A prediction framework based on the evolution of pattern motion probability density is proposed for the output prediction and estimation problem of non-Newtonian mechanical systems, assuming that the system satisfies the generalized Lipschitz condition. As a complex nonlinear system primarily governed by statistical laws rather than Newtonian mechanics, the output of non-Newtonian mechanics systems is difficult to describe through deterministic variables such as state variables, which poses difficulties in predicting and estimating the system’s output. In this article, the temporal variation of the system is described by constructing pattern category variables, which are non-deterministic variables. Since pattern category variables have statistical attributes but not operational attributes, operational attributes are assigned to them by posterior probability density, and a method for analyzing their motion laws using probability density evolution is proposed. Furthermore, a data-driven form of pattern motion probabilistic density evolution prediction method is designed by combining pseudo partial derivative (PPD), achieving prediction of the probability density satisfying the system’s output uncertainty. Based on this, the final prediction estimation of the system’s output value is realized by minimum variance unbiased estimation. Finally, a corresponding PPD estimation algorithm is designed using an extended state observer (ESO) to estimate the parameters to be estimated in the proposed prediction method. The effectiveness of the parameter estimation algorithm and prediction method is demonstrated through theoretical analysis, and the accuracy of the algorithm is verified by two numerical simulation examples.

There is a class of complex industrial processes in modern production processes, such as the sintering process, blast furnace combustion process, and cement rotary kiln production process. Due to their distribution characteristics, nonlinearity, time variation, and parameter perturbation, their dynamic characteristics cannot be accurately described. Moreover, the motion characteristics controlled by statistical laws are common in these complex production processes, and they are collectively referred to as non-Newtonian mechanical systems [

Recently, a dynamic pattern recognition method has been proposed for systems governed by statistical regularities [

Non-Newtonian mechanical systems can essentially be considered as a type of special random system. There are two main research routes for the dynamic analysis of stochastic systems: one is the stochastic differential equation research route formed by combining Einstein’s dissipation diffusion relationship with Ito differential equations [

For a controlled system, the data that can usually be obtained is the system’s input-output data. In recent years, a non-parametric dynamic linearization method based on input-output data has been developed [

In conclusion, this work addresses the problem of predicting and analyzing non-Newtonian mechanical systems in literature [

Firstly, a model of motion system dynamics based on posterior probability density measurement is proposed. Historical data is collected to form a detection sample sequence, and feature extraction and pattern classification are performed to classify similar operating conditions into categories and construct pattern category variables. When the current output of the system is obtained, the system’s pattern category can be discriminated by pattern classification. Then, the posterior probability is used as the measurement of the pattern category variables and mapped to probability space. The law of probability density evolution is analyzed through the principle of probability density conservation in probability space, and a prediction framework of probability density is constructed. As the evolution of probability density in probability space corresponds to the variation of pattern category variables in motion pattern space, a motion trajectory is ultimately formed in motion pattern space over time.

Then, in order to predict and analyze the probability density evolution of pattern motion, a data-driven algorithm for predicting the probability density evolution of pattern motion is proposed based on the characteristics of non-Newtonian mechanical systems that are difficult to describe through mathematical equations. By constructing PPD, the system is transformed into a data-driven form that includes statistical feature terms. Then, the corresponding probability density evolution equation is constructed through probability density evolution theory, and the probability density prediction is finally achieved based on the current input/output of the system.

Finally, to realize the non-parametric dynamic linearization method with the statistical feature term proposed in this paper, a PPD online estimation algorithm is presented, taking advantage of the structure of the ESO. The statistical feature term is used as the extended state, and the statistical feature term in the PPD estimation law is replaced by the output of the ESO. The boundedness of the proposed PPD estimation algorithm and the boundedness of the evolution prediction of the probability density of pattern motion for system output estimation are theoretically analyzed and numerically demonstrated.

This article provides a new theoretical framework for the analysis of patterned motion in non-Newtonian mechanical systems. Unlike previous methods that measure using pattern category centers, the posterior probability density better reflects the statistical properties of the pattern category variable. This paper combines the system analysis theory in the control field with the probability density evolution theory in the area of stochastic structural dynamics analysis. This provides a solution to the difficulty in estimating outputs in non-Newtonian mechanical systems due to being governed by statistical laws.

The rest of the paper is arranged as follows.

For a class of complex industrial production systems governed by statistical regularities, it is considered that they share the following three common features:

The production process involves complicated technological procedures, and its motion mechanism cannot be fully explained at the current stage. During the system’s operation, a series of intertwined physico-chemical changes include combustion thermodynamics, chemical reaction kinetics, phase transitions, and moving boundaries, making it challenging to use existing mathematical and physical equations for description.

The system motion exhibits complex characteristics such as distributed characteristics, nonlinearity, and time-varying parameters.

Some physical processes in the system essentially conform to the laws of statistical movement, such as the physical transformation from liquid phase to solid phase, granularity and fluidity. The corresponding relationships between many variables that characterize operating conditions and product quality cannot be described through deterministic mathematical equations, and only exist in a statistical sense. System dynamics are essentially not governed by Newtonian mechanics but by statistical laws.

It is precisely the characteristic of being governed by statistical regularities that distinguishes these types of systems from other general nonlinear systems and is thus referred to as non-Newtonian mechanical systems [

Assuming that a non-Newtonian mechanical system that satisfies the characteristics 1) to 3) can be represented in the form of

According to the analysis of the system

For a complex industrial production process, input and output data are continuously collected over a sufficient period of time to form a data space. If the data sample is large enough, the information reflecting the motion characteristics of the system can be covered, and it can be considered that the motion of the production process should operate within this data space. This data space is called the motion subspace of the industrial production process. Feature variables are extracted from data on the motion subspace to obtain a pattern sample sequence containing the main statistical characteristics, and the space formed by the feature variables is called the feature variable subspace. The pattern recognition method is used to classify the feature variable subspace, and the pattern categories obtained serve as spatial scales. When the current state data of the system is obtained, the system state can be directed to the corresponding pattern category scale through feature extraction and classification operations. The construction of the pattern category variable is as follows:

According to Definition 1 and Remark 2, the pattern category variable is constructed from working condition samples with the same or similar attributes. Therefore, there must be a corresponding distribution relationship between the various categories corresponding to the pattern category variable and the system data. In other words, different categories have their own class conditional probability density. Estimating the class conditional probability density for each category based on historical working condition data is actually a typical non-parametric estimation problem. It can be estimated using non-parametric estimation methods such as Parzen windows. For the non-Newtonian mechanical system shown in

From the description of pattern motion dynamics based on conditional probability density measurements

However, the system equation of the non-Newtonian mechanical system studied in this paper is difficult to obtain. Therefore, the comprehensive speed

The main idea is to construct corresponding data-driven expression by input-output data of the system in real-time, and then replace the system equations in

To facilitate the analysis and research of the system, the following two assumptions are made for system

According to Cauchy mean value theorem, there must be a point in

Let:

There exists the following equation for each determined time

Since

Let

For

Substituting

From

Theorem 1 has been proven.

According to Theorem 1, a non-Newtonian mechanical system

Combining

According to the principle of conservation of probability density and the generalized probability density evolution equation shown in

According to the Dirac function’s representation of probability density, the solution of

According to the integral properties of Dirac function, the probability density of

Let

Theorem 2 has been proved.

Theorem 1 and Theorem 2 indicate that for a non-Newtonian mechanical system that satisfies Assumption 1 and Assumption 2, the probability density prediction of the system output under historical mode category conditions can be obtained from the input and output data of the system at the current time. Through the analysis of the above theorems, it can be seen that

In summary, probability density evolution prediction and output estimation based on pattern motion are composed of

Due to the unknown

The estimation law of PPD can be obtained from the cost function shown in the minimization

In the estimation algorithm constructed by

Then the ESO of

Combining the PPD parameter estimation law of

The convergence analysis of the whole algorithm consists of the boundedness of the parameter estimation algorithm and the convergence of the system output estimation error, that is, the estimation error of the ESO and the PPD estimation error are bounded, and the bounded convergence of the output estimation error of

Since the input and output of the system are bounded quantities, that is,

Consider the PPD and the output value of ESO in any finite time K. When k = K-1,

Referring to the boundedness analysis of the ESO estimation in reference article [

Assumption 3 shows that the input and output of the system are bounded, and the output of PPD and ESO has been proved to be bounded in finite time. Therefore, as long as the gain

For the boundedness of the estimation error of the PPD, when

When

Let the estimated error of the PPD be:

Take the absolute value on both sides of the above formula, and then the following inequality can be obtained from the triangle inequality:

Due to

Since it has been proved that the output of the PPD and the ESO are bounded, and the input and output of the system are also bounded, the estimation error of the PPD must be bounded.

Let us analyze the boundedness of the minimum-variance unbiased estimator, subtract y (k + 1) from both sides of

Since the estimation error of the PPD is bounded, the statistical characteristics of the overall output estimation of the system must be consistent with the statistical characteristics of the real output of the system, that is, the sum of the mean value of the partial output of the statistical characteristics and the output of the deterministic part must be consistent with the estimated value of the overall output of the system.

In summary, the algorithm for predicting the probability density evolution of the entire pattern movement and estimating system output consists of the following three main parts. The algorithm can be summarized as seven steps:

The following two cases were used to verify the effectiveness of the algorithm. In order to further validate the effectiveness of the method proposed in this article, the traditional nonlinear system analysis method based on pseudo partial derivatives presented in [

Firstly, 2000 sets of historical output data were collected, and according to the process of constructing the pattern category variables defined in definition 1, the classes were divided through ISODATA clustering. The system output is finally divided into 9 classes, and the historical data is shown in

The initial value of the ESO is

It can be observed from

2000 system historical output data were collected and pattern category variables were constructed through the process of Definition 1. The historical data and the distribution of pattern class centers are shown in

Given that the initial value of the ESO is

Based on the results from

This article explores the issue of output prediction in non-Newtonian mechanical systems and proposes an analysis framework based on pattern motion probability density evolution. Within this framework, a data-driven pattern motion probability density prediction method is developed using PPD and probability density evolution theory, which only requires the current input-output data of the system and its statistical characteristics. In addition, based on the system’s output probability density prediction, unbiased estimation of the system’s output value is achieved through minimum variance estimation. Finally, this article combines an ESO to design a corresponding PPD estimation algorithm. The effectiveness of the proposed method is verified through theoretical analysis and numerical simulation.

From the construction of pattern motion probability density evolution prediction method, it can be inferred that this method is an analysis approach that utilizes statistical variables. This implies that the applicability of this method is mainly restricted to systems governed by statistical laws, and when the statistical properties of the system are stronger compared to other dynamic characteristics, this method is more effective compared to general nonlinear analysis methods. In addition, the construction of pattern motion variables requires the prior construction of system motion subspaces and pattern category scale spaces, which require the collection of sufficient historical data and analysis. In fact, the larger the historical dataset, the more complete the constructed motion subspace. Although pattern recognition analysis of historical data usually takes a long time, the advantage is that this process is offline and does not affect the real-time performance of the prediction algorithm. Furthermore, unlike methods that require selecting representative points to solve the generalized probability density evolution equation. In this paper, a data-driven form based on pseudo partial derivatives is constructed to directly solve the corresponding probability density evolution equation, greatly reducing the computational complexity.

The authors wish to express their appreciation to the reviewers and editors for their helpful suggestions which greatly improved the presentation of this paper.

The authors received no specific funding for this study.

The authors confirm contribution to the paper as follows: study conception and design: Cheng Han, Zhengguang Xu; draft manuscript preparation: Cheng Han. All authors reviewed the results and approved the final version of the manuscript.

Not applicable. The experimental results in this article can be verified by computer simulation using the equations given in the example.

The authors declare that they have no conflicts of interest to report regarding the present study.