Finite-time synchronization model and controller design method for chaotic electromechanical gyroscope system

Abstract

The present invention discloses a finite time synchronization model and controller design method for a chaotic electromechanical gyroscope system, which specifically includes the following steps: S1, first constructing an electromechanical gyroscope system, S2, constructing a master-slave finite time synchronization model through two or more electromechanical gyroscope systems constructed in step S1, and the present invention relates to the field of chaotic synchronization technology. The finite time synchronization model and controller design method for a chaotic electromechanical gyroscope system, wherein the function g in the designed controller is r (x r ,z r )’s non-Lipschitz property ensures that the error system is at the origin e r =0 finite time stability can make some state variables synchronized immediately, and at the same time shorten the synchronization time of other state variables, and reduce the dimension of the error system obtained by the master-slave system. The structure combining open loop and closed loop makes the controller more robust. The designed controller is globally continuous, which means that no chattering will occur during the synchronization process, and accurate synchronization can be achieved.

Description

Technical Field

[0001] The invention relates to the technical field of chaos synchronization, in particular to a finite time synchronization model of a chaotic electromechanical gyroscope system and a controller design method. Background Art

[0002] The chaotic electromechanical gyroscope system is a nonlinear system containing nonlinear terms such as cubic terms, cross terms and trigonometric terms. It is used in many practical engineering applications. Under some parameters, the electromechanical gyroscope system will exhibit many dynamic characteristics including chaos.

[0003] As we all know, synchronization is a phenomenon in which two or more systems reach a consensus through some kind of coupling. The earliest research on synchronization originated from Huygens' discovery of pendulum synchronization. Since then, many synchronization phenomena in nature and artificial systems have been discovered, such as glowing fireflies and audience applause.

[0004] In 1990, Pecora and Carroll first confirmed that two chaotic circuit systems could achieve synchronization through a special control method. Since then, chaotic synchronization has attracted widespread attention due to its potential applications in secure communications, psychology, ecology, chemistry and other fields, and some results have been obtained.

[0005] As a typical nonlinear system, the synchronization of chaotic electromechanical gyroscope systems has also attracted the research interest of some scholars. In 2003, Ge and Lin numerically proved that two identical electromechanical gyroscopes coupled by linear, sinusoidal, and exponential state error feedback controllers can achieve asymptotic synchronization. In 2009, Chen et al. used alternative variable control to study the asymptotic synchronization of two identical electromechanical gyroscope systems in a master-slave synchronization model and obtained some synchronization criteria. In 2012, Chen et al. further applied alternative variable controllers to study the asymptotic synchronization of master-slave electromechanical gyroscope systems with time-varying parameter mismatch and proved the corresponding sufficient synchronization criteria. In addition, in 2012, Cai and Zhou applied pulse feedback controllers to study the asymptotic synchronization of electromechanical gyroscope systems and proved some synchronization criteria.

[0006] However, it is worth noting that the above studies mainly focus on the asymptotic synchronization of electromechanical gyroscope systems, which means that two electromechanical gyroscope systems can only achieve synchronization when time approaches infinity. If two electromechanical gyroscope systems coupled in some way achieve synchronization within a finite time or a set time, it is closer to actual engineering applications. Moreover, compared with asymptotic synchronization, finite-time synchronization has been proven to have better robustness and anti-interference performance. In view of this, for electromechanical gyroscope systems, it is more meaningful to study their finite-time synchronization than asymptotic synchronization. However, as far as we know, few studies have involved this topic. Only Yang and Wu applied an adaptive controller to achieve asymptotic synchronization of two electromechanical gyroscope systems. Unfortunately, the controller used in this study is discontinuous, which can easily cause vibration in the slave electromechanical gyroscope system and even destroy the synchronization process of the master and slave electromechanical gyroscope systems. Summary of the invention

[0007] 1. Technical issues to be resolved

[0008] In view of the deficiencies in the prior art, the present invention provides a finite-time synchronization model and a controller design method for a chaotic electromechanical gyroscope system, which solves the problem that the existing controller is discontinuous, easily causing vibration in the slave electromechanical gyroscope system, and even destroying the synchronization process of the master and slave electromechanical gyroscope systems.

[0009] (II) Technical solution

[0010] To achieve the above objectives, the present invention is implemented by the following technical scheme: a design method of a finite time synchronization model of a chaotic electromechanical gyroscope system, specifically comprising the following steps:

[0011] S1. First, build the electromechanical gyroscope system:

[0012] x.=A(t)x+F(x,t),

[0013] where x∈R 4 ,

[0014]

[0015]

[0016] Where all parameters A, B, C, b, f, h i (i=1,2,3), k i (i=1,…,6), K T , K a , K b , R and L are both positive numbers;

[0017] S2. Using two or more electromechanical gyroscope systems constructed in step S1, a master-slave finite time synchronization model is constructed, which is expressed as:

[0018]

[0019] where x,z∈R n They represent the state variables of the master-slave system respectively, and u(x,z) is the coupling controller.

[0020] Preferably, in step S2, if the initial value x(0) of the main system is x0∈R n and from any initial value of the system z(0)=z0∈R n , there exists a constant T(x0,z0)>0 such that the state trajectories x(t) and z(t) of the master-slave system satisfy And when t≥T(x0,z0), |x(t)-z(t)||=0, where |.| represents the Euclidean norm of the vector, then the synchronization model achieves global finite-time synchronization, and T(x0,z0) is the synchronization time.

[0021] The present invention also provides a design method for a finite time controller of a chaotic electromechanical gyroscope system, which specifically comprises the following steps:

[0022] E1. First, construct the state variable x∈R of the main system in the synchronization model. n Decompose into (x d ,x r ) T ,in n d +n r = n, similarly, the state variable z of the slave system in the synchronization model can be decomposed into z d and z r , vector z, z d and z r are n and n respectively. d and n r dimension, the coefficient matrix is ​​decomposed into:

[0023]

[0024] in and

[0025] The nonlinear term is decomposed into

[0026] F(x,t)=(F d (x,t),F r (x,t) T ∈R n ,

[0027] in

[0028] E2. Design the controller according to step E1 and express it as:

[0029]

[0030] E3. Choose the right x d and the feedback gain matrix The finite-time synchronization model can achieve global finite-time synchronization.

[0031] Preferably, the controller designed in step E2 comprises two parts: the first part u d :z d =x d This means extracting some variables x from the main system. d Transmitted to the slave system and replaces the corresponding part of the variable z d , forming an open-loop control;

[0032] Part II r =Kg r (x r ,z r ) is a feedback control, where is the feedback gain matrix to be determined,

[0033]

[0034] Where α∈(0,1), the sign function

[0035]

[0036] Obviously, the function g r (x r ,z r )About x r and z r Continuous and non-Lipschitz, and x r =z r When r (x r ,z r )30.

[0037] Preferably, the specific steps of making the finite time synchronization model achieve global finite time synchronization in step E3 are as follows:

[0038] P1. Based on the controller designed in step E2, the finite time synchronization model is reformulated as:

[0039]

[0040] The non-autonomous system studied meets the following conditions:

[0041]

[0042] in and

[0043] P2, then define an error vector e r =x r -z r , we get the error system of the non-autonomous system with respect to the finite time synchronization model that satisfies the conditions:

[0044]

[0045] P3. The global finite-time synchronization problem of the synchronization model of non-autonomous systems that meet the conditions is equivalent to the error system at the origin e r =0, the global finite-time stability problem of the error system is the synchronization time.

[0046] (III) Beneficial effects

[0047] The present invention provides a finite time synchronization model and controller design method for a chaotic electromechanical gyroscope system. Compared with the prior art, the present invention has the following beneficial effects:

[0048] (1) The finite time synchronization model and controller design method of the chaotic electromechanical gyroscope system. The function g in the designed controller r (x r ,z r )’s non-Lipschitz property ensures that the error system is at the origin e r = 0 is stable for a limited time.

[0049] (2) The finite-time synchronization model and controller design method of the chaotic electromechanical gyroscope system can synchronize some state variables immediately and shorten the synchronization time of the remaining state variables.

[0050] (3) The finite-time synchronization model and controller design method of the chaotic electromechanical gyroscope system can reduce the dimension of the error system obtained by the master-slave system.

[0051] (4) The finite-time synchronization model and controller design method of the chaotic electromechanical gyroscope system make the controller have better robustness through the combination of open-loop and closed-loop structures.

[0052] (5) The finite-time synchronization model and controller design method of the chaotic electromechanical gyroscope system, the designed controller is globally continuous, which means that no jitter phenomenon will occur during the synchronization process, achieving accurate synchronization and avoiding synchronization destruction. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a schematic diagram of the trajectory of the host electric gyroscope system in Example 1 of the present invention;

[0054] Figure 2 is a schematic diagram of the trajectory of the uncontrolled slave electromechanical gyroscope system in Example 1 of the present invention;

[0055] Figure 3 In Example 1 of the present invention, the controller x d = x1 coupled asymptotic synchronization process and the global finite-time synchronization process coupled by a controller with parameters α = 2 / 5 and K = diag{3.0,3.0,3.0};

[0056] Figure 4 It is a schematic diagram of the trajectory of the host electric gyroscope system in Example 2 of the present invention;

[0057] Figure 5 is a schematic diagram of the trajectory of the uncontrolled slave electromechanical gyroscope system in Example 2 of the present invention;

[0058] Figure 6 In Example 2 of the present invention, the controller x d =x2 coupled asymptotic synchronization process and the global finite-time synchronization process coupled by a controller with parameters α = 2 / 5 and K = diag{3.0,3.0,3.0};

[0059] Figure 7 It is a schematic diagram of the trajectory of the host electric gyroscope system in Example 3 of the present invention;

[0060] Figure 8 is a schematic diagram of the trajectory of the uncontrolled slave electromechanical gyroscope system in Example 3 of the present invention;

[0061] Fig. 9 In Example 3 of the present invention, the controller x d =x4 coupled asymptotic synchronization process and the global finite-time synchronization process coupled by a controller with parameters α = 2 / 5 and K = diag{3.0,3.0,3.0};

[0062] Fig.10 It is a schematic diagram of the trajectory of the host electric gyroscope system in Example 4 of the present invention;

[0063] Fig.11 is a schematic diagram of the trajectory of the uncontrolled slave electromechanical gyroscope system in Example 4 of the present invention;

[0064] Fig.12 In Example 4 of the present invention, the controller x dSchematic diagram comparing the asymptotic synchronization process coupled by =x4 and the global finite-time synchronization process coupled by a controller with parameters α = 2 / 5 and K = diag{3.0, 3.0, 3.0}. DETAILED DESCRIPTION

[0065] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0066] See also Figure 1-12 The embodiment of the present invention provides a technical solution: a design method for a finite time synchronization model of a chaotic electromechanical gyroscope system, which specifically includes the following steps:

[0067] S1. First, build the electromechanical gyroscope system:

[0068] x.=A(t)x+F(x,t), (1)

[0069] where x∈R 4 ,

[0070]

[0071] Where all parameters A, B, C, b, f, h i (i=1,2,3), k i (i=1,…,6), K T , K a , K b , R and L are both positive numbers;

[0072] S2. Using two or more electromechanical gyroscope systems constructed in step S1, a master-slave finite time synchronization model is constructed, which is expressed as:

[0073]

[0074] where x,z∈R n They represent the state variables of the master-slave system respectively, and u(x,z) is the coupling controller.

[0075] In the embodiment of the present invention, in step S2, if the initial value x(0) of the main system is x0∈R n and from any initial value of the system z(0)=z0∈R n , there exists a constant T(x0,z0)>0 such that the state trajectories x(t) and z(t) of the master-slave system satisfy And when t≥T(x0,z0), |x(t)-z(t)||=0, where ||.|| represents the Euclidean norm of the vector, then the synchronization model (2) achieves global finite-time synchronization, and T(x0,z0) is the synchronization time.

[0076] The embodiment of the present invention also provides a design method of a finite time controller of a chaotic electromechanical gyroscope system, which specifically includes the following steps:

[0077] E1. First, construct the state variable x∈R of the main system in the synchronization model. n Decompose into (x d ,x r ) T ,in n d +n r = n, similarly, the state variable z of the slave system in the synchronization model can be decomposed into z d and z r , vector z, z d and z r are n and n respectively. d and n r dimension, the coefficient matrix is ​​decomposed into:

[0078]

[0079] in and

[0080] The nonlinear term is decomposed into

[0081] F(x,t)=(F d (x,t),F r (x,t) T ∈R n ,

[0082] in

[0083] E2. Design the controller according to step E1 and express it as:

[0084]

[0085] E3. Choose the right x d and the feedback gain matrix This makes the finite time synchronization model (2) achieve global finite time synchronization.

[0086] In the embodiment of the present invention, the controller designed in step E2 includes two parts: the first part u d :z d =x dThis means extracting some variables x from the main system. d Transmitted to the slave system and replaces the corresponding part of the variable z d , forming an open-loop control;

[0087] Part II r =Kg r (x r ,z r ) is a feedback control, where is the feedback gain matrix to be determined,

[0088]

[0089] Where α∈(0,1), the sign function

[0090]

[0091] Obviously, the function g r (x r ,z r )About x r and z r is continuous and non-Lipschitz, and satisfies x r =z r When r (x r ,z r )30.

[0092] In the embodiment of the present invention, the specific steps of making the finite time synchronization model (2) achieve global finite time synchronization in step E3 are as follows:

[0093] P1. Based on the controller (3) designed in step E2, the finite time synchronization model (2) is re-expressed as:

[0094]

[0095] The non-autonomous system studied meets the following conditions:

[0096]

[0097] in and

[0098] P2, then define an error vector e r =x r -z r , we can get the error system of the non-autonomous system satisfying condition (6) with respect to the finite time synchronization model (5):

[0099]

[0100] The global finite-time synchronization problem of the synchronization model (5) of the non-autonomous system satisfying condition (6) is equivalent to the global finite-time stability problem of the error system (7) at the origin \(e\). r = 0, and the settling time of the globally finite-time stable error system (7) is the synchronization time.

[0101] The global finite-time synchronization theory in the embodiments of the present invention is specifically as follows:

[0102] To obtain the results of this research report, the following lemmas are needed.

[0103] Consider the following non-autonomous system:

[0104] \(x = f(t, x)\), (8)

[0105] where \(f: \mathbb{R}\) + \(\times D \to \mathbb{R}\) n is a continuous function, and \(f(t, 0) = 0\) holds for any \(t \geq t_0\), that is, the origin is an equilibrium point of the system (8).

[0106] Lemma 1 If there exists a continuously differentiable, decreasing positive definite Lyapunov function \(V(t, x): \mathbb{R}\) + \(\times D \to \mathbb{R}\) + in the open neighborhood of the origin such that for any \(t \in \mathbb{R}\) + , \(V(t, 0) = 0\), and there exists a positive definite function \(r: \mathbb{R}\) + \(\to \mathbb{R}\) + , \(r(0) = 0\), and some real number \(\varepsilon > 0\) such that

[0107]

[0108] Then the system (8) is locally finite-time stable at the origin, and the synchronization time satisfies

[0109]

[0110] Furthermore, if the function \(V(t, x)\) is radially unbounded, and \(U = D = \mathbb{R}\) n , then the system (8) is globally uniformly finite-time stable at the origin.

[0111] Lemma 2 For any positive real number \(a\) i , \(i = 1, \ldots, n\) and \(0 < p < 2\), the following inequality holds:

[0112]

[0113] In the process of proving the finite-time synchronization criterion, the following assumption is sometimes used. This assumption is proposed based on the bounded property of the chaotic attractor and is reasonable.

[0114] Assumption 1 For any initial state x(0) in the domain of the chaotic main system, there exists a real constant p i , and p i (i=1,…,n), so that the trajectory of the chaotic main system satisfies:

[0115] or |x i (t,x(0))|≤ρ i ,

[0116] Lemma 3 For any ξ,ζ∈R,ξ 2 +ξζ+ζ 2 ≥0.

[0117] Next, we will prove the global finite-time synchronization criterion for a class of non-autonomous systems that satisfy condition (6) under the synchronization model (5).

[0118] Theorem 1 If there exists a positive definite diagonal matrix and the feedback gain matrix So that for any time t>0,

[0119]

[0120] (A 22 (t)+D(x,z)) T P+P(A 22 (t)+D(x,z))≤0, (14)

[0121] Then the master-slave synchronization model (5) coupled by the controller (3) can achieve global finite-time synchronization, and the corresponding synchronization time satisfies

[0122]

[0123] where e r0 =x r (0)-z r (0), α∈(0,1),

[0124] Proof: Choose the quadratic Lyapunov function Then V(e r ) along the error system (7) with respect to time t is:

[0125]

[0126] According to Lemma 2, it is easy to obtain that for any α∈(0,1) and If we take a positive definite matrix and the feedback gain matrix Make So

[0127]

[0128] Therefore, if (13) and (14) hold, then from (16) and (17) we can get

[0129]

[0130] Based on Lemma 1, it is easy to prove that the master-slave synchronization model (5) of a class of non-autonomous systems that satisfy condition (6) coupled by the controller (3) can achieve global finite-time synchronization, and the corresponding synchronization time satisfies (15).

[0131] The proof is complete.

[0132] Considering the master-slave synchronization model (5) for an electromechanical gyroscope system, the controller in this framework is of the form

[0133]

[0134] At this time, the state variable of the host electric gyroscope system can be decomposed into x = (x d ,x r ) T , where x r =(x2,x3,x4) T The system matrix and nonlinear part of the error system (7) are:

[0135]

[0136] Proposition 1 If there exists a positive definite diagonal matrix P = diag{1,μ1,μ2} and the feedback gain matrix Make

[0137]

[0138] Then the master-slave electromechanical gyroscope system coupled by the controller (19) can achieve global finite time synchronization, and the corresponding synchronization time T(x0,z0) satisfies

[0139]

[0140] where e r0 =(x2(0)-z2(0),x3(0)-z3(0),x4(0)-z4(0)) T , α∈(0,1), Parameter p1 is the bound of the trajectory component x1 of the host electric gyroscope system.

[0141] Proof: Take the diagonal matrix

[0142] P = diag{1, μ1, μ2}, (25)

[0143] Where μ1>0, μ2>0.

[0144] Then, from equations (20)-(22), we can easily get and

[0145]

[0146] By Schur's complement theorem, the above matrix S is negative semidefinite if and only if

[0147]

[0148] and

[0149]

[0150] in

[0151] From Lemma 3, we know that for any x2,z2∈R, S1<0.

[0152] Again, the above matrix S2 is negative semidefinite if and only if

[0153]

[0154] and

[0155]

[0156] According to hypothesis 1, it is easy to get

[0157]

[0158] From Lemma 3, we can see that for any x i ,z i ∈R(i=2,3), S3 defined in (26) satisfies

[0159]

[0160] Therefore, if (23) holds, then inequalities (26) and (27) hold. By Theorem 1, this proposition is proved.

[0161] The proof is complete.

[0162] Considering the master-slave synchronization model (5) for an electromechanical gyroscope system, the controller in this framework is of the form

[0163]

[0164] At this time, the state variable of the host electric gyroscope system can be decomposed into x = (x d ,x r ) T , where x r =(x1,x3,x4) T The system matrix and nonlinear part of the error system (7) are:

[0165]

[0166]

[0167] Similar to the proof process of Proposition 1, it is easy to prove according to Theorem 1:

[0168] Proposition 2 If there exists a positive definite diagonal matrix P = diag{1,μ1,μ2} and the feedback gain matrix Make

[0169]

[0170] Then the master-slave electromechanical gyroscope system coupled by the controller (28) can achieve global finite time synchronization, and the corresponding synchronization time T(x0,z0) satisfies

[0171]

[0172] where e r0 =(x1(0)-z1(0),x3(0)-z3(0),x4(0)-z4(0)), α∈(0,1), Parameter p2 is the bound of the trajectory component x2 of the host electric gyroscope system.

[0173] Considering the master-slave synchronization model (5) for an electromechanical gyroscope system, the controller in this framework is of the form

[0174]

[0175] At this time, the state variable of the host electric gyroscope system can be decomposed into x = (x d ,x r ) T , where x r =(x1,x3,x4) T The system matrix and nonlinear part of the error system (7) are:

[0176]

[0177] Similar to the proof process of Proposition 1, it is easy to prove according to Theorem 1:

[0178] Proposition 3 If there exists a positive definite diagonal matrix P = diag{1, μ1, μ2} and the feedback gain matrix Make

[0179]

[0180] Then the master-slave electromechanical gyroscope system coupled by the controller (34) can achieve global finite time synchronization, and the corresponding synchronization time T(x0,z0) satisfies

[0181]

[0182] where e r0 =(x1(0)-z1(0),x2(0)-z2(0),x4(0)-z4(0)), α∈(0,1), Parameter p3 is the bound of the trajectory component x3 of the host electric gyroscope system.

[0183] Considering the master-slave synchronization model (5) for an electromechanical gyroscope system, the controller in this framework is of the form

[0184]

[0185] At this time, the state variable of the host electric gyroscope system can be decomposed into x = (x d ,x r ) T , where x r =(x1,x2,x3) T The system matrix and nonlinear part of the error system (7) are:

[0186]

[0187] Proposition 4 If there exists a positive definite diagonal matrix P = diag{1,μ1,μ2} and the feedback gain matrix The following conditions are met at the same time:

[0188]

[0189] Then the master-slave electromechanical gyroscope system coupled by the controller (40) can achieve global finite time synchronization, and the corresponding synchronization time T(x0,z0) satisfies

[0190]

[0191] where e r0 =(x1(0)-z1(0),x2(0)-z2(0),x3(0)-z3(0)), α∈(0,1), Parameter p i (i=1,2,3) is the main system trajectory component x i The bounds of (i=1,2,3).

[0192] Proof: Take the positive definite matrix P = diag{1,μ1,μ2}, where the constants μ1>0, μ2>0, if (44) satisfies, then for any have

[0193]

[0194] Furthermore, from equations (41)-(42), we can see that:

[0195] S=(A 22 (t)+D(x,z)) T P+P(A 22 (t)+D(x,z))=(s ij ) 3×3

[0196] in,

[0197]

[0198] make According to the Gale circle theorem, if for any i=1,2,3, s ii +R i ≤0, then the matrix S is negative semidefinite.

[0199] From Assumption 1 and Lemma 3, we know that

[0200]

[0201]

[0202] Therefore, if (45) holds, then the inequality s 11 +R1≤0 holds true.

[0203] Similarly, if (46) and (47) hold, then the inequality s 22+R2≤0 and s 33 +R3≤0 holds true.

[0204] This proposition is proved by Theorem 1.

[0205] Simulation Verification

[0206] In order to verify the validity of the above criteria, we analyzed four examples in the literature [Chen Yun. Research on global synchronization of master-slave non-autonomous chaotic system [D]. Wuhan: Naval University of Engineering, 2009], and compared the results of the present invention with the asymptotic synchronization results in the literature [Chen Yun. Research on global synchronization of master-slave non-autonomous chaotic system [D]. Wuhan: Naval University of Engineering, 2009], see Figure 3 , Figure 6 , Fig. 9 combine Fig.12 In the simulation process, we choose the initial conditions of the master and slave electromechanical gyroscope systems as x(0)=(-0.1,-0.2,-0.3,0.1) T and z(0)=(-10,20,30,-8) T .

[0207] The chaotic trajectory of the master-slave electromechanical gyroscope system in Example 1 is as follows Figure 1-2 As shown in the figure, we can see that P1 = 4. Obviously, Example 1 not only satisfies Corollary 3.9 (iv) in the literature [Chen Yun. Research on global synchronization of master-slave non-autonomous chaotic systems [D]. Wuhan: Naval University of Engineering, 2009], but also satisfies Proposition 1. Therefore, the master-slave electromechanical gyroscope system can be controlled by the controller x d = x1 to achieve global asymptotic synchronization (see Figure 3 ), and global finite-time synchronization can be achieved by the controller (19) with parameters α = 2 / 5 and K = diag{3.0,3.0,3.0} (see Figure 3 The solid line in the figure). From (24), the synchronization time estimate of global finite-time synchronization can be calculated as T E =31.5561.

[0208] Global finite-time synchronization of the master-slave chaotic electromechanical gyroscope system in Example 1, Figure 1 It is the host electric gyro system; Figure 2 is an uncontrolled slave electromechanical gyro system; Figure 3 By controller x d = x1 coupled asymptotic synchronization process (dashed line) and the global finite-time synchronization process (solid line) coupled by controller (19) with parameters α = 2 / 5 and K = diag{3.0, 3.0, 3.0}.

[0209] The chaotic trajectory of the master-slave electromechanical gyroscope system in Example 2 is as follows Figure 4-5 As shown. Obviously, Example 2 not only satisfies Corollary 3.12(iii) in the literature [Chen Yun. Research on Global Synchronization of Master-Slave Non-Autonomous Chaotic Systems [D]. Wuhan: Naval University of Engineering, 2009], but also satisfies Proposition 2. Therefore, the master-slave electromechanical gyroscope system can be controlled not only by the controller x d = x2 to achieve global asymptotic synchronization (see Figure 6 ), and global finite-time synchronization can be achieved by the controller (28) with parameters α = 2 / 5 and K = diag{3.0,3.0,3.0} (see Figure 6 The solid line in the figure). From (33), the synchronization time estimate of global finite-time synchronization can be calculated as T E =4.8706.

[0210] Global finite-time synchronization of the master-slave chaotic electromechanical gyroscope system in Example 2, Figure 4 It is the host electric gyro system; Figure 5 is an uncontrolled slave electromechanical gyro system; Figure 4 By controller x d = x2 coupled asymptotic synchronization process (dashed line) and the global finite-time synchronization process (solid line) coupled by controller (28) with parameters α = 2 / 5 and K = diag{3.0, 3.0, 3.0}.

[0211] The chaotic trajectory of the master-slave electromechanical gyroscope system in Example 3 is as follows Figure 7-8 As shown. Obviously, Example 3 not only satisfies Corollary 3.14(i) in the literature [Chen Yun. Research on Global Synchronization of Master-Slave Non-Autonomous Chaotic Systems [D]. Wuhan: Naval University of Engineering, 2009], but also satisfies Proposition 3. Therefore, the master-slave electromechanical gyroscope system can be controlled not only by the controller x d = x3 to achieve global asymptotic synchronization (see Fig. 9 ), and global finite-time synchronization can be achieved by the controller (34) with parameters α = 2 / 5 and K = diag{3.0, 3.0, 3.0} (see Fig. 9 The solid line in the figure). From (39), the synchronization time estimate of global finite-time synchronization can be calculated as T E =27.5583 (take μ2=0.01).

[0212] Global finite-time synchronization of the master-slave chaotic electromechanical gyroscope system in Example 3. Figure 7 It is the host electric gyro system; Figure 8 is an uncontrolled slave electromechanical gyro system; Fig. 9 By controller x d= x3 coupled asymptotic synchronization process (dashed line) and the global finite-time synchronization process (solid line) coupled by controller (34) with parameters α = 2 / 5 and K = diag{3.0, 3.0, 3.0}.

[0213] The chaotic trajectory of the master-slave electromechanical gyroscope system in Example 4 is as follows Figure 10-11 As shown. Obviously, Example 4 not only satisfies Corollary 3.15(i) in the literature [Chen Yun. Research on global synchronization of master-slave non-autonomous chaotic systems [D]. Wuhan: Naval University of Engineering, 2009], but also satisfies Proposition 4. Therefore, the master-slave electromechanical gyroscope system can be controlled not only by the controller x d = x4 achieves global asymptotic synchronization (see Fig.12 ), and global finite-time synchronization can be achieved by the controller (40) with parameters α = 2 / 5 and K = diag{3.0, 3.0, 3.0} (see Fig.12 The solid line in the figure). From (48), the synchronization time estimate of global finite-time synchronization can be calculated as T E =4.9067.

[0214] Global finite-time synchronization of the master-slave chaotic electromechanical gyroscope system of Example 4. Fig.10 It is the host electric gyro system; Fig.11 is an uncontrolled slave electromechanical gyro system; Fig.12 By controller x d =x4 coupled asymptotic synchronization process (dashed line) and the global finite-time synchronization process (solid line) coupled by controller (40) with parameters α=2 / 5 and K=diag{3.0,3.0,3.0}.

[0215] Meanwhile, the contents not described in detail in this specification belong to the prior art known to those skilled in the art.

[0216] It should be noted that, in this article, relational terms such as first and second, etc. are only used to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Moreover, the terms "include", "comprise" or any other variants thereof are intended to cover non-exclusive inclusion, so that a process, method, article or device including a series of elements includes not only those elements, but also other elements not explicitly listed, or also includes elements inherent to such process, method, article or device.

[0217] While the embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that many changes, modifications, substitutions and variations can be made to the embodiments without departing from the principles and spirit of the invention.

Claims

1. A design method for a finite-time synchronization model of a chaotic electromechanical gyroscope system, characterized in that: The specific steps include: S1. First, build the electromechanical gyroscope system: where x∈R 4 , Where all parameters A, B, C, b, f, h i ,i=1,2,3,k i ,i=1,...,6,K T , K a , K b , R and L are both positive numbers; S2. Using two or more electromechanical gyroscope systems constructed in step S1, a master-slave finite time synchronization model is constructed, which is expressed as: where x,z∈R n They represent the state variables of the master-slave system respectively, and u(x,z) is the coupling controller.

2. The design method of the finite time synchronization model of the chaotic electromechanical gyroscope system according to claim 1 is characterized in that: In step S2, if the initial value x(0) = x0∈R n and from any initial value of the system z(0)=z0∈R n , there exists a constant T(x0,z0)>0 so that the state trajectories x(t) and z(t) of the master-slave system satisfy And when t≥T(x0,z0), ||x(t)-z(t)||≡0, where ||·|| represents the Euclidean norm of the vector, then the synchronization model achieves global finite-time synchronization, and T(x0,z0) is the synchronization time.

3. A design method for a finite time synchronization controller for a chaotic electromechanical gyroscope system, characterized in that: The specific steps include: E1. First, construct the state variable x∈R of the main system in the synchronization model in claim 1. n Decompose into (x d ,x r ) T ,in n d +n r = n, similarly, the state variable z of the slave system in the synchronization model constructed in claim 1 can also be decomposed into z d and z r , vector z, z d and z r are n and n respectively. d and n r dimension, the coefficient matrix is ​​decomposed into: in and The nonlinear term is decomposed into F(x,t)=(F d (x,t),F r (x,t)) T ∈R n , in E2. Design the controller according to step E1 and express it as: E3. Choose the right x d and the feedback gain matrix The finite-time synchronization model can achieve global finite-time synchronization.

4. The design method of the finite time synchronization controller of the chaotic electromechanical gyroscope system according to claim 3 is characterized in that: The controller designed in step E2 includes two parts: the first part u d :z d =x d This means extracting some variables x from the main system. d Transmitted to the slave system and replaces the corresponding part of the variable z d , forming an open-loop control; Part II r =Kg r (x r ,z r ) is a feedback control, where is the feedback gain matrix to be determined, Where α∈(0,1), the sign function Obviously, the function g r (x r ,z r )About x r and z r is continuous and non-Lipschitz, and satisfies x r =z r When r (x r ,z r )≡0.

5. The design method of the finite time synchronization controller of the chaotic electromechanical gyroscope system according to claim 3 is characterized in that: The specific steps for making the finite time synchronization model achieve global finite time synchronization in step E3 are as follows: P1. Based on the controller designed in step E2, the finite time synchronization model is reformulated as: The non-autonomous system studied meets the following conditions: in and P2, then define an error vector e r =x r -z r , we get the error system of the non-autonomous system with respect to the finite time synchronization model that satisfies the conditions: P3. The global finite-time synchronization problem of the synchronization model of non-autonomous systems that meet the conditions is equivalent to the error system at the origin e r =0, the global finite-time stability problem of the error system is the synchronization time.

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