Periodic intermittent control method based on dynamic event triggering

Abstract

The invention discloses a periodic intermittent control method based on dynamic event triggering, and belongs to the technical field of control. According to the method, the problems that errors exist in measurement output, different time delays exist in a system and resources are limited are considered, and the actual engineering situation is reflected more truly and objectively. Meanwhile, the method has the advantages of being simple and easy to implement, and the stability and the overall performance of the nonlinear hybrid time-delay random system can be effectively improved. Through a dynamic event triggering mechanism and an intermittent controller, an expected control effect can be realized on the premise of effectively saving resources. The method comprises the following steps: S1, establishing a time-delay stochastic system dynamic model with a measurement error; s2, designing a dynamic event triggering strategy; s3, designing a periodic intermittent controller based on a dynamic event triggering strategy; and S4, obtaining sufficient conditions for ensuring the controlled system to realize the p-order moment index stability.

Description

Technical Field

[0001] The invention belongs to the field of control technology, and in particular relates to a feedback control based on a dynamic trigger strategy. Background Art

[0002] As an important dynamic model for describing time delay, nonlinear characteristics, and sudden changes in system parameters and structures, nonlinear hybrid time-delay stochastic systems have been widely used in the fields of economics and finance, physics, biology, and engineering control. Stability is the key to ensuring the safe and effective operation of the system. Therefore, it is particularly important to design a controller that can stabilize such systems. Periodic intermittent control can divide time into "working time" and "sleeping time", which can effectively reduce the usage rate of the controller and thus effectively extend the service life of the equipment. It has attracted much attention from the industry. Therefore, it is of practical significance to study the use of intermittent control methods based on dynamic event triggering to stabilize unstable nonlinear hybrid time-delay stochastic systems.

[0003] In actual operation, due to reasons such as equipment aging, human error, and data rounding errors, it is very easy to generate errors when obtaining measurement output information. Designing controllers using measurement values ​​containing errors may reduce system performance and even make it difficult to achieve the desired control effect. On the other hand, with the development of technology, more and more information is transmitted through the network, which is prone to congestion and collision. In addition, the limitation of network bandwidth is also one of the factors that need to be considered. Therefore, designing effective and low-consumption control mechanisms to meet these challenges is a key link in improving the stability and overall performance of nonlinear hybrid time-delay stochastic systems.

[0004] Existing intermittent control methods are difficult to solve the stabilization problem of nonlinear hybrid time-delay stochastic systems that are simultaneously subject to measurement errors and network resource constraints. If traditional control methods are used to control such systems, the desired performance requirements may not be achieved. Summary of the invention

[0005] In order to cope with the problem that a nonlinear hybrid time-delay stochastic system is difficult to achieve stable operation in the presence of measurement loss and limited network bandwidth, the present invention provides a periodic intermittent control method based on dynamic event triggering. The method takes into account the phenomenon of errors in the measurement output, the different time delays in the system, and the problem of resource constraints, so as to more realistically and objectively reflect the actual engineering situation. At the same time, the method has the advantages of being simple and easy to implement, and can effectively improve the stability and overall performance of the nonlinear hybrid time-delay stochastic system. Through the dynamic event triggering mechanism and the intermittent controller, the expected control effect can be achieved under the premise of effectively saving resources.

[0006] The technical solution adopted by the present invention is:

[0007] A periodic intermittent control method based on dynamic event triggering comprises the following steps:

[0008] S1. Establish a dynamic model of a time-delay stochastic system with measurement errors;

[0009] S2. Design dynamic event triggering strategy based on the dynamic model of time-delay stochastic system;

[0010] S3. Design of periodic intermittent controller based on dynamic event triggering strategy;

[0011] S4. Based on the periodic intermittent controller, sufficient conditions are obtained to ensure that the controlled system achieves p-order moment exponential stability.

[0012] Compared with the prior art, the present invention has the following beneficial effects:

[0013] 1. The present invention takes into account the existence of measurement errors and the impact of limited network resources on dynamic systems, and designs a periodic intermittent control strategy based on a dynamic event triggering strategy. Compared with the traditional state feedback control strategy, the control method proposed in the present invention can effectively reduce the utilization rate of the controller and extend the service life of the equipment. The proposed periodic intermittent control strategy based on the dynamic event triggering strategy can reduce the transmission frequency of the output signal, solving the problem that the existing control method is difficult to simultaneously handle measurement errors and output feedback control under limited network bandwidth.

[0014] 2. This method has the advantages of saving network resources and extending the service life of equipment, and effectively improves the stability and overall performance of nonlinear hybrid time-delay random systems. Through the dynamic event trigger mechanism, the transmission of redundant signals can be effectively prevented, ensuring cost savings while achieving the expected control effect.

[0015] 3. The present invention establishes sufficient conditions to ensure that the controlled system achieves p-order moment exponential stability by means of comparison principle and M-matrix technology. The above method realizes that the designed periodic intermittent control strategy can still achieve the expected stability in the presence of measurement errors and limited network resources.

[0016] 4. The periodic intermittent control method based on dynamic event triggering strategy proposed in the present invention can effectively control unstable nonlinear hybrid time-delay random systems. BRIEF DESCRIPTION OF THE DRAWINGS

[0017] Figure 1 It is a flow chart of a control method for stabilizing a nonlinear hybrid time-delay stochastic system by periodic intermittent control based on a dynamic event triggering strategy according to the present invention;

[0018] Figure 2 It is a random state trajectory diagram without control;

[0019] Figure 3 It is a periodic intermittent control input trajectory diagram based on a dynamic event trigger mechanism;

[0020] Figure 4 It is the state trajectory diagram of the closed-loop system under the periodic intermittent control of the dynamic event trigger mechanism;

[0021] Figure 5 It is a control input trajectory diagram based on the dynamic event trigger mechanism;

[0022] Figure 6 It is a state trajectory diagram of a closed-loop system under the control of a dynamic event trigger mechanism;

[0023] Figure 7 is the trajectory diagram of periodic intermittent control input;

[0024] Figure 8 It is the state trajectory diagram of the closed-loop system under periodic intermittent control. DETAILED DESCRIPTION

[0025] In order to better understand the purpose, structure and function of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings.

[0026] Reference Figure 1 As shown, the present invention provides a periodic intermittent control method based on dynamic event triggering. First, a state space model of a nonlinear hybrid time-delay random system is constructed, and a dynamic event triggering strategy related to time delay is designed considering the existence of measurement errors. Secondly, the measurement information obtained by transmitting the event triggering strategy is used to design a periodic intermittent control strategy. Finally, a judgment basis is given to ensure that the controlled nonlinear hybrid time-delay random system achieves p-order moment exponential stability. Figure 1 As shown, the method comprises the following steps:

[0027] Step 1: Establish a dynamic model of a nonlinear hybrid time-delay stochastic system with different time delays.

[0028] In this step, the dynamic model of the nonlinear hybrid time-delay stochastic system with measurement errors is established as follows:

[0029] dx(t)=(f(x(t-η1(t)),m(t),t)+u(t))dt+g(x(t-η2(t)),m(t),t)dB(t)

[0030]

[0031] x(t)=ξ(t),t∈[-η,0]

[0032] Where x(t), x(t-η1(t)) and x(t-η2(t)) represent the state vectors of the time-delay random system at time t, t-η1(t) and t-η2(t), respectively; u(t) represents the control input of the time-delay random system at time t; y(t) represents the measured output of the time-delay random system at time t; f(x(t-η1(t)), m(t), t) is the nonlinear function of the drift part related to the time-delay random system at time t, t-η1(t) and the mode at time t; g(x(t-η2(t)), m(t), t) is the nonlinear function of the diffusion part related to the time-delay random system at time t, t-η2(t) and the mode at time t; η1(t) and η2(t) respectively describe the time lag of the drift part and the diffusion part. Characterize the measurement error of the output at time t, satisfying is the maximum value of the measurement error, h(x(t), m(t), t) represents the output function related to the system state x(t) and the mode at time t, t∈[-η,0] represents that time t is an element in the closed interval formed by the left endpoint η and the right endpoint 0, the constant η is used to maximize η1(t) and η2(t), ξ(t) represents the initial value function of the state of the time-delay random system, and B(t) is defined in the complete probability space (Ω,F,{F t} t≥0 ,P) in standard Brownian motion, where t represents time, Ω, F, {F t} t≥0 , P represent the sample space set, set family, filter and probability measure respectively, and the filter is right continuous and F0 contains all P empty sets, d is the differential operator, m(t) describes the mode of the time-delay stochastic system at time t, and takes values ​​in the state space M = {1,2,…,r}, {1,2,…,r} represents the set composed of elements 1,2,…,r, r represents the number of modes of the Markov chain, and the generator of the Markov chain Θ = (θ ij ) r×r Determined by

[0033]

[0034] In the formula, P{m(t+l)=j|m(t)=i} describes the probability of transferring from mode i to mode j, l is a constant greater than 0, and θ ij represents the rate of transfer from mode i to mode j, o(l) is an infinitesimal quantity, and the nonlinear function of the drift part, the nonlinear function of the diffusion part and the output function satisfy

[0035] |f(x(t),m(t),t)|≤ε1|x(t)|

[0036] |g(x(t),m(t),t)|≤ε2|x(t)|

[0037] |h(x(t),m(t),t)|≤ε3|x(t)|

[0038] x(t) T f(x(t),m(t),t)+0.5|g(x(t),m(t),t)| 2 ≤ε 4,m(t) |x(t)| 2

[0039] Where ε1, ε2 and ε3 are the linear growth coefficients of the nonlinear functions f, g and h, ε 4,m(t) represents the coefficients of the nonlinear functions f and g that satisfy the amplification conditions at time t, and |g| represents the Euclidean norm of the vector g.

[0040] Step 2: Based on the nonlinear hybrid time-delay stochastic system dynamic model established in step 1, a time-delay-related dynamic event triggering strategy is designed. The time-delay-related dynamic event triggering strategy is as follows:

[0041]

[0042] In the formula, represents the measured output value that will be transmitted to the controller at time t, express The measured output value at the time, and Respectively represent Second and The time sequence of releasing the output signal through the dynamic event trigger strategy is determined by the following method:

[0043]

[0044] In the formula, min means taking the minimum value operation, E means performing the expected operation, |y(t)| 2 represents the square of the Euclidean norm of the vector y(t), σ1 and σ2 are the parameters in the event generation function, and χ(t) is the dynamic variable related to time t in the dynamic event triggering mechanism. Its update equation is

[0045]

[0046] Where σ3 and σ4 are parameters in the dynamic variables;

[0047] Step 3: According to the time-delay-related dynamic event triggering strategy described in step 2, a periodic intermittent controller based on the dynamic event triggering strategy is designed. The specific form of the periodic intermittent controller is:

[0048]

[0049] In the formula, Represents the measured output obtained at time t and the control input nonlinear function designed for the mode m(t), [lγ, (l+μ)γ) represents the time interval of intermittent controller operation, [(l+μ)γ, (l+1)γ) represents the time interval of intermittent controller sleep, l = 0, 1, 2, L represents the number of cycles, μ represents the intermittent control rate, γ represents the cycle length, and the nonlinear function satisfy

[0050]

[0051] Where ε5 is the linear growth coefficient of the nonlinear function k. The input without dynamic event triggering transmission mechanism and intermittent control satisfies

[0052] x(t) T (f(x(t),m(t),t)+u(y(t),m(t),t))+0.5|g(x(t),m(t),t)| 2 ≤-ε 6,m(t) |x(t)| 2

[0053] In the formula, ε 6,m(t) Represents the coefficients of the nonlinear functions f, g and u that satisfy the amplification conditions at time t.

[0054] Step 4: Based on the periodic intermittent controller designed in step 3, obtain sufficient conditions to ensure that the controlled system achieves p-order moment exponential stability.

[0055] The sufficient condition to ensure the controlled system achieves p-order moment exponential stability is:

[0056] The upper bound of the time delay in the system satisfies η∈(0,η * ), η* is the root of the equation J(η) = 0, and the intermittent control rate satisfies The period of the intermittent controller is

[0057] in,

[0058]

[0059]

[0060] Where J(η) is a nonlinear function related to the upper bound of the time lag η. is the maximum value parameter, π is an adjustable parameter, H is an arbitrary positive integer, δ1, δ2, δ3, δ4 and δ5 are the five transition parameters in the judgment condition, χ0 is the initial value of the dynamic variable, |ξ| is the maximum value of the norm of the initial value function, e is a natural constant, log represents the logarithmic operation, λ is the order of exponential decay, and κ are the sets {κ1,κ2,…,κ r}, κ1,κ2,…,κ r is the element of the matrix obtained by multiplying the inverse of the M matrix H with a column vector of all 1s, that is, The superscript T means taking the transpose of the corresponding matrix, and the superscript -1 means taking the transpose of the corresponding matrix. The definition of the M matrix H is Represented by the element ε 4,1 ,ε 4,2 ,…,ε 4,r The diagonal matrix is ​​composed of θ∈(0,1), θ∈(0,1) is a free parameter, and the parameter max is the maximum value operation.

[0061] Example:

[0062] This embodiment takes a nonlinear hybrid time-delay random system affected by measurement errors and different time delays as an example, and uses the method of the present invention to perform simulation:

[0063] The correlation coefficient of the nonlinear hybrid time-delay stochastic system is set as follows:

[0064] The value space of Markov chain m(t) is M = {1,2,3}, and the generator is

[0065] f(x(t-η1(t)),1,t)=0.22x(t-η1(t))|cos(x(t-η1(t)))|, h(x(t),1,t)=0.35x(t)

[0066] f(x(t-η1(t)),2,t)=0.2x(t-η1(t))|cos(x(t-η1(t)))|, h(x(t),2,t)=0.33x(t)

[0067] f(x(t-η1(t)),3,t)=0.19x(t-η1(t))|cos(x(t-η1(t)))|, h(x(t),3,t)=0.3x(t)

[0068] g(x(t-η2(t)),1,t)=0.14|x(t-η2(t))|,k(y(t),1,t)=-2.261y(t)

[0069] g(x(t-η2(t)),2,t)=0.15|x(t-η2(t))|,k(y(t),2,t)=-2.338y(t)

[0070] g(x(t-η2(t)),3,t)=0.11|x(t-η2(t))|, k(y(t),3,t)=-2.52y(t),

[0071] The values ​​of other simulation parameters are selected as follows: Measurement error Intermittent control rate μ = 0.9, intermittent control period γ = 0.1042, moment stability order p = 1, free parameters The attenuation order λ = -0.7870, the time lag η1(t) = 0.0001|sint| and η2(t) = 0.0001|cost|, an arbitrary positive integer H = 1, the initial value ξ(t) = 10, and the parameters in the dynamic event triggering mechanism are: χ(0) = 1, σ1 = 50, σ2 = 300 and σ4 = 80.

[0072] Effect of periodic intermittent controller based on dynamic event triggering:

[0073] Depend on Figure 2 to Figure 8 It can be seen that the control strategy based on the dynamic event trigger mechanism, the periodic intermittent control strategy and the periodic intermittent control strategy based on the dynamic event trigger mechanism designed by the present invention can all effectively stabilize unstable nonlinear hybrid random systems. Compared with the first two control strategies, the control method of the present invention can not only effectively reduce the transmission frequency of the output signal, but also reduce the utilization rate of the controller.

[0074] It is to be understood that the present invention is described by some embodiments, and it is known to those skilled in the art that various changes or equivalent substitutions may be made to these features and embodiments without departing from the spirit and scope of the present invention. In addition, under the teachings of the present invention, these features and embodiments may be modified to adapt to specific circumstances and materials without departing from the spirit and scope of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed herein, and all embodiments falling within the scope of the claims of this application are within the scope of protection of the present invention.

Claims

1. A periodic intermittent control method based on dynamic event triggering, characterized in that: The following steps are involved: S1. Establish a dynamic model of a time-delay stochastic system with measurement errors; S2. Design dynamic event triggering strategy based on the dynamic model of time-delay stochastic system; S3. Design of periodic intermittent controller based on dynamic event triggering strategy; S4. Based on the periodic intermittent controller, sufficient conditions are obtained to ensure that the controlled system achieves p-order moment exponential stability.

2. The periodic intermittent control method based on dynamic event triggering according to claim 1 is characterized in that: The dynamic model of the time-delay stochastic system in S1 is: dx(t)=(f(x(t-η1(t)),m(t),t)+u(t))dt+g(x(t-η2(t)),m(t),t)dB(t) x(t)=ξ(t),t∈[-η,0] Where x(t), x(t-η1(t)) and x(t-η2(t)) represent the state vectors of the time-delay random system at time t, t-η1(t) and t-η2(t), respectively; u(t) represents the control input of the time-delay random system at time t; y(t) represents the measured output of the time-delay random system at time t; f(x(t-η1(t)), m(t), t) is the nonlinear function of the drift part related to the time-delay random system at time t, t-η1(t) and the mode at time t; g(x(t-η2(t)), m(t), t) is the nonlinear function of the diffusion part related to the time-delay random system at time t, t-η2(t) and the mode at time t; η1(t) and η2(t) respectively describe the time lag of the drift part and the diffusion part. It describes the measurement error of the output value at time t, h(x(t), m(t), t) represents the output function related to the system state x(t) and the mode at time t, m(t) describes the mode of the time-delay random system at time t, m(t) is a right-continuous Markov chain, t∈[-η,0] means that time t is an element in the closed interval formed by the left endpoint η and the right endpoint 0, the constant η is used to describe the maximum value of η1(t) and η2(t), ξ(t) represents the initial value function of the state of the time-delay random system, B(t) is defined in the complete probability space (Ω,F,{F t } t≥0 ,P) in standard Brownian motion, where t represents time, Ω, F, {F t } t≥0 , P represent sample space set, set family, filter and probability measure respectively, and the filter is right continuous and F0 contains all P empty sets, d is the differential operator.

3. The periodic intermittent control method based on dynamic event triggering according to claim 2 is characterized in that: In the time-delay stochastic system dynamic model of S1, The nonlinear functions f, g and h satisfy |f(x(t),m(t),t)|≤ε1|x(t)| |g(x(t),m(t),t)|≤ε2|x(t)| |h(x(t),m(t),t)|≤ε3|x(t)| x(t) T f(x(t),m(t),t)+0.5|g(x(t),m(t),t)| 2 ≤ε 4,m(t) |x(t)| 2 Where ε1, ε2 and ε3 are the linear growth coefficients of the nonlinear functions f, g and h, ε 4,m(t) represents the coefficients of the nonlinear functions f and g that satisfy the amplification condition at time t, and |g| represents the Euclidean norm of the vector g; Measurement error of the measured output at time t satisfy is the maximum value of the measurement error.

4. The periodic intermittent control method based on dynamic event triggering according to claim 3 is characterized in that: In the time-delay stochastic system dynamic model of S1, the Markov chain m(t) takes values ​​in the state space M = {1,2,…,r}, where {1,2,…,r} represents a set composed of elements 1,2,…,r, r represents the number of modes of the Markov chain, and the generator Θ of the Markov chain = (θ ij ) r×r Determined by In the formula, It describes the probability of transferring from mode i to mode j. is a constant greater than 0, θ ij represents the rate of transfer from mode i to mode j, for An infinitesimal amount.

5. The periodic intermittent control method based on dynamic event triggering according to claim 4 is characterized in that: In S2, a dynamic event triggering strategy is designed, and the specific steps are as follows: In the formula, represents the measured output value that will be transmitted to the controller at time t, express The measured output value at the time, and Respectively represent Second and The time sequence of releasing the output signal through the dynamic event trigger strategy is determined by the following method: In the formula, min means taking the minimum value operation, E means performing the expected operation, |y(t)| 2 represents the square of the Euclidean norm of the vector y(t), σ1 and σ2 are parameters in the event generation function, and χ(t) is the dynamic variable related to time t in the dynamic event triggering mechanism.

6. The periodic intermittent control method based on dynamic event triggering according to claim 5, characterized in that: In S2, the update equation of the dynamic variable χ(t) is Where σ3 and σ4 are parameters in the dynamic variables.

7. The periodic intermittent control method based on dynamic event triggering according to claim 6 is characterized in that: In S3, the specific form of the periodic intermittent controller is: In the formula, Represents the measured output obtained at time t And the control input nonlinear function designed for the mode m(t), [lγ,(l+μ)γ) represents the working time interval of the intermittent controller, [(l+μ)γ,(l+1)γ) represents the sleeping time interval of the intermittent controller, l=0,1,2, L represents the number of cycles, μ represents the intermittent control rate, and γ represents the cycle length.

8. The periodic intermittent control method based on dynamic event triggering according to claim 7, characterized in that: In the periodic intermittent controller of S3, the nonlinear function satisfy Where ε5 is the linear growth coefficient of the nonlinear function k. The input without dynamic event triggering transmission mechanism and intermittent control satisfies x(t) T (f(x(t),m(t),t)+u(y(t),m(t),t))+0.5|g(x(t),m(t),t)| 2 ≤-ε 6,m(t) |x(t)| 2 In the formula, ε 6,m(t) Represents the coefficients of the nonlinear functions f, g and u that satisfy the amplification conditions at time t.

9. The periodic intermittent control method based on dynamic event triggering according to claim 8, characterized in that: In S4, the sufficient condition to ensure that the controlled system achieves p-order moment exponential stability is: The upper bound of the time delay in the system satisfies η∈(0,η * ), η * is the root of the equation J(η)=0, and the intermittent control rate satisfies The period of the intermittent controller is Where J(η) is a nonlinear function related to the upper bound of the time lag η. is the maximum value parameter, π is an adjustable parameter, and H is an arbitrary positive integer.

10. The periodic intermittent control method based on dynamic event triggering according to claim 9, characterized in that: In S4, J(η), and π is defined as in, In the formula, δ1, δ2, δ3, δ4 and δ5 are five transition parameters in the discrimination conditions, χ0 is the initial value of the dynamic variable, |ξ| is the maximum value of the norm of the initial value function, e is a natural constant, log represents the logarithmic operation, λ is the order of exponential decay, and κ are the sets {κ1,κ2,…,κ r }, max is the maximum value operation, κ1,κ2,…,κ r is the element of the matrix obtained by multiplying the inverse of the M matrix H with a column vector of all 1s, that is, The superscript T means taking the transpose of the corresponding matrix, and the superscript -1 means taking the transpose of the corresponding matrix. The definition of the M matrix H is Represented by the element ε 4,1 ,ε 4,2 ,…,ε 4,r The diagonal matrix is ​​composed of θ∈(0,1) and θ∈(0,1) is a free parameter.

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