CN-120540077-B - Fixed time synchronization method for heterogeneous four-pendulum system based on pulse control

Abstract

The invention discloses a fixed time synchronization method of a heterogeneous four-pendulum system based on pulse control. The method firstly designs a fixed time controller based on pulse control for a four-pendulum system with heterogeneous characteristics. Then, a novel Lyapunov functional is designed, pulse intervals are partitioned, a convex combination technology is used, and a fixed time synchronization sufficient condition represented by a matrix inequality is established, so that four-pendulum system synchronization is realized. A comparison system is constructed to estimate settling time in the case of sync pulses, inactive pulses, and non-sync. When the method is applied to the heterogeneous four-pendulum system, the stable time for realizing synchronization of the system is reduced, and the actual efficiency is improved.

Inventors

• SHEN MOUQUAN
• YE YUFEI
• ZHANG ZHIHAO

Assignees

• 南京工业大学

Dates

Publication Date

20260512

Application Date

20250520

Claims (1)

1. 1. The fixed time synchronization method of the heterogeneous four-pendulum system based on pulse control is characterized by comprising the following steps of: The fixed time controller based on pulse control is designed for the four-pendulum system with heterogeneous characteristics, and specifically comprises the following steps: the four-pendulum system of heterogeneous character and the desired target dynamics model are as follows: , , In the middle of , The total number of the nodes is represented, Representing a non-linear activation function, Represent the first The number of system state variables that are to be used, Representing the desired target system state variable, Represent the first The individual system control inputs the variables, Representing the in-system coupling matrix, Representing the system out-coupling matrix, , , , Representing a matrix of known parameters of appropriate dimensions; Has the following properties: , , wherein Representing the 1-norm of the vector, , , , , , , , , , , , , , ; Definition of the first embodiment The synchronization error of each node is An error system can be obtained ; The controller is constructed as follows: , In the formula, Indicating that the control gain constant is to be applied, , And Is satisfied with Is a positive odd number of (a), Representing hyperbolic sine functions The number: Number of pulses Is a non-negative integer number, A matrix of pulse gains is represented and, Wherein For an n-dimensional identity matrix, Is the pulse gain, dependent on the pulse time Sum node ; Is a dirac function of Time of day And has ; The method meets the following conditions: , , , , , wherein , , , , , At this point, the error system can be rewritten as if Time of day When (when) Time of day ; Definition of the definition , , , , , , , The error system can be rewritten as ; The novel Lyapunov functional is designed, pulse intervals are partitioned, a convex combination technology is used, a fixed time synchronization sufficient condition expressed by matrix inequality is established, and four-pendulum system synchronization is realized, specifically: B001 selecting the following energy functions: , Wherein the method comprises the steps of Representing the positive definite diagonal matrix, Representing a transpose; B002 taking into account a positive integer Pulse interval is to Divided into The sub-interval defines the first Starting time of subinterval Wherein Then there is a time when Time of day I.e. the starting time of the first subinterval is the same as the starting time of the original pulse interval, when In the time-course of which the first and second contact surfaces, I.e. during the passage of After the sub-interval is divided, the obtained specific moment; B003 definition (th) Subinterval Then for There is Wherein Is a positive definite matrix of the matrix and the matrix, , Is a description in subintervals Internal time of Relative position parameter, time The positions in the subintervals are normalized, and the value range is that Reflecting time of The relative progress in the subinterval, Is a time interval of minimum Partitioning parameters The related parameters, for each subinterval, the minimum time interval under the normalized time scale, are derived from , ; B004 when , Is fixed as a positive definite matrix Thus, it is obtained that for , ; B005 for , Definition of , Thus, it is obtained that for , ; B006 utilizing the above-mentioned specific time-varying variables And Combining with convex technology, can obtain: ; B007 for When (when) At this time calculate Is the first derivative of (a): ; B008 due to the reason In the time-course of which the first and second contact surfaces, And is also provided with Thus, using global Lipschitz conditions, one can obtain: ; B009 due to In combination with B007-B008, we can obtain: , Wherein the method comprises the steps of , ; B010 when , Based on And Is defined in the definition of (a), Establishment; b011, combining B009-B010, can obtain: , , ; B012 for When (when) At the time, can be obtained: ; b013 when At the time, can be obtained: , ; b014, in combination with B011 and B013, for The method can obtain: ; B015 using Taylor expansion , The method can obtain: ; b016 by From the Taylor expansion of (2), for a given Exists in the presence of Make the following , Definition of The method can obtain: , Wherein the method comprises the steps of , , , , , , , Therefore, the method can be used for manufacturing the optical fiber, ; B017 using the inequality properties, it is possible to: ; b018, in combination with B001-B017, obtainable: , Wherein the method comprises the steps of , , , , , And Respectively represent Is set to the minimum feature value and the maximum feature value of (a), Representing the maximum lower bound of a set or function, Representing a minimum upper bound for a set or function; b019 due to Thus, when At the time, can be obtained: ; B020 to , When (when) At the time, can be obtained: ; b021, combining B018 and B020, according to the fixed time stability definition, implementing fixed time synchronization by the system; A comparison system is constructed, and the estimated stable time under the synchronous pulse, the inactive pulse and the asynchronous condition is specifically: The following system was constructed for comparison: ; When (when) Definition of time As can be seen from the comparison system, when In the time-course of which the first and second contact surfaces, When (when) In the time-course of which the first and second contact surfaces, Thus, it is possible to obtain: , Wherein the method comprises the steps of At this time, the first and second electrodes are connected, Is equivalent to ; When (when) Definition of time As can be seen from the comparison system, when In the time-course of which the first and second contact surfaces, When (when) In the time-course of which the first and second contact surfaces, Thus, it is possible to obtain: , Wherein the method comprises the steps of At this time, the first and second electrodes are connected, Is equivalent to ; Estimated settling time in case of sync pulse, inactive pulse and non-sync pulse: , And ; Consider case 1: Corresponding to the synchronous pulse, there are And ; For the following The method can obtain: , Wherein the method comprises the steps of ; Due to And When then When there is So that And Thus, it is possible to obtain: ; Further can be obtained: ; Taking into account that The method can obtain: ; the inequality is obtained by: ; Will be Substituting the above inequality, we can obtain: ; Next, calculate So that Transition from 1 to 0, and calculate Similarly, it is possible to obtain: , At the position of Monotonically decreasing due to The method can obtain: Order-making , The method can obtain: ; Definition of the definition Due to , And is also provided with Thus there is a unique So that Wherein Thus, the optimum settling time ; Consider case 2: For the case of inactive pulses, there are ; Similar to the sync pulse case, it is possible to obtain: , thus, the optimum settling time ; Consider case 3: for the case of asynchronous pulses, there are And ; Due to The method can obtain: ; Due to Can be obtained when The right side of the inequality tends to infinity, and therefore, a fixed time cannot be ensured Make the following steps Approaching 0 from 1.

Description

Fixed time synchronization method for heterogeneous four-pendulum system based on pulse control Technical Field The invention relates to a method for realizing fixed time synchronization of a four-pendulum system, in particular to a heterogeneous four-pendulum system fixed time synchronization method based on pulse control. Background Many kinetic models for four pendulum systems only consider homogeneous network models. In reality heterogeneous networks with individual compositions of different dynamics are ubiquitous. For example, generators in the power grid are controlled by swing equations with different inertia coefficients, different viscous damping, and different natural frequencies. A manipulator of the plurality of manipulators having lagrangian dynamics is described by a second order differential equation having different inertial matrices characterizing its coriolis and centrifugal forces. Since isomerism may reduce the synchronisation performance of the system, it is more challenging to study synchronisation behaviour in heterogeneous dynamic networks. Based on the above ideas, researchers have developed a great deal of research on the output synchronization of the four-pendulum system and have achieved a series of achievements. In addition, the fixed time controller used in the dynamic model of the four-pendulum system is the sum of power functions of signed functions, so that the four-pendulum system has numerous parameters and complex structure, and can cause buffeting. Therefore, it is particularly critical and important to adopt hyperbolic sine functions as the fixed time controller part to reduce parameters, simplify the structure and avoid the buffeting phenomenon. Disclosure of Invention The invention aims to provide a fixed time synchronization method of a heterogeneous four-pendulum system based on pulse control, which effectively improves the stability and convergence time of the system. The specific technical scheme of the invention is as follows, a fixed time synchronization method of a heterogeneous four-pendulum system based on pulse control comprises the following steps: a fixed time controller based on pulse control is designed for a four-pendulum system with heterogeneous characteristics. The four-pendulum system of heterogeneous character and the desired target dynamics model are as follows: Where i=1, 2,..n, N represents the total number of nodes, g (·) represents a nonlinear activation function, w i (t) represents the i-th system state variable, w 0 (t) represents the desired target system state variable, u i (t) represents the i-th system control input variable, Γ represents the system in-coupling matrix, s= (S ij)N×N represents the system out-coupling matrix, C i,Hi, C, H represents a known parameter matrix of appropriate dimensions; g (·) has the following properties: |gi(ζ1)-gi(ζ2)|≤δi|ζ1-ζ2|,ζ1≠ζ2,δi>0, Where |·| represents the 1-norm of the vector. Wherein the method comprises the steps of Defining the synchronization error of the ith node as e i(t)＝wi(t)-w0 (t) to obtain an error system The controller is constructed as follows: where η i >0 denotes the control gain constant, AndIs satisfied withIs, sh (·) represents a hyperbolic sine function: r 0 is a non-negative integer number, A matrix of pulse gains is represented and,Wherein I n is an n-dimensional identity matrix,Is the pulse gain, delta (·) is a dirac function, delta (x) =0, (x+.0),T r has the following properties: τmin＜tr-tr-1≤τmax,r∈R0,t0≥0,τmin>0,τmax>0,R0＝{1,…,r0} Where ζ=0.5, ηi＝3.2,τmin＝0.04,τmax=0.06, At this point, the error system can be rewritten as Definition e (t) = (e 1T(t),e2T(t),...,eNT(t))T,,G(e(t))＝(gT(e1(t)),gT(e2(t)),...,gT(eN(t)))T,Λ=diag(η1,η2,...,ηN),The error system can be rewritten as The novel Lyapunov functional is designed, pulse intervals are partitioned, a convex combination technology is used, a fixed time synchronization full necessary condition expressed by matrix inequality is established, and four-pendulum system synchronization is realized, and the method specifically comprises the following steps: c001 selecting the following energy function: V(t)=V(e(t))=eT(t)P(t)e(t), where P (T) represents a positive definite diagonal matrix and T represents a transpose. C002 the pulse interval [ t r,tr+1 ] is divided into Θ+1 subintervals taking into account a positive integer Θ > 0. Defining the start time of the (i+1) th subintervalWhere i=0, 1,..a.Θ -1, there is t r,0＝tr when i=0, i.e. the start instant of the first subinterval is the same as the start instant of the original pulse interval, when i=Θ, t r,Θ＝tr+τmin, i.e. a specific time obtained after division by Θ subintervals, C003 defining the (i+1) th subintervalThen for the followingThere is P (t) =p (t r,i+ρF)＝(1-ρ)Pi+ρPi+1＝Pi (P), where P i is a positive definite matrix,Is a description in subintervalsThe parameter of the relative position of the time t in the subinterval is normalized, the value range of the parameter is [0,1