Construction of finite-time synchronization model and controller design method for heterogeneous chaotic systems

Abstract

The present invention discloses a method for constructing a finite-time synchronization model of heterogeneous chaotic systems and designing a controller. By constructing a model of two completely synchronized heterogeneous chaotic systems and designing a simple and continuous controller, the vibration phenomenon caused by the discontinuity of the controller is overcome, and the loss of original system information due to simply eliminating the nonlinear terms of the master-slave system is avoided, and the characteristics of the original system are retained as much as possible. In addition, the synchronization criterion and the analytical expression of the synchronization time estimation are proved. The criterion can determine in advance whether the two heterogeneous chaotic systems can achieve synchronization, and the analytical expression of the synchronization time estimation can perform analytical estimation of the synchronization time. The present invention relates to the field of chaos synchronization technology. The method for constructing a finite-time synchronization model of heterogeneous chaotic systems and designing a controller solves the problem of complete synchronization of heterogeneous systems, constructs a finite-time synchronization model, and can estimate the synchronization time through an analytical expression.

Description

Technical Field

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

[0002] Synchronization is the process in which two or more systems, under the action of external drive or mutual coupling, achieve the same state of the dynamic system under different initial conditions. The study of synchronization can be traced back to 1673 when the Dutch physicist, astronomer, and mathematician Huygens discovered and studied the phase synchronization of pendulums. Subsequently, people discovered and studied various forms of synchronization phenomena in nature and artificial systems, such as the chirping of crickets and the applause of the audience.

[0003] In 1990, two scholars from the U.S. Naval Research Laboratory, LMPecora and TLCarroll, further expanded the study of synchronization to chaotic systems, which are a type of nonlinear system with initial value sensitivity, fractal dimension and strange attractors. Since then, the synchronization of chaotic systems has attracted great attention due to its potential application prospects in secure communications, physiology, neural networks and other fields, and has therefore become a research hotspot in the nonlinear field.

[0004] Initially, the research on the synchronization of chaotic systems focused on asymptotic synchronization, which is a type of synchronization achieved when time tends to infinity. However, people later found that in practical applications, it is often necessary to achieve synchronization in the shortest possible time or within a set time, such as the synchronization of neural networks, the synchronous tracking control of robots, and the attitude control of satellites. As a result, the finite time synchronization of chaotic systems has attracted widespread attention and obtained many research results.

[0005] However, it is worth noting that the current research results on finite-time synchronization are mainly limited to two homogeneous chaotic systems, and there are few relevant results on the finite-time synchronization theory and its application research of heterogeneous chaotic systems. In the real world, the two chaotic systems synchronized in finite time are often heterogeneous, and the finite-time synchronization theory and methods applicable to homogeneous chaotic systems are not completely applicable to heterogeneous chaotic systems.

[0006] At present, the research results on the finite-time synchronization of heterogeneous chaotic systems are only found in the synchronization models of heterogeneous (chaotic) systems under observer-based control, sliding mode control, and adaptive control.

[0007] References: [XTTran, HJKang. A novel observer-based finite-time control method for modified function projective synchronization of uncertain chaotic (hyperchaotic) systems [J]. Nonlinear Dynamics, 2015, 80: 905-916] By designing a continuous observer-based finite-time controller, the finite-time synchronization of two heterogeneous systems of the same dimension but different structures with model uncertainty and external disturbances is realized, and the analytical expression of synchronization time estimation is derived. References: [MPAghababa, S. Khanmohammadi, G. Alizadeh. Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique [J]. Applied Mathematical Modelling, 2011, 35: 3080-3091] By designing a non-singular sliding mode controller, the synchronization of two heterogeneous chaotic systems of the same dimension but different structures is realized and the synchronization time is estimated. The literature [XTTran, HJKang. Robust adaptive chatter-free finite-time control method for chaos control and (anti-) synchronization of uncertain (hyper) chaotic systems [J]. Nonlinear Dynamics, 2015, 80: 637-651] constructs an adaptive sliding mode controller, which achieves the synchronization of two hyperchaotic systems with the same dimension but different structures in the presence of model uncertainty and external disturbances within a finite time, and gives an estimation formula for the synchronization time.Reference [J.Sun, Y.Shen, X.Wang, J.Chen. Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control [J]. Nonlinear Dynamics, 2014, 76: 383-397.] takes the combination of two chaotic systems with unknown parameters as the master system and the combination of another two chaotic systems with unknown parameters as the slave system. The non-singular terminal sliding mode control technology is used to realize the finite-time synchronization of these two heterogeneous combination systems with the same dimension but different structures, and the criterion for finite-time synchronization and the expression for synchronization time estimation are derived. The literature [M.Kazerooni, S.Pourdehi, A.Soltani Sarvestani, R.Soltani.Sarvestani.Adaptive finite-time controller for synchronization ofnon-identical chaotic systems with nonlinear input[C].3rd International Conference on Control,Instrumentation and Automation,Tehran,Iran,2013,107-112] designed an adaptive nonlinear controller to achieve finite-time synchronization of two heterogeneous chaotic systems with the same dimension but different structures under the conditions of unknown parameters, external disturbances, and dead-zone input nonlinearity. An estimated expression for the synchronization time was given, but the controller used was discontinuous and prone to vibration, which in turn destroyed the synchronization. Reference [J.Zhao, Y.Wu, Y.Wang.Generalized finite-time synchronization between coupled chaotic systems of different orders with unknown parameters[J].Nonlinear Dynamics, 2013, 74:479-485] Based on an adaptive finite-time controller, the finite-time synchronization problem of two heterogeneous chaotic systems of different dimensions under unknown parameters is studied, and the synchronization time estimation is given through a mathematical expression.Reference [Y. Wei. Finite-time adaptive synchronization of chaotic systems with known or unknown parameters [C]. 2010 Second WRI Global Congress on Intelligent Systems, 2010, 404-407] constructs an adaptive finite-time control model and studies the finite-time synchronization problem of two systems with the same dimensions but different structures, but does not give an estimated expression for the synchronization time. Reference [D. Zhang, J. Mei, P. Miao. Global finite-time synchronization of different dimensional chaotic systems [J]. Applied Mathematical Modelling, 2017, 48: 303-315] studies the generalized finite-time synchronization of two heterogeneous chaotic systems with different dimensions where the state variables have a certain functional relationship.

[0008] It is worth noting that in the existing research results, some designed controllers achieve synchronization by adding and subtracting nonlinear terms to eliminate all nonlinear terms of the error system obtained by the master-slave system. The designed controllers are too complex, even more complex than the master-slave system to be synchronized. Such controllers are not easy to implement in engineering and have high synchronization costs. In addition, completely eliminating the nonlinear terms in the master-slave system may lead to a large loss of information in the original system, thereby losing its original system characteristics; some designed controllers are discontinuous and prone to vibration, which in turn destroys synchronization; some do not derive an estimated expression for synchronization time, while synchronization time estimation is often very important in practical applications. In addition, in the existing research on synchronization of heterogeneous chaotic systems, most of the research focuses on generalized synchronization in which there is a functional relationship between the state variables of the master-slave chaotic system, rather than complete synchronization of the state variables of the master-slave heterogeneous chaotic system. In practical applications, two heterogeneous chaotic systems are often required to achieve complete synchronization.

[0009] Based on the analysis of the above research status, the present invention constructs a master-slave finite-time synchronization model composed of two heterogeneous chaotic systems; designs a relatively simple and continuous controller to reduce the synchronization cost, overcome the oscillation phenomenon caused by the discontinuity of the controller, avoid the loss of original system information due to simply eliminating the nonlinear terms of the master-slave system, and retain as many characteristics of the original system as possible. Summary of the invention

[0010] 1. Technical issues to be resolved

[0011] In view of the shortcomings of the prior art, the present invention provides a method for constructing a finite-time synchronization model of heterogeneous chaotic systems and designing a controller, constructs a model of two completely synchronized heterogeneous chaotic systems, and designs a relatively simple and continuous controller to overcome the oscillation phenomenon caused by the discontinuity of the controller, avoids the loss of original system information due to the simple elimination of the nonlinear terms of the master-slave system, and retains as many characteristics of the original system as possible. In addition, the synchronization criterion and the analytical expression of the synchronization time estimation are proved. The criterion can pre-determine whether the two heterogeneous chaotic systems can achieve synchronization, and the analytical expression of the synchronization time estimation can perform analytical estimation of the synchronization time.

[0012] (II) Technical solution

[0013] To achieve the above objectives, the present invention is implemented by the following technical scheme: a method for constructing a finite-time synchronization model of a heterogeneous chaotic system, specifically comprising the following steps:

[0014] S1. First, select the following two heterogeneous chaotic systems:

[0015] Main system:

[0016] From the system: where X∈R m , Z∈R m+n , A(t)∈R m×m , B(t)∈R (m+n)×(m+n) , continuous function F(X,t):R m ×R + →R m and G(Z,t):R m+n ×R + →R m+n are the nonlinear parts of the two systems respectively, and m≥2, n≥0. When n=0, the two chaotic systems are isodimensional heterogeneous chaotic systems, and when n>0, the two chaotic systems are different-dimensional heterogeneous chaotic systems.

[0017] S2. Since the dimension of the master system is less than or equal to that of the slave system, select m state components of the slave system to synchronize with m state components of the master system, and synchronize the remaining state components to 0. Then expand the dimension of the master system as follows:

[0018]

[0019] in is a 0 vector, 0 m×n ∈R m×n , 0 n×m ∈R n×m , 0 n×n ∈R n×n is a 0 matrix;

[0020] S3. At the same time, the system will be split as follows:

[0021]

[0022] Where Z m ∈R m is the selected partial state component to be synchronized with the main system, Z n ∈R n , B 11 (t)∈R m×m , B 12 (t)∈R m×n , B 21 (t)∈R n×m , B 22 (t)∈R n×n , continuous function G m (Z,t):R m+n ×R + →R m and G n (Z,t):R m+n ×R + →R n are the nonlinear components from the system respectively;

[0023] S4. The master-slave finite time synchronization model composed of the above master-slave heterogeneous chaotic systems is expressed as:

[0024]

[0025] where U(t)∈R m+n For coupling controller.

[0026] Preferably, if the initial value of the main system X(0) = X0∈R m and from any initial value of the system Z(0)=(Z m (0),Z n (0)) T =Z0∈R m+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 And when t≥T(X0,Z0), ||Z m (t)-X(t)|≡0 and ||Z n (t)||≡0, where ||·|| represents the Euclidean norm of the vector. The above synchronization model achieves global finite-time synchronization, and T(X0,Z0) is the synchronization time.

[0027] The present invention also provides a design method for a finite time controller of a heterogeneous chaotic system, which specifically comprises the following steps:

[0028] P1, let E = (Z m (t)-X(t),Z n (t)) T ∈R m+n , then the following error system can be obtained from the master-slave finite time synchronization model in step S4:

[0029]

[0030] In general, the following holds true:

[0031]

[0032] where Q(t)∈R (m+n)×(m+n) Bounded, ψ(t)∈R m+n .

[0033] P2. According to the above two formulas, we can get:

[0034]

[0035] Therefore, the finite time synchronous control is designed as follows:

[0036]

[0037] where K∈R (m+n)×(m+n) and L∈R (m+n)×(m+n) is the diagonal coupling matrix to be determined,

[0038]

[0039] where α∈(0,1),

[0040]

[0041] P3. Under the designed controller, from the error system, we get:

[0042]

[0043] Preferably, the global finite-time synchronization problem of the synchronization model of heterogeneous chaotic systems that meets the conditions is equivalent to the error system at the origin E＝0 The global finite-time stability problem of the error system is solved, and the rest time of the global finite-time stability of the error system is the synchronization time.

[0044] (III) Beneficial effects

[0045] The present invention provides a method for constructing a finite-time synchronization model of a heterogeneous chaotic system and a controller design method. Compared with the prior art, the present invention has the following beneficial effects:

[0046] (1) The construction of the finite-time synchronization model of heterogeneous chaotic systems and the controller design method, by constructing a model of two completely synchronized heterogeneous chaotic systems and designing a relatively simple and continuous controller, overcomes the oscillation phenomenon caused by the discontinuity of the controller, avoids the loss of the original system information due to the simple elimination of the nonlinear terms of the master-slave system, and retains as many characteristics of the original system as possible. In addition, the synchronization criterion and the analytical expression of the synchronization time estimation are proved. The criterion can determine in advance whether the two heterogeneous chaotic systems can achieve synchronization, and the analytical expression of the synchronization time estimation can estimate the synchronization time analytically.

[0047] (2) The construction of the finite-time synchronization model and controller design method of the heterogeneous chaotic system solves the complete synchronization problem of the heterogeneous system and constructs a finite-time synchronization model, and the synchronization time can be estimated through analytical expressions.

[0048] (3) The construction of the finite-time synchronization model of the heterogeneous chaotic system and the controller design method. The designed controller structure is relatively simple, avoiding the loss of original system information due to the elimination of all nonlinear terms of the master-slave system and retaining as many characteristics of the original system as possible. BRIEF DESCRIPTION OF THE DRAWINGS

[0049] Figure 1 A schematic diagram of a chaotic attractor of a main chaotic gyroscope system in an embodiment of the present invention;

[0050] Figure 2 Schematic diagram of a chaotic attractor of a first uncontrolled slave chaotic electromechanical gyroscope system in an embodiment of the present invention;

[0051] Figure 3 Schematic diagram of a chaotic attractor of a second uncontrolled slave chaotic electromechanical gyroscope system in an embodiment of the present invention;

[0052] Figure 4 Schematic diagram of the finite time synchronous evolution of the master chaotic gyroscope system and the slave chaotic electromechanical gyroscope system in an embodiment of the present invention. DETAILED DESCRIPTION

[0053] 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.

[0054] See also Figure 1-4 The embodiment of the present invention provides a technical solution: a method for constructing a finite-time synchronization model of a heterogeneous chaotic system, which specifically includes the following steps:

[0055] S1. First, select the following two heterogeneous chaotic systems:

[0056] Main system:

[0057] From the system: where X∈R m , Z∈R m+n , A(t)∈R m×m , B(t)∈R (m+n)×(m+n) , continuous function F(X,t):R m ×R + →R m and G(Z,t):R m+n ×R + →R m+n are the nonlinear parts of the two systems respectively, and m≥2, n≥0. When n=0, the two chaotic systems are isodimensional heterogeneous chaotic systems, and when n>0, the two chaotic systems are different-dimensional heterogeneous chaotic systems.

[0058] S2. Since the dimension of the master system is less than or equal to that of the slave system, select m state components of the slave system to synchronize with m state components of the master system, and synchronize the remaining state components to 0. Then expand the dimension of the master system as follows:

[0059]

[0060] in is a 0 vector, 0 m×n ∈R m×n , 0 n×m ∈R n×m , 0 n×n ∈R n×n is a 0 matrix;

[0061] S3. At the same time, the system will be split as follows:

[0062]

[0063] Where Z m ∈R m is the selected partial state component to be synchronized with the main system, Z n ∈R n , B 11 (t)∈R m×m , B 12 (t)∈R m×n , B 21 (t)∈R n×m , B 22 (t)∈R n×n , continuous function G m (Z,t):R m+n ×R+ →R m and G n (Z,t):R m+n ×R + →R n are the nonlinear components from the system respectively;

[0064] S4. The master-slave finite time synchronization model composed of the above master-slave heterogeneous chaotic systems is expressed as:

[0065]

[0066] where U(t)∈R m+n For coupling controller.

[0067] In the embodiment of the present invention, if the initial value X(0) of the main system = X0∈R m and from any initial value of the system Z(0)=(Z m (0),Z n (0)) T =Z0∈R m+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 And when t≥T(X0,Z0), ||Z m (t)-X(t)||≡0 and ||Z n (t)||≡0, where ||·|| represents the Euclidean norm of the vector. The above synchronization model achieves global finite-time synchronization, and T(X0,Z0) is the synchronization time.

[0068] The embodiment of the present invention also provides a design method of a finite time controller for a heterogeneous chaotic system, which specifically includes the following steps:

[0069] P1, let E = (Z m (t)-X(t),Z n (t)) T ∈R m+n , then the master-slave finite time synchronization model (5) in step S4 can obtain the following error system:

[0070]

[0071] In general, the following holds true:

[0072]

[0073] where Q(t)∈R (m+n)×(m+n) Bounded, ψ(t)∈R m+n .

[0074] P2. According to the above two equations (6) and (7), we can get:

[0075]

[0076] Therefore, the finite time synchronous control is designed as follows:

[0077]

[0078] where K∈R (m+n)×(m+n) and L∈R (m+n)×(m+n) is the diagonal coupling matrix to be determined,

[0079]

[0080] where α∈(0,1),

[0081]

[0082] P3. Under the designed controller (9), from the error system (8), we get:

[0083]

[0084] In the embodiment of the present invention, the global finite-time synchronization problem of the synchronization model (5) of the heterogeneous chaotic system satisfying condition (7) is equivalent to the global finite-time stability problem of the error system (11) at the origin E=0, and the global finite-time stability rest time of the error system (11) is the synchronization time.

[0085] In the embodiment of the present invention, the finite time synchronization criterion of heterogeneous chaotic systems proves that:

[0086] Before proving the synchronization criterion, we first give a lemma to be used in the proof:

[0087] Lemma 1 For any positive real number a i , i=1,...,n and 0<p<2, the following inequality holds:

[0088]

[0089] Next, we prove the global finite-time synchronization criterion of the synchronization model (5) for heterogeneous chaotic systems that satisfy conditions (7) and (9).

[0090] Theorem 1 If there exists a positive definite diagonal matrix P = diag{p1,…,p m+n}∈R (m+n)×(m+n) , feedback gain matrix K = diag{k1,…,k m+n}∈R (m+n)×(m+n) and L = diag{l1,…,l m+n}∈R (m+n)×(m+n) , so that for any time t>0,

[0091]

[0092] [B(t)+Q(t)-K] T P+P[B(t)+Q(t)-K]≤0, (13) then the master-slave synchronization model (5) of the heterogeneous chaotic system coupled by the controller (9) can achieve global finite-time synchronization, and the corresponding synchronization time satisfies

[0093]

[0094] Where E=(Z m (0)-X(0),Z n (0)) T ∈R m+n , α∈(0,1), β=min{l1p1 (1-α)2 ,…,l m+ n p m+n (1-α)2}.

[0095] Proof: Choose the quadratic Lyapunov function V(E) = E T PE, then the derivative of V(E) with respect to time t along the error system (12) is:

[0096]

[0097] According to Lemma 1, for any α∈(0,1) and E∈R m+n , if we take the positive definite matrix P = diag{p1,…,p m+n}∈R (m+n)×(m+n) And the feedback gain matrix L = diag{l1,…,l m+n}∈R (m+n)×(m+n) So that β = min{l1p1 (1 -α)2 ,…,l m+n p m+n (-1α / )}2, then

[0098]

[0099] Therefore, if (12) and (13) hold, then from (15) and (16) we can get

[0100]

[0101] According to the finite time stability theorem in the literature [Bhat SP, Bernstein DS. Finite time stability of continuous autonomous systems [J]. SIAM Journal on Control and Optimization., 2000, 38(3): 751-766.], the master-slave synchronization model (5) of heterogeneous chaotic systems coupled by the controller (9) can achieve global finite time synchronization, and the corresponding synchronization time satisfies (14).

[0102] In the embodiment of the present invention, the theoretical application and simulation verification are as follows:

[0103] Taking the third-order gyroscope chaotic system and the fourth-order electromechanical gyroscope chaotic system as examples, the finite-time synchronization of two heterogeneous chaotic systems of different dimensions is studied.

[0104] The matrix form of the gyroscope system can be described as:

[0105]

[0106] Where X = (x1, x2, x3) T ∈R 3 ,

[0107]

[0108]

[0109] Where A, B and C are the main inertial potential differences, h1, h2 and h3 are the angular moments of the rotor, ω r It is the projection of the angular velocity of the gyroscope on the X, Y and Z axes. All parameters A, B, C, b, f, h i (i=1,2,3),q i (i=1,…,6) are all positive numbers.

[0110] The electromechanical gyroscope system can be described as:

[0111]

[0112] Where Z = (z1, z2, z3, z4) T ∈R 4 ,

[0113]

[0114] Where all parameters A, B, C, b, f, h i (i=1,2,3),q i (i=1,…,6), KT , K a , K b , R and L are both positive, and the electromechanical gyroscope system is chaotic under certain parameters.

[0115] Taking the gyroscope chaotic system (18) as the master system and the electric gyroscope chaotic system (21) as the slave system, the following finite-time synchronization model of heterogeneous chaotic systems can be constructed from (5):

[0116]

[0117] Where Z m =(z1,z2,z3) T ∈R 3 is the selected partial state component to be synchronized with the main system, Z n =z4∈R,U(t)∈R 4 is the coupling controller to be designed,

[0118]

[0119] Based on formula (7), from formula (20) and (27), we can get:

[0120]

[0121] Therefore, we know that:

[0122]

[0123] Based on equation (9), the controller of the finite time synchronization model (24) of the third-order gyroscope chaotic system and the fourth-order electric gyroscope chaotic system can be designed by equations (19), (25), (26) and (30) as follows:

[0124]

[0125] Where K = diag{k1, k2, k3, k4} and L = diag{l1, l2, l3, l4} are the diagonal coupling matrices to be determined, k i ≥0 and l i >0(i=1,2,3,4),

[0126]

[0127] where α∈(0,1),

[0128]

[0129] Therefore, we can use Theorem 1 to prove the following proposition:

[0130] Proposition 1 If there exists a feedback gain matrix K = diag{k1, k2, k3, k4} and L = diag{l1, l2, l3, l4} such that for any time t>0,

[0131] A=B=C, (33)

[0132] k1≥0, (34)

[0133] k2≥0, (35)

[0134] k3≥0, (36)

[0135] k4≥0, (37)

[0136] Then the gyroscope chaotic system and the electric gyroscope chaotic system in the master-slave synchronization model (24) can achieve finite time synchronization, and the synchronization time satisfies:

[0137]

[0138] where α∈(0,1),

[0139] Proof: Take

[0140]

[0141] Therefore, if formula (33) holds, then:

[0142]

[0143] From (22), (28) and (39), we can know:

[0144]

[0145] in:

[0146] Where θ(w,v)=w 2 +wv+v 2 .

[0147] According to Shur's lemma: S is semi-negative definite if the following two equations

[0148]

[0149] Established at the same time.

[0150] Obviously, formula (42) holds if the following two formulas hold:

[0151]

[0152] Obviously, θ(w,v)=w 2 +wv+v 2 ≥0, and parameters A, B, C, b, f, h i (i=1,2,3),q i (i=1,…,6), K T , K a , K b , R and L are both positive numbers, k i ≥0 and l i ≥0(i=1,2,3,4), so equation (44) and equation (45) always hold, and thus equation (42) holds.

[0153] Furthermore, formula (43) holds if the following two formulas hold:

[0154]

[0155] Obviously, θ(w,v)=w 2 +wv+v 2 ≥0, and parameters A, B, C, b, f, h i (i=1,2,3),q i (i=1,…,6), K T , K a , K b , R and L are both positive numbers, k i ≥0 and l i >0(i=1,2,3,4), so equation (46) and equation (47) always hold, and thus equation (43) holds.

[0156] In summary, Theorem 1 shows that the gyroscope chaotic system and the electromechanical gyroscope chaotic system in the master-slave synchronization model (24) can achieve finite-time synchronization, and the synchronization time satisfies equation (38).

[0157] Next, we use an example to verify the theoretical results. In this example, the initial value of the master chaotic gyroscope system and the initial condition of the slave chaotic electromechanical gyroscope system are arbitrarily selected as x(0) = (1, -1, -1.8) T and z(0)=(-0.1,-0.2,-0.3,0.1) T The parameters of the main chaotic gyroscope system are A = 1000, B = 1000, C = 1000, h1 = 200, h2 = 200, h3 = 200, b = 200, f = 12.9, ω = 1, ω r =0,q i(i=1,…,6)=1, the parameters of the chaotic electromechanical gyroscope system are A=1000, B=1000, C=1000, h1=200, h2=200, h3=200, b=200, f=12.9, ω=1, ω r =0,q i (i=1,…,6…)=1,K T =300,K a =50,K b =1.3,R=100,L=2.

[0158] Under the given initial values ​​and parameters, both the master chaotic gyroscope system and the slave chaotic electromechanical gyroscope system are chaotic, such as Figure 1-3 As shown. The control parameters of (31) are selected as L = diag{0.3, 0.1, 0.5, 0.9} and K = diag{0, 0, 0, 0}, then this example satisfies Proposition 1. We can conclude that the master chaotic gyroscope system and the slave chaotic electromechanical gyroscope system can achieve finite time synchronization, and the synchronization time T ≤ T can be predicted by equation (38). E =57.5771. In addition, we simulate the synchronization process of the master chaotic gyroscope system and the slave chaotic electromechanical gyroscope system, as shown in Figure 4 As shown in the figure, it can be seen that each state component can achieve synchronization, and the actual synchronization time is within the estimated time. Therefore, we can verify the effectiveness of the obtained criterion.

[0159] Meanwhile, the contents not described in detail in this specification belong to the prior art known to those skilled in the art.

[0160] 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.

[0161] 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 method for constructing a finite-time synchronization model for heterogeneous chaotic systems, characterized by: The specific steps include: S1. First, select the following two heterogeneous chaotic systems: Main system: From the system: where X∈R m , Z∈R m+n , A(t)∈R m×m , B(t)∈R (m+n)×(m+n) , continuous function F(X,t):R m ×R + →R m and G(Z,t):R m+n ×R + →R m+n are the nonlinear parts of the two systems respectively, and m≥2, n≥0. When n=0, the two chaotic systems are isodimensional heterogeneous chaotic systems, while when n>0, the two chaotic systems are different-dimensional heterogeneous chaotic systems. S2. Since the dimension of the master system is less than or equal to that of the slave system, select m state components of the slave system to synchronize with m state components of the master system, and synchronize the remaining state components to 0. Then expand the dimension of the master system as follows: in is a 0 vector, 0 m×n ∈R m×n , 0 n×m ∈R n×m , 0 n×n ∈R n×n is a 0 matrix; S3. At the same time, the system will be split as follows: where Z m ∈R m is the selected partial state component to be synchronized with the main system, Z n ∈R n , B 11 (t)∈R m×m , B 12 (t)∈R m×n , B 21 (t)∈R n×m , B 22 (t)∈R n×n , continuous function G m (Z,t):R m+n ×R + →R m and G n (Z,t):R m+n ×R + →R n are the nonlinear components of the slave system respectively; S4. The master-slave finite-time synchronization model composed of the above master-slave heterogeneous chaotic systems is expressed as: where U(t)∈R m+n is a coupling controller; The finite time synchronous controller is as follows: where ψ(t)∈R m+n , K∈R (m+n)×(m+n) and L∈R (m+n)×(m+n) is the diagonal coupling matrix to be determined, where α∈(0,1), 2. The method for constructing a finite-time synchronization model of heterogeneous chaotic systems according to claim 1, characterized in that: If the initial value of the main system X(0) = X0∈R m and from any initial value of the system Z(0)=(Z m (0),Z n (0)) T =Z0∈R m +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 And when t≥T(X0,Z0), Z m (t)-X(t)≡0 and Z n (t)≡0, where · represents the Euclidean norm of the vector. The above synchronization model achieves global finite-time synchronization, and T(X0,Z0) is the synchronization time.

3. A design method for a finite-time controller for heterogeneous chaotic systems, characterized by: The specific steps include: P1, let E = (Z m (t)-X(t),Z n (t)) T ∈R m+n , then the following error system can be obtained from the master-slave finite time synchronization model in step S4 of claim 1: In general, the following holds true: where Q(t)∈R (m+n)×(m+n) Bounded, P2. According to the above two formulas, we can get: P3. Under the designed controller, the error system is obtained from the synchronization model:

4. The design method of a finite-time controller for a heterogeneous chaotic system according to claim 3 is characterized by: The global finite-time synchronization problem of the synchronization model of heterogeneous chaotic systems that meet the conditions is equivalent to the global finite-time stability problem of the error system at the origin E=0, and the global finite-time stability stopping time of the error system is the synchronization time.

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