A hybrid singular system modeling and stability analysis method based on multi-sampling rates

Abstract

The present invention belongs to the field of industrial Internet and control technology. In view of the phenomenon that continuous dynamic processes, discrete events, and the diversity of controlled objects and multiple time scales coexist in industrial Internet systems, a hybrid singular system modeling and stability analysis method based on multiple sampling rates is proposed to solve the stability analysis and controller design problems of the proposed model, while considering the influence of multiple sampling rates and singular perturbation parameters on system stability. The present invention can achieve the complex characteristics of multiple system couplings such as continuous dynamics, multi-sampling, discrete events and multiple time scales, that is, effectively improve the system operation performance and anti-interference performance. At the same time, the hybrid singular perturbation system model established by the Lyapunov function method is applicable to more complex actual systems and also provides a better compromise between system performance and execution cost, that is, it has strong feasibility, which means that it has good practical application prospects.

Description

Technical Field

[0001] The present invention belongs to the field of industrial Internet and control technology. More specifically, in view of the existence of continuous dynamic processes, discrete events, and the diversity of controlled objects and the coexistence of multiple time scales in industrial Internet systems during industrial operation, a hybrid singular system modeling and stability analysis method based on multiple sampling rates is proposed to solve the stability analysis and controller design problems of the proposed model, while considering the influence of multiple sampling rates and singular perturbation parameters on system stability. Background Art

[0002] In industrial production, the increasing variety of controlled objects, coupled with their multi-scale temporal characteristics, time lags, and mismatches between controllers and sensors, leads to significant variations in the rate of change of individual signals. The sampling periods of detection devices also vary, making the use of a single sampling period impractical, if not impossible. Furthermore, a large number of research objects in engineering exhibit significant multi-timescale characteristics. Due to the system's inherent multi-process nature and the influence of complex external environmental factors, it is essential to appropriately describe dynamics at different times. Almost all large-scale systems experience the coexistence of dynamics with significantly different timescales. For example, in systems such as metallurgy, chemical engineering, power generation, and robotics, relatively fast industrial production dynamics, measured on minute-to-second timescales, coexist with relatively slow scheduling dynamics, measured on lunar-day timescales.

[0003] In order to solve the multi-time scale modeling in industrial interconnected systems, many scholars have begun to devote themselves to the study of hybrid singular perturbation systems. Hybrid singular perturbation systems are a type of hybrid system with fast-changing and slow-changing dynamics. Its essence is that it has both multiple time scale characteristics and continuous and discrete dynamic hybrid characteristics. Regarding its performance analysis, early research work mainly focused on the pulse hybrid singular perturbation system model, the switching hybrid singular perturbation system model and the general hybrid singular perturbation system model. For example, in the literature "JBRejeb,TC In A. Girard et al., "Stability analysis of a general class of singularly perturbed linear hybrid systems" [J], Automatica, vol. 90, pp. 98-108, 2018, the authors considered a class of singularly perturbed linear systems with switching and impulses. The sampling of the fast and slow variables in such systems is dynamically dependent. Using singular perturbation decomposition techniques, they calculated an upper bound on the minimum dwell time required to ensure system stability. However, this method is not applicable to general hybrid singularly perturbed systems. Furthermore, since the sampling requirements for fast and slow states differ, multi-rate sampling is crucial in the study of hybrid singularly perturbed systems to achieve cost savings, improve efficiency, and enhance system performance and anti-interference capabilities. For example, in the paper "WH Chen, HH He, XMLu. Multi-rate sampled-data composite control of linear singularly perturbed systems" [J], Journal of the Franklin Institute, vol. 357, no. 4, pp. 2028-2048, 2020." The authors discussed the multi-rate stabilization problem of a class of singularly perturbed systems, in which the sampling times of the slow-rate and fast-rate state variables are asynchronous and non-uniform. A new time-varying Lyapunov function was introduced, and sufficient conditions for the exponential stability of the closed-loop system were derived based on linear matrix inequalities. However, the authors mainly considered linear systems with asynchronous and non-uniform sampling times of the slow-rate and fast-rate state variables.

[0004] In the existing technology, although great progress has been made in the theory of singularly perturbed systems, little consideration has been given to specific implementation issues. In order to reduce the conservatism of the results and propose more accurate optimization algorithms, there are still many open issues worth considering and solving:

[0005] (1) Traditional modeling methods focus on modeling single systems such as pure discrete, pure continuous, and single sampling rate systems. They cannot be applied to multi-sampling rate systems where continuous dynamics and discrete events coexist. Therefore, new modeling methods need to be considered.

[0006] (2) Regarding the stability analysis of a system with multiple sampling rates and singular perturbations, only a single sampling is given, ignoring the case of multiple sampling rates or not accurately giving the upper bound of the multi-sampling period. Different sampling rates have different effects on the stability of the system.

[0007] (3) For general hybrid singular perturbation systems, only their semi-global asymptotic stability has been studied, and few criteria for exponential stability and robust stability have been given. Stability and robustness are extremely important for the performance indicators of the system.

[0008] In summary, traditional modeling and stability analysis methods for industrial interconnected systems cannot simultaneously meet factors such as pure discreteness, pure continuousness, and single sampling rate. Therefore, a new hybrid system modeling method that can describe fast and slow dual time scales and multiple sampling rates is urgently needed. Summary of the Invention

[0009] In response to the existing phenomenon of asynchronous sampling of fast and slow variables in singular perturbation systems, the present invention proposes a modeling, performance analysis and controller design method for a class of multi-sampling rate hybrid singular systems, establishes a class of hybrid singular perturbation system models with different sampling rates, then combines multi-sampling rate technology and hybrid theory technology to design a novel multi-sampling rate controller for the established model, and uses the Lyapunov function method to achieve global asymptotic stability of the established model; in addition, the allowable sampling interval and the value of the singular perturbation parameter are estimated, and the quantitative relationship between the singular perturbation parameter and the sampling interval is analyzed. These theorems are applied to the analysis of a DC motor model, and the effectiveness is verified through simulation experiments. Compared with traditional singular models, the hybrid singular perturbation system model established in the present invention includes both continuous and discrete systems, and can also reflect the singular characteristics of fast and slow state changes under different sampling rates.

[0010] In order to achieve the above object, the technical solution of the present invention is as follows:

[0011] A hybrid singular system modeling and stability analysis method based on multiple sampling rates. First, a controller with multiple sampling rates is designed under different sampling rates for fast-changing and slow-changing states, and a new performance indicator is constructed, which can be applied to more complex practical systems. Secondly, based on the designed controller, a hybrid singular perturbation system model with multiple sampling rates is established by introducing auxiliary variables and parameter transformation techniques. Finally, an auxiliary function is constructed, and the Lyapunov function method is used to give the sufficient conditions for the uniform global exponential stability of the system. The maximum allowable parameter is then calculated and used to analyze the relationship between the sampling interval and the perturbation parameter. The specific steps are as follows:

[0012] Step 1: Build a hybrid singular system model with multiple sampling rates

[0013] S1.1 In practical systems, the dynamic equations of a general nonlinear hybrid singular perturbation system can be expressed as:

[0014]

[0015] in and Represent the slow state, fast state and control input respectively, It means the derivative of x1, It represents the derivative of x2, n1, n2, n3 represent the dimension of the vector, R n represents the n-dimensional vector space, ε represents the singular perturbation parameter, and f and g are continuously differentiable functions.

[0016] S1.2 Design of multi-sampling rate controller:

[0017] In a mixed singular perturbation system, there are different fast and slow states. If sampling is based on the fast-changing state, the sampling rate is high, resulting in wasted resources and high costs. If sampling is based on the slow-changing state, the sampling rate is slow, resulting in performance degradation or even instability in the fast-changing subsystem. In addition, the sampling periods of the detection devices in the system are also different. To solve these problems, it is necessary to appropriately sample the dynamics at different times, so multiple sampling rates are considered.

[0018] For the dynamic equations of the nonlinear hybrid singular perturbation system in step S1.1, the zero-order hold principle is used to design a controller with multiple sampling rates as follows:

[0019]

[0020] Where x1 represents the slow state and x2 represents the fast state. and Respectively represent the slow state and the fast state at t k Moment and s j Sampling is performed at the moment, h is a continuously differentiable function, t k With s j is a positive real number, k and j are positive integers, and the influence of multiple sampling rates and singular perturbation parameters on the system is fully considered.

[0021] S1.3 Establish a hybrid singular perturbation system model based on multiple sampling rates:

[0022] In step S1.2, the controller design problem for a hybrid singular perturbation system with multiple sampling rates is studied. In this step, the controller designed in step S1.2 is introduced to address the asynchronous sampling of fast and slow variables in industrial processes. Auxiliary parameters are added, and a class of hybrid singular perturbation system models with different sampling rates is established using hybrid theory. The details are as follows:

[0023] make x=(x1 T ,x2 T ),ξ1=(x1 T ,e1 T ) T ,ξ2=(x2 T,e2 T ) T , e=(e1 T ,e2 T ),ξ=(ξ1 T ,ξ2 T ) T ,Combining Equation (1) in step S1.1 and Equation (2) in step S1.2, the hybrid singular perturbation system model with multiple sampling rates can be described as:

[0024]

[0025]

[0026] Where τ is an auxiliary variable, t∈[t k ,t k+1 ), k,j=0,1,2,3…,κ represents the number of fast state sampling,κ|N represents that κ can be divided by N, Expressing that κ cannot be divided by N, τ limits the sampling interval change of the fast state, represents the maximum allowable sampling period, f1 and g1 are continuously differentiable functions, T1 and T2 are the sampling periods of the slow-changing state and the fast-changing state respectively, and satisfy T1=NT2, N represents the multiple. On this basis, construct and s J are the sampling sequences of the slow state x1 and the fast state x2, respectively, where k∈Z + , t0=s0,Z + is a set of non-negative integers.

[0027] Step 2: Stability analysis of singularly perturbed systems

[0028] S2.1 Obtain the maximum allowable sampling parameters

[0029] First, by solving the ordinary differential equation H(τ)=φ(τ)exp(pτ) and estimating the value of T2, satisfy:

[0030]

[0031] Where L and ε are positive parameters given by the system, p is a positive constant, φ is a function, and φ(τ)∈(0,+∞). Then there exists T2>0 such that for any τ∈[0,T2], H(τ)>φ(0)λ, where λ∈(0,1),

[0032] Then, according to When , solve the maximum parameter that satisfies T2 in S2.1 in

[0033] S2.2 Constructing W and V functions

[0034] First, construct a local Lipschitz function W about the error function e: Such that there exists a continuous function A positive constant L and λ∈(0,1) satisfy:

[0035] W(κ + ,e + ,τ + )≤λW(κ,e,τ),

[0036]

[0037] in

[0038] Then, construct the local Lipischitz function with respect to the state x Make

[0039] Where m∈[1-l,l+1], p is a positive constant.

[0040] S2.3 Based on the W function and V function in S2.2, let The Lyapunov function is constructed as follows:

[0041]

[0042] S2.4 In order to verify the stability of the system (3)-(4), it is necessary to search for a positive integer K so that for T2 obtained in S2.1 * and given positive constants T2, l1, l2 and l3, when Sometimes, there are and

[0043] S2.5 Based on the structure of the constructed Lyapunov function (5), we need to verify the stability of system (3)-(4) in two cases. First, consider case 1: τ∈[0,T2]. For case 1, only system (3) is meaningful, and the discrete subsystem (4) does not occur. Therefore, we only need to consider the change of function U in system (3). Then, from S2.2, we can get:

[0044]

[0045] in p = 2l3 is a positive constant; when inequality (6) holds, it means that the energy of function U decays along system (3), proving that when τ∈[0,T2], the system is stable;

[0046] S2.6 Consider Case 2: For case 2, both systems (3) and (4) make sense, so we need to analyze the energy change of function U in two sub-cases:

[0047] Subcase 2.1: The energy variation of function U along system (3) is analyzed as follows:

[0048]

[0049] From S2.4, inequality (7) can be transformed into:

[0050]

[0051] Subcase 2.2: The energy of function U along the energy of system (4) is analyzed as follows: Using S2.2, we can get:

[0052]

[0053] When inequality (8) holds, it means that the energy of function U is attenuated along the system (3)-(4), which proves that when The system is stable;

[0054] S2.7 Order Combined with the situation of S2.5-S2.6, it is calculated that:

[0055]

[0056] From S2.1, we know that there exist positive numbers T2 and λ∈(0,1) such that for any τ∈(0,T2), H(τ)>φ(0)λ, i.e., when φ(T2)exp(pT2)>φ(0)λ, so

[0057] Therefore, when inequality (9) holds, the function U decays exponentially along the entire system (3)-(4), indicating that the multi-rate hybrid singular perturbation system (3)-(4) is uniformly globally exponentially stable.

[0058] The present invention is designed for sampling control of hybrid singular perturbation systems. In order to design and improve the controller of formula (2), a singular perturbation system that can describe both fast and slow time scales is adopted, and the sampling moments are all different, i.e., multiple sampling rates. Then, based on the model established by the hybrid singular perturbation system formulas (3)-(4), a Lyapunov function is constructed, and the exponential stability of the constructed model is verified in two cases. Finally, sufficient conditions for the model to be stable and the allowable range of variation of the singular perturbation parameters and the multiple sampling periods are given, and the judgment conditions for global asymptotic stability are solved. Through the above rigorous theoretical analysis, the feasibility of the new method for judging the asymptotic stability of the hybrid singular perturbation system in the technical solution proposed by the present invention is proved, and the influence of the singular perturbation parameters and the multiple sampling rates on the overall system is also considered.

[0059] Compared with the prior art, the present invention has the following beneficial effects:

[0060] (1) In view of the coexistence of continuous dynamics and discrete events in industrial operation and the diversity of sampling periods of controlled objects, the present invention studies the mechanism that can simultaneously describe the integration of continuous dynamics, discrete events, multi-sampling and multi-time scale characteristics, breaks through the traditional modeling methods (such as pure discrete, pure continuous, single sampling rate, singular perturbation and other single system modeling methods), and proposes a multi-sampling rate mixed singular perturbation system modeling method, which is a beneficial aspect of the present invention.

[0061] (2) Compared with previous research results, the present invention constructs a Lyapunov function with a novel structure, and ensures the stability of the established model by constructing a piecewise Lyapunov function containing singular parameters. It proposes a consistent global asymptotic stability judgment condition for nonlinear hybrid singular perturbation systems, breaking through the result that general hybrid singular perturbation systems are limited to semi-realistic global asymptotic stability. For the constructed model, combined with singular perturbation theory and hybrid theory, the judgment criteria of the global asymptotic stability and exponential stability of the model are given, the tolerance of the system to uncertain parameters under the condition of model uncertainty is studied, and the robust stability of the model is achieved. The proposed method and results are more practical and universal.

[0062] (3) The present invention studies the influence of multiple sampling rates on system performance and estimates the maximum allowable fast and slow sampling intervals under the condition of system stability. It considers multiple sampling rates instead of the traditional single sampling rate, so that there are two or more sampling rates in the system. The sampling control system is applicable to complex practical systems and analyzes the stability of the system. At the same time, it also considers the influence of multiple sampling rates and singular perturbation parameters on the system, and proposes a new method for determining the asymptotic stability of a mixed singular perturbation system.

[0063] In summary, the hybrid singular perturbation system modeling based on multi-sampling rates and its stability analysis method proposed in the present invention can simultaneously describe the complex characteristics of multiple system couplings such as continuous dynamics, multi-sampling, discrete events and multiple time scales, that is, effectively improve the system operation performance and anti-interference performance. At the same time, the hybrid singular perturbation system model established using the Lyapunov function method is suitable for more complex actual systems, and also provides a better compromise between system performance and execution cost, that is, it has strong feasibility, which means that it has good practical application prospects. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 It is a process framework diagram of the present invention. DETAILED DESCRIPTION

[0065] The specific implementation of the present invention is further described below in conjunction with the accompanying drawings and technical solutions.

[0066] The basic process of the present invention is as follows Figure 1 To further verify the effectiveness and advancement of the proposed technology, a simulation comparison was conducted with the traditional modeling method for hybrid singular perturbation systems. The hybrid singular perturbation system based on multiple sampling rates can appropriately describe the dynamics at different times. Designed based on the Lyapunov function method and ordinary differential equation theory, this control scheme is easy to adjust and implement, making it widely used in the industrial Internet field.

[0067] Simulation: An armature-controlled DC motor is selected for speed regulation performance comparison.

[0068] This experiment addresses the modeling of hybrid singular perturbation systems at multiple sampling rates. A majority-rate sampling controller was designed and the control performance of the closed-loop system was verified. The DC motor model selected was an armature-controlled DC motor. The sampling rates for current x1 and speed x2 varied with different parameters. Typically, current sampling is fast, while speed sampling is slow.

[0069] First, a DC motor model is described as follows:

[0070]

[0071] where i a 、i f 、R a and L a The sub-tables represent current, voltage, resistance and inductance, ω and J represent speed and moment of inertia respectively, and c1, c2 and c3 represent voltage constant, torque constant and friction constant respectively. Then introduce the variable x1=ω, and The DC motor model can be described by the following singular perturbation system:

[0072]

[0073] in

[0074] Then, design the controller as follows:

[0075]

[0076] in t∈[t k ,t k+1 )and k,j∈Z + , then the DC motor model system and controller can be described as the hybrid singular perturbation system (3)-(4) in step S1.3, where δ1-δ4 is given by the following proof. Next, take Among them, according to the characteristics of the Lyapunov function construction, the theorem result is always valid for the discrete case in step S2.6, case 2, so it is only necessary to verify the continuous case in step S2.5, case 1.

[0077] To verify case 1, first calculate the following formula:

[0078]

[0079]

[0080] Then, combining (13)-(14) we can get

[0081]

[0082] use and For any τ∈(0,T2), we have

[0083]

[0084] where u j ,j=1,2…5 is a positive constant. Let μ1=1,select parameter δ j ,j=1,2,…4 satisfies the following conditions:

[0085]

[0086] and ε<ε * ,in

[0087]

[0088] Therefore, we can get

[0089]

[0090] and

[0091] Finally, according to the above parameter conditions and formula (15), we can obtain that there exists a constant p>0, satisfying Therefore, it can be seen from the results of the present invention that the system is consistent and stable, which verifies the effectiveness of the method of the present invention.

[0092] In summary, the simulation results demonstrate that, compared to traditional singular perturbation system modeling methods commonly used by researchers, the proposed technical solution offers superior response and stable control, enabling more accurate sampling of system parameters. Furthermore, the proposed technical solution offers advantages in modeling methods, effectively and accurately characterizing the system. This suggests that the proposed method is more practical and suitable for application in real-world systems.

Claims

1. A hybrid singular system modeling and stability analysis method based on multiple sampling rates, characterized by: The specific steps are as follows: Step 1: Build a hybrid singular system model with multiple sampling rates S1.1 In practical systems, the dynamic equations of nonlinear hybrid singular perturbation systems are expressed as: in and Represent the slow state, fast state and control input respectively, It means the derivative of x1, It represents the derivative of x2, n1, n2, n3 represent the dimension of the vector, R n represents the n-dimensional vector space, ε represents the singular perturbation parameter, f and g are continuously differentiable functions; S1.2 Design of multi-sampling rate controller: For the dynamic equations of the nonlinear hybrid singular perturbation system in step S1.1, the zero-order hold principle is used to design a controller with multiple sampling rates as follows: Where x1 represents the slow state and x2 represents the fast state; and and Respectively represent the slow state and the fast state at t k Moment and s j Sampling is performed at the moment, h is a continuously differentiable function, t k With s j Belongs to positive real numbers, k and j are positive integers; S1.3 Establish a hybrid singular perturbation system model based on multiple sampling rates: Introduce the controller designed in step S1.2, add auxiliary parameters, and use hybrid theory to establish a hybrid singular perturbation system model with different sampling rates, as follows: make x=(x1 T ,x2 T ),ξ1=(x1 T ,e1 T ) T ,ξ2=(x2 T ,e2 T ) T , e=(e1 T ,e2 T ),ξ=(ξ1 T ,ξ2 T ) T ,Combining Equation (1) in step S1.1 and Equation (2) in step S1.2, the model of the hybrid singular perturbation system with multiple sampling rates is described as: Where τ is an auxiliary variable, t∈[t k ,t k+1 ), k,j=0,1,2,3…,κ represents the number of fast state sampling,κ|N represents that κ can be divided by N, Expressing that κ cannot be divided by N, τ limits the sampling interval change of the fast state, represents the maximum allowable sampling period, f1 and g1 are continuously differentiable functions, T1 and T2 are the sampling periods of the slow-changing state and the fast-changing state respectively, and satisfy T1=NT2, N represents the multiple; on this basis, construct and s J are the sampling sequences of the slow state x1 and the fast state x2, respectively, where k∈Z + , t0=s0,Z + is a set of non-negative integers; Step 2: Stability analysis of singularly perturbed systems S2.1 Obtain the maximum allowable sampling parameters First, by solving the ordinary differential equation H(τ)=φ(τ)exp(pτ) and estimating the value of T2, satisfy: Where L and ε are positive parameters given by the system, p is a positive constant, φ is a function, and φ(τ)∈(0,+∞); then there exists T2>0 such that for any τ∈[0,T2], H(τ)>φ(0)λ, where λ∈(0,1), Then, according to When , solve the maximum parameter that satisfies T2 in S2.1 in S2.2 Constructing W and V functions First, construct a local Lipschitz function W about the error function e: Such that there exists a continuous function A positive constant L and λ∈(0,1) satisfy: W(k + ,e + ,t + )≤λW(κ,e,τ) in Then, construct the local Lipischitz function with respect to the state x Make Where m∈[1-l,l+1], p is a positive constant; S2.3 Based on the W function and V function in S2.2, let The Lyapunov function is constructed as follows: S2.4 In order to verify the stability of the system (3)-(4), it is necessary to search for a positive integer K so that for T2 obtained in S2.1 * and the given positive constants T2, T2 * , l1, l2 and l3, when Sometimes, there are and S2.5 Based on the structure of the constructed Lyapunov function (5), the stability of system (3)-(4) is verified in two cases: First, consider case 1: τ∈[0,T2]. For case 1, only system (3) is meaningful, and the discrete subsystem (4) does not occur. Therefore, we only need to consider the change of function U in system (3); then from S2.2, we can get: Case 1: When τ∈[0,T2], combining S2.1-S2.4, we can get: in F(x, e, ε) is as shown in step S2.2, and p = 2l3 is a positive constant. When inequality (6) holds, it means that the energy of function U is decaying along the system (3). It can be proved that when τ∈[0,T2], the system is stable. S2.6 Consider Case 2: For Case 2, both systems (3) and (4) are meaningful, so it is necessary to analyze the energy change of function U in two sub-cases: Sub-case 2.1: The energy change of function U along system (3) is analyzed as follows: From S2.4, inequality (7) can be transformed into Subcase 2.2: The energy of function U along the energy of system (4) is analyzed as follows: Using S2.2, we can get: When inequality (8) holds, it means that the energy of function U is attenuated along the system (3)-(4). Then it can be proved that when The system is stable; S2.7 Order Combined with the situation of S2.5, it is calculated that: From S2.1, we know that there exist positive numbers T2 and λ∈(0,1) such that for any τ∈(0,T2), H(τ)>φ(0)λ, i.e., when φ(T2)exp(pT2)>φ(0)λ, so Therefore, when inequality (9) holds, the function U decays exponentially along the entire system (3)-(4), indicating that the multi-rate hybrid singular perturbation system (3)-(4) is uniformly globally exponentially stable.

Images