic stability is defined by system matrix and output class partition. Firstly, “pattern moving space” is constructed by a new maximum entropy clustering method. Multi-dimensional pattern class variables measured by pattern class centers are defined in this new space. Then the nonlinear state space model of the relevant control system is derived and established through two-step prediction model. On the basis of the original stability in Euclidean space, the multi-dimensional pattern moving state space is defined and Lyapunov stability is given in this space. Finally, experimental results show the effectiveness of the proposed conditions.
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1 Introduction
For the modeling of dynamic system, we usually use neural network [1], transfer function [2] and other methods in Euclidean space. However, for many complex production processes in process industries such as metallurgy and chemical industry, we cannot establish an exact mathematical model describing the dynamic characteristics of the controlled object in Euclidean space because of the complexity of the technological process, the moving mechanism cannot be fully explained and the numerous parameters describing the working conditions and product quality. And system dynamics properties are not dominated by the law of Newton’s mechanics but mainly controlled by the statistical law. Because of the common characteristics of statistical dynamics, this kind of complex production process is called Non-Newton Mechanical Systems [3].
For this type of system, modeling and control methods based on moving pattern were put forward in [4,5,6,7,8]. In these references, pattern class variable and “pattern moving space” have been defined. The control method and modeling method of the system are researched in “pattern moving space”. The pattern class variable itself is a statistical variable, which can better reflect the statistical characteristics. Due to patterns (or pattern classes) do not have the operation attribute (that is, pattern 1 ± pattern 2 \(\ne\) pattern 3), several methods such as interval number [4,5,6], pattern class center [7] and cell [8] have been proposed to measure pattern class variables. A pattern moving trajectory mapping from “pattern moving space” to Euclidean space is completed. The definition and analysis of stability of control system based on moving patterns was also involved in [9]. However, the actual working condition patterns are generally multi-dimensional. In order to describe better the dynamic characteristics of system and realize directly the data based on multi-dimensional working condition patterns, the model of multi-dimensional moving pattern is studied by using the pattern class centers to measure the multi-dimensional pattern class variables, and then the stability of multi-dimensional moving pattern is studied in this paper.
Different from the traditional dynamic model of the system established in Euclidean space, the modeling based on multi-dimensional moving pattern describes the dynamic characteristics of the system in multi-dimensional “pattern moving space”. Unlike the traditional “point” trajectory of variables moved from one “point” to another “point” in Euclidean space, in multi-dimensional “pattern moving space”, variables move from one pattern class to another pattern class, which movement trajectory reflects the change of the classes of working pattern over time. In view of the measurement problem of multi-dimensional pattern class variables, multi-dimensional pattern class centers is used to represent the class.
Dynamic Stability of systems in engineering and physics is widely followed topic from a couple of viewpoints using various approaches [10, 11]. In particular, procedures based on entropy maximum is very important in the Stochastic Dynamics, when using Fokker-Planck equation of Probability Density Function as a system of auxiliary conditions of extreme searching [12]. Dynamic stability is used in many fields, such as multi-body systems, large deformation systems, flow stability and so on. Lyapunov stability is mainly used in this paper. This applies to nonlinear systems and can be used to study the most general theorems and to study special systems.
Considering the system stability of a kind of complex production process based on multi-dimensional moving pattern description, system Lyapunov stability is defined in multi-dimensional pattern moving state space and a method to judge system asymptotic stability is proposed. Firstly, maximum entropy clustering algorithm [13, 14] based on krill herd [15, 16] is used to divide the output of the system into several classes, and these output classes are used to construct the multi-dimensional “pattern moving space”. The dynamic description method of control system based on multi-dimensional moving pattern is given. Then the system output class centers are used to measure multi-dimensional pattern class variables. The system dynamics is described by statistical space mapping, which is a two-step method. The initial control model of the AutoRegressive eXogenous (ARX) of system is established by using the modeling method in Euclidean space in [17, 18]. And a discrete-time and discrete-state nonlinear state space model of the system is deduced and established based on this initial control model. Then the output of system in Euclidean space is classified to multi-dimensional “pattern moving space”. The judgment method of asymptotic stability of the control system is studied by using Lyapunov theory. That is, the asymptotic stability of system is judged by the classification of system output and system matrix of the linear part of state space model. The key of this method is that there is a one-to-one mapping relationship between the class scales and class centers in multi-dimensional “ pattern moving space”. Finally, a practical input–output data and nonlinear numerical example are used to verify the feasibility of the proposed method.
The main contribution of this paper is that the modeling and control of multi-dimensional moving pattern are studied, and the asymptotic stability of system is judged by using the output class information of system. And a new maximum entropy clustering method is put forward to describe the dynamic characteristics of a kind of complex production process based on multi-dimensional moving pattern, which overcomes the difficulty in selecting the initial clustering centers and the number of classes in the original K-means [19, 20] clustering algorithm.
This paper is organized as follows: In Sect. 2, maximum entropy clustering method based on krill herd is put forward to divide pattern classes. The description method of system dynamics based on multi-dimensional moving pattern is given. Based on the initial prediction model, the method of establishing nonlinear state space model of control system is proposed. In Sect. 3, the system stability described by multi-dimensional pattern class variables are defined, the characteristics of output class partition is extracted and the sufficient conditions for asymptotic stability of system are deduced by Lyapunov theory based on the nonlinear state space model. In Sect. 4, simulation examples of sintering process and nonlinear mathematical model data are used to verify the proposed stability discrimination method. Finally, conclusions of this paper are summarized in Sect. 5.
2 Dynamics description based on multi-dimensional moving pattern and the nonlinear state space model
A method based on pattern recognition for describing a complex dynamic model is given. The main application of this method concerns sintering production process. The main idea consists in describing the variation of patterns over time through the introduction of “Pattern class variables”. Take the sintering production process as an example, the temperature and pressure of the last three bellows are taken as feature variable, then pattern classes are obtained by a new maximum entropy clustering method in Subsection 2.1, class centers are as the measurement value. System dynamics description based on multi-dimensional moving pattern is given in Subsection 2.2. and the nonlinear state space model of the relevant control system is given in Subsection 2.3.
2.1 Maximum entropy clustering method based on krill herd optimization
Common clustering methods, such as K-means [19, 20] and ISODATA [21, 22], are generally difficult to determine the number of classes and select the class centers, and even easy to fall into the local optimal solution. Therefore, maximum entropy clustering method based on krill herd algorithm was proposed to settle these problems.
The maximum entropy clustering algorithm [14] uses potential function method [23] to determine the initial clustering centers, takes entropy as the index to evaluate the clustering results, and krill herd algorithm [15, 16] is used to optimize the clustering results. Then the change of entropy rate is analyzed in order to determine whether the clustering process is over.
2.1.1 The krill herd algorithm
The krill herd (KH) algorithm was a kind of swarm intelligent optimization algorithm proposed by Gandomi et al. inspired by the simulation experiment of the living environment and living habits of the Antarctic krill herd. It was a global optimization algorithm, which simulates krill herd searching for food and individual interaction. Krill migration is a multi-objective process that involves increasing krill density and reaching the food. And the fitness of each krill individual should be a combination of the distance from the krill individual to the food and the distance from the krill individual to the peak density of the krill herd in a natural system. Therefore, fitness is the objective function value. \(X_{i}\) represents the position vector for individual krill i, and its evolution is synergically influenced by motion induced \(N_{i}\) by other krill individuals, foraging motion \(F_{i}\) and physical diffusion \(D_{i}\). The krill herd algorithm uses the Lagrange model to search effectively, and the position change rate of krill individuals is described as follows:
where, \(X_{i}=\{X_{i,1},X_{i,2},\cdots ,X_{i,NV}\}\), \(dX_{i}/ds\) represents the speed at which individual krill move. NV is the total number of variables, \(N_{i}\) is krill i induced by other krill movement, \(F_{i}\) is the foraging movement of krill i, \(D_{i}\) is the random diffusion motion of krill i.
The motion of Krill i induced by other krill \(N_{i}^{new}\) is defined as
where \(\omega _{n}\) is the inertia weight of the motion induced in the range [0, 1], \(N_{i}^{old}\) is the last induced motion, \(N^{max}\) represents the maximum induced speed and \(N^{max}=0.01(ms^{-1})\). \(\alpha _{i}^{local}\) is the local effect provided by the neighbors and \(\alpha _{i}^{target}\) is the target direction effect provided by the best krill individual.
The foraging motion is formulated by the food location and the previous experience about the food location. This motion can be expressed for the ith krill individual as follows:
where \(\omega _{f}\) is the inertia weight of the foraging motion in the range [0, 1], \(F_{i}^{old}\) is the last foraging motion, \(V^{max}_{f}\) represents the foraging speed and \(V^{max}_{f}=0.02(ms^{-1})\). \(\beta _{i}^{food}\) is the food attractive, \(\beta _{i}^{best}\) is the effect of the best fitness of the ith krill so far.
The random diffusion \(D_{i}\) of krill individuals can be defined as:
where \(D^{max}\) is the maximum diffusion random speed and \(D^{max}=0.005(ms^{-1})\), I is the number of iterations, \(I_{max}\) is the maximum number of iterations, \(\delta\) is the direction of random diffusion and \(\delta \in [-1,1]\).
The particle updating process of KH algorithm is as follows:
where \(\Delta {s}\) is a scaling factor of the velocity vector at time s, c is the step factor and \(c\in [0,2]\), NV is the dimension of the decision variable, \(UB_{j}\) and \(LB_{j}\) are the upper and lower bounds of the jth dimension variable, \(j=1,2,\cdots ,NV\).
However, the traditional krill herd algorithm was coded by uses continuous value vector for individual position and position update operation in continuous space was completed, which is suitable for optimization problems in the continuum domain. According to the discrete nature of problem, the discrete krill herd optimization algorithm [24] is used to optimize the clustering results.
The average fitness of krill herd was defined as the virtual food, and the position update was carried out by comparing the individual fitness with the virtual food. The update strategy was shown in (6)
where, f(i), f(food) represent fitness of krill individual i and virtual food respectively. s is the current number of search. The update process is as follows: in the process of \(s+1\) search, the fitness of the ith individual and the virtual food is compared. If \(f(i)\prec {f(food)}\), a new individual position \(X_{i}(s+1)\) is formed by the linear sequence cross operation(LOX) between individual position \(X_{i}(s)\) and \(X_{i}^{food}\). If \(f(i)\succ {f(food)}\), the partial mapping cross operation(PMX) is performed between the individual position \(X_{i}(s)\) and the optimal individual of the current herd \(X^{gbest}\) to form a new position \(X_{i}(s+1)\). Otherwise, the mutation operation(MU) is performed on the current individual \(X_{i}(s)\) to produce a new position \(X_{i}(s+1)\).
In formula (6), if the fitness of krill individual i is be inferior to the virtual food in the optimization process, it obtains the good fragments of the virtual food by crossing with the virtual food. If the fitness of individual i is superior to the virtual food, the optimal fragment is obtained by crossing with the current optimal herd. Otherwise, it indicates that the current krill individual i has been in a good position, the mutation operation is carried out on individual i to prevent falling into the local extreme value.
In addition to the influence of virtual food, the herd in the neighborhood also has an impact on individuals. The neighborhood range is defined by the distance between krill individuals. The distance \(d_{ij}\)
where \(X_{i,l},X_{j,l}\) are codes of krill individuals i and j in the l dimension. n is number of individual. N is calculated as follows
Position update is carried out by comparing with the fitness of individuals in the neighborhood. The update strategy is shown in (7)
where ngb represents neighborhood range. \(X_{j}(s)\) is the spatial position of individual j in the neighborhood range in the s search.
In the process of \(s+1\) search, the neighborhood range is determined according to the distance between krill individuals. In the neighborhood individuals are selected, individual \(X_{i}(s)\) is crossed with the selected individual \(X_{j}(s)\), and the optimal individual is selected as the new individual \(X_{i}(s+1)\). If there is no qualified krill individual in the neighborhood, individual \(X_{i}(s)\) crosses with the population optimal \(X^{gbest}\) to form a new individual.
According to the above method, the position update formula of the discrete krill herd algorithm can be expressed as (8)
2.1.2 Maximum Entropy Clustering Method
Entropy is a thermodynamic concept that measures the disorder of a physical system. In thermodynamics, it is called Boltzmann entropy or Gibbs entropy. Gibbs entropy is defined as
where \(A_{i}\) is the states, \(p(A_{i})\) is the probability of \(A_{i}\) in all possible states, and they usually satisfy
Tsallis entropy [25] is the extension and generalization of Gibbs entropy. According to Tsallis entropy, the threshold is selected and the optimal threshold is obtained by maximizing the information between the target class and the background class. It can solve the non-generalization problem of the system. The Tsallis entropy is defined as
where, k is a conventional positive constant, q is the undetermined coefficient, which is used to describe the non-extensive problem. When \(q=1\), \(S_{q}\) is the standard entropy form in statistics [26], that is
Thus, Gibbs entropy is a special case of Tsallis entropy at \(q=1\).
Assume that the state is finite, entropy will have the maximum value when all possibilities are equal.
The procedure of maximum entropy clustering method is as follows:
Step 1. Let the initial number of classes \(N_{c}=1\) and the initial entropy \(E(1)=0\). The initial class center is obtained by formula (9) of the potential function method:
where \(N_{s}\) is the total number of samples in the sample set, \(x_{i}\) is a value of the class center to which sample ith belongs and \(x_{i}=[x_{i1},x_{i2}]^{T}\), \(\alpha\) is the neighborhood radius.
Step 2. The potential of each sample is obtained by updating formula (10) in the potential function method, and the sample corresponding to the largest potential serves as a new initial class center. Let the number of classes \(N_{c}=N_{c}+1\).
where \(P^{*}_{s}=\max _{i}\{P^{i}_{s-1}\}\), \(x^{*}_{s}\) is the sth initial class center and \(x^{*}_{s}=[x^{*}_{s1},x^{*}_{s2}]^{T}\), \(\beta\) is the neighborhood radius and \(\beta \ge \alpha\).
Step 3. On the basis of obtaining the new initial class centers, the k-means method is used for clustering in order to accelerate the convergence speed of algorithm, and a group of better class centers are obtained than before. Then, the class centers obtained by k-means clustering are used as the random initial position of krill herd. The fitness rule is set as the entropy of sample set classification.
Step 4. The krill herd was optimized iteratively. The location of krill \(X_{i}\) was updated under the synergistic influence of the movement induced by other krill, the foraging movement of krill individual, and the random diffusion of krill individual. A new set of individuals are obtained by each optimization. All krill individuals are classified into \(N_{c}\) classes based on the nearest neighbor rule.
Step 5. The probabilities for each category \(p(A_{i})\) are calculated according to division of each category. Entropy \(E(N_{c})\) after the sample set is divided into \(N_{c}\) classes was used as fitness for all krill, that is
where, \(p(A_{i})=N_{i}/N\), \(N_{i}\) is the number of samples in the ith class of the nearest neighbor classification. The individual with the highest entropy is found as a group of optimal class center from all individuals.
Step 6. The optimization process ends and is entered to Step 7 when the algorithm reaches the maximum number of iterations \(I_{max}\). Otherwise, it is transferred to Step 4.
Step 7. The formula (11) is used to calculate the entropy rate when the number of classes are \(N_{c}\)
Step 8. Check whether the entropy rate \(\Delta {E}(N_{c})\) is less than the given threshold \(\varepsilon _{E}\). If \(\Delta {E}(N_{c})<\varepsilon _{E}\), the clustering process ends. Otherwise, return to Step 2.
The process of maximum entropy clustering method based on krill herd is shown in the Fig. 1.
The flow of maximum entropy clustering method
The classes are obtained by the above maximum clustering method. The “pattern moving space” is constructed by these classes as its scale.
2.2 System dynamics description based on multi-dimensional moving pattern
This section introduces the system dynamics description method based on multi-dimensional moving pattern. For actual industrial process of each working condition, there is also a pattern \(p_{i}(i=1,2,...,n)\) that describes the current production working condition at each sampling time \(s_{i}\). With the change of time, the working pattern of system also changes regularly. At this time, the movement of system can be described by the movement of pattern. Firstly, enough operating data of working conditions should be collected in a long enough time to form a data space. In order to obtain the output classes, the clustering method is used to divide the pattern classes. The system output could be divided into C classes \(Y_{i}\), \(i{\in }\{1, 2, \cdots , C\}\), which considered as scales to establish multi-dimensional “pattern moving space”. Multi-dimensional pattern class variables that describe working moving pattern are defined in this space as follow.
Definition 1
Let tr(s) and pr(s) represent detection sample sequence and pattern sample sequence respectively, the pattern class variable dr(s) should meet the following transformation process:
where T and F represent the process of feature variable extraction and pattern classification respectively. And pattern class variable satisfy
-
a)
It is a functions of time;
-
b)
It has the class attribute.
In this paper, 2-dimensional pattern is used as an example to illustrate the modeling method of multi-dimensional moving pattern. 2-dimensional pattern class variables is represented by dr(s), where
Although the moving trajectory of working condition pattern can be established in 2-dimensional “pattern moving space”, it is difficult to give the computable rules between pattern classes because pattern does not have the operation attribute. Considering this problem, 2-dimensional pattern class variables are mapped to 2-dimensional points (such as interval numbers [4,5,6], pattern class centers [7], and cell [8] etc.) in the Euclidean space. At moment s, the measurement values of 2-dimensional pattern class variables \({\widetilde{dr}}(s-n+1),{\widetilde{dr}}(s-n+2),\cdots ,{\widetilde{dr}}(s)\) have been obtained at the first s moments in Euclidean space. In this paper, \({\widetilde{dr}}(s-i)\) are the measure value of the pattern class variables \(dr(s-i)\) in Euclidean space, pattern class centers are used as the measure value of the pattern class variables, where
According to the initial prediction model constructed in the computable space, at the moment \(s+1\) the initial prediction output \(\widehat{{\widetilde{dr}}}(s+1)\) is calculated by using the measure value of 2-dimensional pattern class variables \({\widetilde{dr}}(s-i)\) for the first s moments. And then the output is mapped back to 2-dimensional “pattern moving space” by classification mapping F. The moving trajectory of 2-dimensional “pattern moving space” are obtained at the moment \(s+1\), which is denoted as
The prediction model based on 2-dimensional moving pattern is established, and then the moving characteristics of system is described in the 2-dimensional “pattern moving space”. The prediction model based on moving pattern described by 2-dimensional pattern class variables can be expressed as follows
where F is a classification mapping. f represents an initial prediction model in Euclidean space. D is a measurement mapping. \(\widehat{{\widetilde{dr}}}(s+1)=f(\cdot )\) is an initial prediction model. u represents input of system. m and n are the order of system input and output respectively. The dynamic characteristic description method based on moving pattern is shown in the Fig. 2.
System dynamics description based on moving pattern
According to formula (13), it can be concluded that the process of dynamics description is composed of three parts: D, f and F. The dynamic description process is as follows:
Step 1. Due to the incomputability of patterns and pattern classes, at the sampling moment s, the pattern class variable dr(s) is represented by the corresponding measurement value \({\widetilde{dr}}(s)\) in Euclidean space.
Step 2. At the sampling time s, \(dr(s-n+1),dr(s-n+2), \cdots ,dr(s)\) are mapped to Euclidean space through measurement mapping D, and the initial prediction model f is established in Euclidean space, and the initial prediction output \(\widehat{{\widetilde{dr}}}(s+1)=f(\cdot )\) can be obtained
where f is a linear autoregressive model. u is input of system, it represents the ignition temperature in the given sintering data. (16) can be obtained by identifying model parameters.
where
The input–output order of model can be determined by calculating the residual error of model, and when the residual is less than a minimal positive number, the order of the model can be determined. At this time, the deviation between the real output class center of system and \(\widehat{{\widetilde{dr}}}(s+1)\) is small enough, and both of them belong to the same output class.
Step 3 Finally, \(\widehat{{\widetilde{dr}}}(s+1)\) is mapped back to multi-dimensional “pattern moving space” through classification mapping F, that is
Thus, a new trajectory can be obtained in 2-dimensional “pattern moving space”.
2.3 Construction of nonlinear state space model of control systems
According to formulas (14) - (17), a system model without input can be established in Euclidean space as follow
Thus, the prediction model based on moving pattern described by 2-dimensional pattern class variables can be expressed as follows
Based on the formula (18),
where \(e(s+1)\) is the deviation produced by the classification-measurement mappings that contained in the nonlinear part and
Supposed \(\widehat{{\widetilde{dr}}}(s+1)\in {Y_{i}}\), \(i\in \{1,2,\cdots ,C\}\), \(r_{i}\) is the radius of the class \(Y_{i}\), then \(r_{i}=\max _{y_{ij}}\Vert y_{ij}-c_{i}\Vert _{2}\), where \(y_{ij}=[y_{ij_{1}},y_{ij_{2}}]^{T}\) is jth 2-dimensional point in class \(Y_{i}\), \(c_{i}=[c_{i_{1}},c_{i_{2}}]\) is ith 2-dimensional center. According to the classification-measurement mapping, we can get its deviation
Then the state of the control system constructed as follow:
And let
The formula (22) can be obtained based on the formula (21)
For a system (16) without input, that is \(u=0\),
Therefore, nonlinear state space model of the system can be obtained by (21), (22) and (23)
where
and
The matrix A is called as a system matrix.
The model (24) consists of two parts: a linear part and a nonlinear part. And the classification-measurement mappings are contained in the nonlinear part \([\Xi ^{T}(X_{1}(s)),\Xi ^{T}(X_{2}(s))]^{T}\). In this part, \(e(X_{1}(s)),e(X_{2}(s))\) are caused by the classification-measurement mapping \(\widehat{{\widetilde{dr}}}(s+1)\). In other words, them depend on system dynamics and the partition of output \(\widehat{{\widetilde{dr}}}(s+1)\). Therefore, stability of the system (24) is concerned with the system matrix A and output partition. A nonlinear state space model (24) has been established in Euclidean space. Based on the state of the model (24), system stability definition is defined in the following section.
3 Stability definition and analysis based on pattern moving
3.1 Stability definition based on moving pattern
In the Subsection 2.3, a nonlinear state space model is established based on the measurement values of 2-dimensional pattern class variables. In this section, a 2-dimensional pattern moving state space of the control system is defined in order to study the stability of system. The method of defining system stability is given in this space.
The system state will be constructed in this section. Suppose \(L_{{\widetilde{\theta }}}\) is a single-valued mapping, it exists to construct system states from the measure value of 2-dimensional pattern class variables in Euclidean space, that is
where \([{\widetilde{\theta }}^{T}_{1}(s),{\widetilde{\theta }}^{T}_{2}(s)]^{T}\) is the pattern that describes the dynamic characteristics of system at the current time.
Similarly, suppose \(L_{\theta }\) is a single image mapping, a system state constructed by 2-dimensional pattern class variables can be denoted, that is
where \([\theta ^{T}_{1}(s),\theta ^{T}_{2}(s)]^{T}\) is the moving trajectory of describing 2-dimensional pattern class variables and is the sequence of pattern class scale at the moment s.
And
Thus, a map \(\tau\) can exist, such that \(\left[dr_{1}(s),dr_{2}(s)\right]^{T}=\) \(\tau \left(\left[DX^{T}_{1}(s),DX^{T}_{2}(s)\right]^{T}\right)\). The set composed of all system states constructed by pattern class variables \(L_{\theta }\) is denoted as \(\Omega _{DX}\), which is called system state space based on 2-dimensional moving pattern.
According to the system dynamics description based on 2-dimensional moving pattern, it can be obtained that an one-to-one mapping exists between \(dr(s-i)\) and \({\widetilde{dr}}(s-i)\). Thus, there is also an one-to-one mapping K between \(\left[\theta ^{T}_{1}(s),\theta ^{T}_{2}(s)\right]^{T}\) and \(\left[{\widetilde{\theta }}^{T}_{1}(s),{\widetilde{\theta }}^{T}_{2}(s)\right]^{T}\), such that
And then
Therefore, it can be concluded that \(\left[X^{T}_{1}(s),X^{T}_{2}(s)\right]^{T}\) and \(\left[DX^{T}_{1}(s),DX^{T}_{2}(s)\right]^{T}\) are the unique states by \(\left[\theta ^{T}_{1}(s),\theta ^{T}_{2}(s)\right]^{T}\) in Euclidean space and 2-dimensional moving pattern state space respectively at the sampling time s. And \(\left[X^{T}_{1}(s),X^{T}_{2}(s)\right]^{T}\) is considered as the measurement value of \(\left[DX^{T}_{1}(s),DX^{T}_{2}(s)\right]^{T}\) in Euclidean space, which is denoted by \(\left[{\widetilde{DX}}^{T}_{1}(s),{\widetilde{DX}}^{T}_{2}(s)\right]^{T}\). However, this measure value is not unique and is related to the construction method \(L_{{\widetilde{\theta }}}\). If the mapping \(L_{{\widetilde{\theta }}}\) is given, the measure value of system state \(\left[{\widetilde{DX}}^{T}_{1}(s),{\widetilde{DX}}^{T}_{2}(s)\right]^{T}\) is unique when 2-dimensional pattern class variables are measured by the class centers.
It is assumed that the distance between any two states \(\left[DX^{T}_{i1},DX^{T}_{i2}\right]^{T}\) and \(\left[DX^{T}_{j1},DX^{T}_{j2}\right]^{T}\) in 2-dimensional pattern moving state space is defined as the distance between their corresponding measures, that is
The system state measurement space based on 2-dimensional moving pattern is formed. The system stability is defined and analyzed in this space.
According to the above discussion, suppose that the system model (19) based on 2-dimensional pattern class variables description can be denoted as state space model (26)-(27)
where \(\Gamma\) represents a mapping from \(\left[DX^{T}_{1}(s),DX^{T}_{2}(s)\right]^{T}\) to \(\left[DX^{T}_{1}(s+1),DX^{T}_{2}(s+1)\right]^{T}\). \(\tau\) is a mapping from \(\left[DX^{T}_{1}(s),DX^{T}_{2}(s)\right]^{T}\) to \(\left[dr_{1}(s),dr_{2}(s)\right]^{T}\). Based on the model (26) - (27), the definition of system stability can be given by [27].
Definition 2
The system (26)-(27) is Lyapunov stable at equilibrium state \(\left[DX^{T}_{e1},DX^{T}_{e2}\right]^{T}\), at \(s=s_{0}\) if \(\forall \varepsilon >0\), \(\exists \delta (s_{0},\varepsilon )>0\) such that
where \(\forall {s}>s_{0}\), \(\left[DX^{T}_{1}(s_{0}),DX^{T}_{2}(s_{0})\right]^{T}\ne {\left[DX^{T}_{e1},DX^{T}_{e2}\right]^{T}}\).
Definition 3
The system (26)-(27) is Lyapunov asymptotically stable at equilibrium state \(\left[DX^{T}_{e1},DX^{T}_{e2}\right]^{T}\), at \(s=s_{0}\) if
-
1.
\(\left[DX^{T}_{e1},DX^{T}_{e2}\right]^{T}\) is stable, and
-
2.
\(\exists \delta >0\) such that
$$\begin{gathered} \left| {\left[ {\begin{array}{*{20}c} {DX_{1} (s_{0} )} \\ {DX_{2} (s_{0} )} \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {DX_{{e1}} } \\ {DX_{{e2}} } \\ \end{array} } \right]} \right|{ < }\delta \hfill \\ \;\;\;\; \Rightarrow \mathop {\lim }\limits_{{s \to \infty }} \left| {\left[ {\begin{array}{*{20}c} {DX_{1} (s)} \\ {DX_{2} (s)} \\ \end{array} } \right] - \left[ {\begin{array}{*{20}c} {DX_{{e1}} } \\ {DX_{{e2}} } \\ \end{array} } \right]} \right| = 0, \hfill \\ \end{gathered}$$
The system (26)-(27) is Globally asymptotically stable if all states \(\left[DX^{T}_{1}(s),DX^{T}_{2}(s)\right]^{T}\) are asymptotically stable.
Definition 4
The system (26)-(27) is Exponential asymptotically stable at equilibrium state \(\left[DX^{T}_{e1},DX^{T}_{e2}\right]^{T}\), at \(s=s_{0}\) if \(\forall \varepsilon >0\), \(\exists \delta (\varepsilon )>0\) and \(\nu >0\) such that
where \(\forall {s}>s_{0}\), \(\left[DX^{T}_{1}(s_{0}),DX^{T}_{2}(s_{0})\right]^{T}\ne {\left[DX^{T}_{e1},DX^{T}_{e2}\right]^{T}}\).
According to Definition 2-4, the system stability based on 2-dimensional moving pattern description is related to the measurement method of 2-dimensional pattern class variables. When the system (24)-(25) is stable in Euclidean space, the system (26)-(27) is also stable in the 2-dimensional pattern moving state space. Therefore, the stability of system described by 2-dimensional pattern class variables can be studied by the model in Euclidean space.
3.2 Stability analysis based on moving pattern
Considering the influence of output partition on stability, feature of class partition is defined before stability analysis.
In order to judge the system stability by using the output class information, the class information is represented as the ratio of the class radius \(r_{i}\) and the 2-norm of the class center \([c_{i1},c_{i2}]^{T}\), that is
where \([c_{i1},c_{i2}]^{T}\) is the class center for the ith output class that does not contain the origin, that is \(\left\| {[c_{{i1}} ,c_{{i2}} ]^{T} } \right\|_{2} \ne 0\), and the center of the class including the origin is \([0,0]^{T}\). \(r_{i}\) is the class radius of the ith output class.
And let
where \(\mu\) is called the feature of class partition of the system output. It can be concluded that \(\mu <1\). If not, the error caused by the classification mapping will be so large.
According to stability definition in Subsection 3.1, we only need to study the system stability in Euclidean space.
Suppose that the system operates within a bounded region \(\Omega\), all subregions \(\Omega _{i}\) by partitioned \(\Omega\) satisfies \(\Omega =\bigcup _{i}\Omega _{i}\). And all class radii are bounded.
Lemma 1
For \({\forall }A{\in }{\mathbb {C}}^{2n{\times }2n}\), supposed that \(V{\in }{\mathbb {C}}^{2n{\times }2n}\) is an invertible matrix, \({\exists }B{\in }{\mathbb {C}}^{2n{\times }2n}\), such that \(B=VAV^{-1}\). That is \(\Vert A\Vert _{\rho {\varsigma }}=\Vert VAV^{-1}\Vert _{\varsigma }=\Vert B\Vert _{s}\), where \(\varsigma =\{1,2,\infty \}\). \(\Vert \cdot \Vert _{\rho {\varsigma }}\) is a matrix norm. And
\({\forall }\begin{bmatrix} X_{1} \\ X_{2} \end{bmatrix}{\in }{\mathbb {C}}^{2n}\), let \(\left\| {\left[ {\begin{array}{*{20}c} {X_{1} } \\ {X_{2} } \\ \end{array} } \right]} \right\|_{{\upsilon \varsigma }} = \left\| {V\left[ {\begin{array}{*{20}c} {X_{1} } \\ {X_{2} } \\ \end{array} } \right]} \right\|_{\varsigma }\), \(\Vert \cdot \Vert _{\upsilon {\varsigma }}\) is a vector norm. And \(\Vert \cdot \Vert _{\rho {\varsigma }}\) is an induced matrix norm of \(\Vert \cdot \Vert _{\upsilon {\varsigma }}\).
Proof
According to the definition of vector norm and matrix norm, it is obvious that \(\Vert \cdot \Vert _{\upsilon {\varsigma }}\) and \(\Vert \cdot \Vert _{\rho {\varsigma }}\) are vector norm and matrix norm respectively. The proof procedure is omitted. Here \(\Vert \cdot \Vert _{\rho {\varsigma }}\) is an induced matrix norm of \(\Vert \cdot \Vert _{\upsilon {\varsigma }}\) is discussed.
We are only discuss the case of \(\varsigma =2\).
Suppose \(\varsigma =2\), we have
Therefore
For \(\forall {j=1,2,\cdots ,n},\) there exist \(\left[X^{T}_{1},X^{T}_{2}\right]^{T}\in {\mathbb {C}}^{2n}\) such that
where \(e_{j}\) is jth 2-dimensional unit vector. Then
Based on the above discussion,
Similarly, \(\forall \varsigma =\{1,\infty \}\), it can be proved that \(\Vert \cdot \Vert _{\rho {\varsigma }}\) is an induced matrix norm of \(\Vert \cdot \Vert _{\upsilon {\varsigma }}\). \(\square\)
In order to obtain the influence of the classification-measurement deviation on system stability Theorem 1 is presented:
Theorem 1
For the nonlinear system (24), \(\forall {\left[X_{1}(0),X_{2}(0)\right]^{T}}\ne {[0,0]^{T}}\), if the spectral radius of A satisfies \(\rho (A)<1\), we have
where \(c>0\) and \(r_{max}=\max \{r_{i}\}\),
Proof
According to \(\rho (A)<1\) and Lemma 1, it can be obtained that \(\Vert A\Vert _{\rho }<1\).
Because
By iterative methods, its solution is
Based on the norm equivalence theorem and the formula (20), it can be obtained
where \(\alpha >0\), \(r_{max}=\max \{r_{i}\}\). Thus
Because \(\Vert A\Vert _{\rho }<1\), if \(s\rightarrow \infty\), it can be obtained that
Therefore
Let
it can be obtained that
\(\square\)
According to Theorem 1, it can be obtained that the output of nonlinear system (24) is bounded. The system eventually converges to the neighborhood with radius \(cr_{max}\) instead of asymptotically converges to 0 even if \(\rho (A)<1\). That is, it is hard to say that the system described by the 2-dimensional pattern class variables is asymptotically stable. Therefore, it is necessary to study the asymptotic stability conditions of systems described by 2-dimensional moving pattern.
Theorem 2-3 are proposed to judge the nonlinear system (24) stability via the feature of class partition of the system output \(\mu\) and the system matrix A.
Theorem 2
Considering the nonlinear system (24), an equilibrium point \(\left[X^{T}_{e1},X^{T}_{e2}\right]^{T}=[0,0]^{T}\) is asymptotically stable if there exist \(\theta >0\), a symmetric matrix \(P>0\) and satisfy
such that
where \(A=\begin{bmatrix} A_{11} &{} A_{12} \\ A_{21} &{} A_{22} \end{bmatrix}\) is a nonsingular matrix,
and I is a unit matrix of appropriate dimensions.
Proof
Suppose \(V=\begin{bmatrix} V_{11} &{} V_{12} \\ V_{21} &{} V_{22} \end{bmatrix}\) is a nonsingular matrix. According to Lemma 1,
where
The Lyapunov function is defined as follows
Thus,
Let
then
According to the value of \({\widetilde{dr}}(s+1)=\left[{\widetilde{dr}}_{1}(s+1),{\widetilde{dr}}_{2}(s+1)\right]^{T}\), the stability analysis is partitioned into two parts: \(\left\| {\mathop {dr}\limits^{ \sim } (s + 1)} \right\| \ne 0\) and \(\Vert {\widetilde{dr}}(s+1)\Vert =0\).
1)\(\Vert {\widetilde{dr}}(s+1)\Vert \ne 0\).
Considering the nonlinear system (24), it can be obtained that
that is
Substitute formula (39) into formula (40), we have
thus
Substitute formula (41) into formula (38), it can be obtained that
where
And
Thus
\(\Delta {W(s)}<0\) can be deduced because
2)\(\left\| {\mathop {dr}\limits^{ \sim } (s + 1)} \right\| = 0\)
If \(\left\| {\mathop {dr}\limits^{ \sim } (s + 1)} \right\| = 0\), the relationship between \(\left[X^{T}_{1}(s+1),X^{T}_{2}(s+1)\right]^{T}\) and \(\left[X^{T}_{1}(s),X^{T}_{2}(s)\right]^{T}\) can be turned into
where
Thus
Based on we already know that \(A^{T}H^{T}PHA-P\le {0}\), the conclusion \(\Delta {W(s)}\le {0}\) can be obtained.
Sum up, for \(\forall {\widetilde{dr}}(s+1)\), we have \(\Delta {W(s)}\le {0}\). So the equilibrium point \(\left[X^{T}_{e1},X^{T}_{e2}\right]^{T}=[0,\,0]^{T}\) is asymptotic stable. \(\square\)
Theorem 3
Considering the nonlinear system (24), an equilibrium point \([X^{T}_{e1},X^{T}_{e2}]^{T}=[0,0]^{T}\) is asymptotically stable if there exist \(\theta \in (0,1)\), a symmetric matrix \(P>0\) and satisfy
such that
where
Proof
The proof is similar to Theorem 2, where
\(\square\)
Theorem 4 is put forward to discuss the relationship between spectral norm of the system matrix \(\rho (A)\), the feature of class partition of the system output \(\mu\) and the system stability:
Theorem 4
For the nonlinear system (24), \(\rho (A)\ne {0}\), an equilibrium point \([X^{T}_{e1},X^{T}_{e2}]^{T}=[0,0]^{T}\) is asymptotically stable if there exists an invertible matrix V satisfying
such that
where
Proof
Suppose V is a nonsingular matrix. According to Lemma 1,
The Lyapunov function is defined as follows
Thus,
Let
then
According to the value of \({\widetilde{dr}}(s+1)\) the stability analysis is partitioned into two parts: \(\Vert {\widetilde{dr}}(s+1)\Vert \ne {0}\) and \(\Vert {\widetilde{dr}}(s+1)\Vert =0\).
1)\(\Vert {\widetilde{dr}}(s+1)\Vert \ne {0}\)
Considering the nonlinear system (24), it can be obtained that
that is
Thus
and
Substitute formula (47) into formula (45), it can be obtained that
According to the formula (43) in Theorem 2,
Based on the condition that
we can obtain
Substitute formula (49) into formula (48), \(\Delta {W(s)}<0\) can be deduced.
2)\(\Vert {\widetilde{dr}}(s+1)\Vert =0\)
If \(\Vert [{\widetilde{dr}}(s+1)\Vert =0\), the relationship between \([X^{T}_{1}(s+1),X^{T}_{1}(s+1)]^{T}\) and \([X^{T}_{1}(s),X^{T}_{1}(s)]^{T}\) can be turned into
where
Thus
Based on we already know that \(A^{T}H^{T}PHA-P\le {0}\), the conclusion \(\Delta {W(s)}\le {0}\) can be obtained.
Sum up, for \(\forall {\widetilde{dr}}(s+1)\), we have \(\Delta {W(s)}\le {0}\). So the equilibrium point \([X^{T}_{e1},X^{T}_{e2}]^{T}=[0,0]^{T}\) is asymptotic stable. \(\square\)
According to [28, 29], we give the related concept of exponential stability. For an arbitrary matrix A, we use the matrix norm \(|A|= (\lambda _{max}(A^{T}A))^{1/2}\). Moreover, denote by \(\lambda _{max}(B)\), \(\lambda _{min}(B)\) the maximum and minimum eigenvalues metric matrix B. Also, we define \(\varphi (B)=\lambda _{max}(B)/\lambda _{min}(B)\). System (24) is Lyapunov stable in \(X(s)=0\) if arbitrary \(\varepsilon >0\), there exists \(\delta (\varepsilon )>0\), such that for any other solution X(s), we have \(\Vert X(s)\Vert <\varepsilon\) for \(s=0,1,\cdots\), and \(\Vert X(0)\Vert _{a}<\delta (\varepsilon )\). The system (24) is Lyapunov exponentially stable if there exist constants \(N>0\) and \(\theta \in (0,1)\) such that
The idea of the method is to construct the Lyapunov function \(W(s)= X(s)^{T}PX(s)\), and to estimate its forward difference by applying the norms and eigenvalues of systems matrices, where P is the unique positive definite solution
for a pre-specified positive definite matrix Q. Obviously, we have
Theorem 5
Let \(\rho (A)<1\), Q is a positive definite matrix, matrix P solve the corresponding Lyapunov matrix equation (50),
where
Then, the system (24) is exponentially stable. And for an arbitrary X(s), the estimate
where
Proof
Let
a is a constant. We get
According to formula (51),
From equation (52), we know that
Thus,
We definite \(v_{0}=max\{W(X(0)),\cdots ,W(X(-a))\}\), \(v_{0}{\le }\lambda _{max}(P)\Vert X(0)\Vert _{a}^{2}\). Let \(s=0\),
Thus,
And
According to formula (51) and (55),
\(\square\)
4 Simulation verification
In Sect. 4, the correlated process of the multi-dimensional moving pattern is illustrated by a 2-dimensional case as an example. The raw input–output data collected from a sintering process in Anyang iron and steel plant is used to verify the presented method of stability analysis. Here 1905 sets of production conditions data were collected including temperature and pressure of bellows, ignition temperature. The ignition temperature is used as raw input data to the sintering process. The temperature of the last three bellows on the east and west sides and pressure of the last three bellows are taken as raw output data.
4.1 Construction of multi-dimension ”pattern moving space”
The collected raw input–output data is disposed by filter before constructing the 2-dimensional “pattern moving space” in order to remove noise and reduce fluctuation degree. And then the data is normalized by the z-score. At last, the principal component analysis (PCA) is used to extract feature variables as the output of system model to reduce the dimension, and the 2-dimensional system output data can be obtained. The input–output data after preprocessing is shown in Fig. 3.
Normalized input–output data
For the working condition pattern after dimension reduction, maximum entropy clustering method based on krill herd is used for partition pattern classes. And the 2-dimensional “pattern moving space” is constructed based on output classes. In this method, parameters are set to \(\alpha =0.01\), \(\beta =0.02\), \(\varepsilon _{E}=0.018\) respectively. The maximum iterations in the clustering process \(I_{max}=100\). The clustering process ends when \(\Delta {E}(N_{c})<\varepsilon _{E}\) and the number of output classes \(N_{c}=41\). The 2-dimensional clustering results as shown in Fig. 4.
The results of 2-dimensional clustering
The 2-dimensional “pattern moving space” is constructed by these pattern classes. The pattern class variables that describe pattern class changes are defined in the 2-dimensional “pattern moving space”. In this experiment, the class centers C and class radii R obtained by clustering are listed in Table 1.
After constructing 2-dimensional “pattern moving space”, the 2-dimensional pattern class variables are mapped to the trajectory points of Euclidean space. The measured output sequence is the measurement value sequence of pattern class variables after the pattern classes are measured to Euclidean space.
All samples belonging to the same pattern classes can be represented by the same measure values, that is, the class centers of the output pattern, as shown in Fig. 5.
Actual output data and measurement values of pattern class variables
4.2 Prediction model and the nonlinear state space model of control system
Before the nonlinear state space model based on moving pattern is established, the initial control model of system is established in Euclidean space by the linear AutoRegressive eXogenous (ARX) structure. And the least square method is used for the identification model parameters. The root mean square error [7] of system identification is expressed by the formula (56)
In this section, 1500 input–output data are used to establish the initial prediction model of sintering process when pattern class variables are 2-dimensional. Let \(n=2\), that is, the first two moments is used for predicting the next moment. The established prediction model is presented by the formula (57). And the model is verified using the remaining 405 data. Simulation results are showed in Fig. 6.
Validity verification of the initial control model
Therefore, model parameters can be expressed as
In addition, the classification process may also affect the stability analysis, so we also compared the prediction results before classification with the results after classification to verify the effectiveness of the classification process. Results are showed in Fig. 7.
Comparison of prediction models before and after classification
Considering the related process of establishing nonlinear state space model,system matrix A can be obtained by formula (57), that is
The nonlinear state space model of the system can be shown as
4.3 The stability analysis of system
Considering the nonlinear state space model (58)-(59), the asymptotic stability of system will be discussed. According to formula (28)-(29), the feature of class partition of the system output \(\mu =0.8414\). Let \(\theta =0.95\), it can be obtained that
where
Thus, it can be verified that A, P, \(\theta\) satisfy the conditions of theorem 3, so the system with the system matrix A is asymptotically stable. Let any output \([y_{1}(s_{0}),y_{2}(s_{0})]^{T}=[0.1,0.2]^{T}\), \([y_{1}(s_{0}-1),y_{2}(s_{0}-1)]^{T}=[-0.8,0.6]^{T}\). Their measurement values after classification are \([{\widetilde{dr}}_{1}(s_{0}),{\widetilde{dr}}_{2}(s_{0})]^{T}=[0,0]^{T}\) and \([{\widetilde{dr}}_{1}(s_{0}-1),{\widetilde{dr}}_{2}(s_{0}-1)]^{T}=[-0.8277,0.6869]^{T}\) respectively.
The initial state \([X_{11}(s_{0}),X_{21}(s_{0}),X_{12}(s_{0}),X_{22}(s_{0})]^{T}=\) \([-0.8227,0,0.6869,0]^{T}\) can be constructed in Euclidean space. The system output curves of burning processes are shown in Fig. 8. And in Fig. 9, we use the unclassified prediction results to establish the model and verify the stability conditions. It can be seen from the Fig. 9 that the classification process will have a certain impact on the stability of the system, such that the stable point is not reached.
System output curves of burning processes
System output curves of burning processes before classification
According to Fig. 8, the blue line is actual output of the model, and green line represents measurement values. The system is asymptotic stable that it can be seen from Fig. 8. And the output rapidly reaches the stable point \([0,0]^{T}\) through classification-measurement mapping when the output of system enters the class belonging to the stable point.
Remark: In the early stage, we have done the optimal control based on moving pattern [30]. Next, we will study the stability analysis of the optimal control law based on adaptive dynamic programming control algorithm. And some other ways of construction of the objective function will be studied to improve convergence and numerical stability of relevant numerical process such as in [31].
5 Conclusion
Aiming at the stability problem of complex production process based on moving pattern description, this paper presents a method to discuss the stability problem of system by system matrix. Firstly, the system dynamics description method based on moving pattern is introduced, and the maximum entropy clustering method based on krill herd is proposed to divide pattern classes and construct multi-dimensional pattern moving “space”. Then, the pattern class centers are used to measure the pattern class variables. The initial prediction model is established in Euclidean space, and the nonlinear state space model of the system is established based on the prediction model. The system stability based on moving pattern is defined in the pattern moving state space, and the asymptotic stability of the system is discussed by the feature of class partition of output \(\mu\) and system matrix A. Finally, the validity of the condition is verified by the actual production work condition data.
Availability of data
The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.
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This work was supported by the National Natural Science Foundation of China (No.61903167).
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Wu, J., Li, J. System asymptotic stability analysis of a kind of complex production processes based on multi-dimensional moving pattern. SN Appl. Sci. 5, 32 (2023). https://doi.org/10.1007/s42452-022-05254-y
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DOI: https://doi.org/10.1007/s42452-022-05254-y