Non-fragile master-slave synchronization control method of chaos Lur'e system based on linear matrix inequality

Abstract

The invention discloses a non-fragile master-slave synchronization control method of a chaos Lur'e system based on a linear matrix inequality. The method comprises the following steps: S1, establishing a master-slave system of the chaos Lur'e system; s2, constructing a lagging non-fragile controller of the chaotic Lur'e system; s3, enabling the lagging non-fragile controller to act on the slave system, and building an error system of the chaotic Lur'e system; s4, constructing an augmented Lyapunov generic function of the state information and the sector function information; s5, solving the derivative of the Lyapunov ubiquitous function along the error system; according to the method disclosed by the invention, the synchronization of the slave system and the master system can still be realized under the condition of relatively large lagging.

Description

Technical Field

[0001] The invention belongs to the field of intelligent control of chaotic systems, and in particular relates to a non-fragile master-slave synchronization control method of a chaotic Lur'e system based on linear matrix inequality. Background Art

[0002] Since Pecora and Carroll first proposed the master-slave synchronization characteristics of chaotic systems, in the past few decades, chaotic synchronization has been widely used in many fields such as physics, chemical reactors, ecosystems, and secure communications. The so-called master-slave synchronization refers to the use of the output variables of the master-slave system to control the slave system so that the state vector of the slave system can asymptotically approach the state vector of the master system. Lur'e system is a widely existing type of nonlinear system. Cai's circuit, hyperchaotic attractor, etc. can be modeled as Lur'e system.

[0003] On the other hand, due to factors such as signal transmission, time lag is common in real systems. The existence of time lag may cause the system to oscillate, or make the corresponding closed-loop system performance worse, or even make the system unstable. Therefore, it is of great significance to study the master-slave synchronization of chaotic Lur'e system with lag. At present, the research on the master-slave synchronization control of chaotic Lur'e system with lag mainly focuses on constructing appropriate Lyapunov functionals, estimating the derivatives of Lyapunov functionals along the error system, and finding that the error system can still be stabilized within a certain range of lag, that is, master-slave synchronization is achieved. Generally speaking, the larger the allowed lag range, the less conservative the result. It is noted that when the controller acts on the system, the controller parameters may also change to a certain extent due to environmental changes or aging of components. In this case, how to design the controller so that the chaotic Lur'e system can still achieve master-slave synchronization has practical significance. Summary of the invention

[0004] In view of the problems existing in the prior art, the present invention provides a non-fragile master-slave synchronization control method for a chaotic Lur'e system based on a linear matrix inequality. The proposed non-fragile master-slave synchronization controller for a chaotic Lur'e system constructs an augmented Lyapunov functional containing more information. In order to reduce the conservatism of the result, when estimating the derivative of the Lyapunov functional along the error system, the generalized free weight matrix integral inequality, inverted convex combination and other methods are used. The second-order convex combination is used to express the obtained result as a linear matrix inequality, which can be conveniently solved using a toolkit in Matlab to achieve the master-slave synchronization of the chaotic Lur'e system.

[0005] In order to solve the above technical problems, the present invention provides the following technical solutions: a non-fragile master-slave synchronization control method of a chaotic Lur'e system based on linear matrix inequality, comprising the following steps:

[0006] S1. Establish the master-slave system of the chaotic Lur'e system;

[0007] S2, construct a hysteresis non-fragile controller for chaotic Lur'e system;

[0008] S3, the delayed non-fragile controller acts on the slave system to establish the error system of the chaotic Lur'e system;

[0009] S4, Augmented Lyapunov Function for constructing state information and fan function information;

[0010] S5. Find the derivative of the Lyapunov functional along the error system, and based on the linear matrix inequality, derive the calculation method of the controller gain matrix to stabilize the error system and achieve master-slave system synchronization.

[0011] Furthermore, in the aforementioned step S1, the master-slave system of the chaotic Lur'e system is established as follows:

[0012] Main system:

[0013]

[0014] From the system

[0015]

[0016] Where x(t),y(t)∈R n are the state vectors of the master system and the slave system respectively, p(t),q(t)∈R n are the output vectors of the master system and the slave system respectively, A, A1, W, D, C are constant matrices with appropriate dimensions, and the lag function d(t) satisfies f(·) satisfies the fan-shaped condition, that is,

[0017]

[0018] Among them, f i (·) represents the i-th component of f(), are constants, u∈R,v∈R.

[0019] Furthermore, the hysteresis non-fragile controller in the aforementioned step S2 is as follows:

[0020] U:u(t)=(K+ΔK(t))(p(td(t))-q(td(t))) (4)

[0021] Where K is the controller gain matrix, ΔK(t) = HF(t)E, H, E is a constant matrix of preset dimensions, and F(t) satisfies F T (t)F(t)≤I, I is the identity matrix.

[0022] Furthermore, in the aforementioned step S3, the error system of the chaotic Lur'e system is established, and e(t)=x(t)-y(t). The error system is as follows:

[0023]

[0024] Where g(De(t))=f(Dx(t))-f(Dy(t)), then

[0025]

[0026] D i Represents the i-th row vector of matrix D.

[0027] Furthermore, in the aforementioned step S4, an augmented Lyapunov function of the state information and the fan function information is constructed as follows:

[0028] V(t)=V1(t)+V2(t)+V3(t)+V4(t) (7)

[0029] In the formula,

[0030]

[0031] P i ,R i ,(i=1,2),Z a ,Z b , Q is a constant symmetric positive definite matrix with a preset dimension, δ ki (k=1,2,3,4,i=1,2,…,m…,) is a positive scalar.

[0032] Furthermore, the aforementioned step S5 specifically involves taking the derivative of V(t) along the error system, and using methods such as generalized free weight matrix integral inequality and inverted convex combination to obtain The single integral term is estimated and the calculation method of the controller gain is obtained.

[0033] η1(t)=col(e(t)e(td(t))e(td)v1(t)v2(t)v3(t)v4(t)),

[0034]

[0035] η(t)=col(η1(t)η2(t)η3(t)),

[0036]

[0037] e i =(0 n,(i-1)n I n 0n,(10-i)n+3m ),i=1,2,…,10,e j =(0 m,10n+(j-11)m I m 0 m,(13-j)m ),j=11,12,13,

[0038]

[0039] e d =col(e1 e2 e3 e4 e5 e6 e7 e8 e 10 ).

[0040] For the preset constants d, μ, a1, a2, the error system (5) is stable if there exists a symmetric positive definite matrix P i ,R i (i=1,2),Z a ,Z b ,Q, positive definite diagonal matrix Θ i (i=1,2,3,4), F bi ,F di (i=1,2),H b1 ,H d1 , symmetric matrices T1, T2, and a matrix M with a preset dimension ij (i,j=1,2),S kj (k=1,2,3),F,G,Y, the positive definite matrix S>0 makes the following linear matrix inequalities (8)-(10) hold, and the corresponding controller gain matrix is ​​K=G -1 Y,

[0041]

[0042] Ξ(d(t))=Ξ1(d(t))+Ξ2+Ξ3+Ξ 41 (d(t))+Ξ5(d(t))+Ξ6(d(t))+Ξ7+(ECe2) T SECe2,

[0043]

[0044] Γ1=diag{-Z a2 -3Z a2 -Z b -3Z b -5Z b},

[0045] Γ2=diag{-Z a1 -3Z a1 -Z b -3Z b -5Zb},

[0046]

[0047]

[0048] Ξ3=sym((e 11 -L1De1) T Θ1De8+(L2De1-e 11 ) T θ2De8)

[0049] +sym((e 12 -L1De3) T Θ3De9+(L2De3-e 12 ) T Θ4De9),

[0050]

[0051] Ξ7(d(t))=sym{(a1e1+a2e8) T G(-e8+Ae1+A1e2+We 11 -YCe2)},

[0052]

[0053] V i (t) is derived along the error system (5) to obtain

[0054]

[0055]

[0056] right Estimate the integral term in . If T1 and T2 are symmetric matrices, the following formula holds.

[0057]

[0058] By the generalized free weight matrix integral inequality, we get

[0059]

[0060] The following formula is obtained from the inverted convex combination inequality:

[0061]

[0062] For a positive definite diagonal matrix F bi ,F di (i=1,2),H b1 ,H d1The following inequality holds true i (s) = 2(g(De(s)) - L1De(s)) T (d(t)F bi +(dd(t))F di )(L2De(s)-g(De(s))≥0,q i (s1,s2)=2((g(De(s1))-g(De(s2)))-L1D(e(s1)-e(s2))) T (d(t)H b1 +(dd(t))H d1 )

[0063] ×(L2D(e(s1)-e(s2))-(g(De(s1))-g(De(s2)))≥0,

[0064] For any matrix F, G with a preset dimension, the following equation holds:

[0065] 2(e d η(t)) T F(d 2 (t)e6+(dd(t)) 2 e7(t)+(dd(t))d(t)e4-e 10 )η(t)=0,2η T (t)(a1e1+a2e8) T G(-e8+Ae1+A1e2+We 11 -(K+ΔK(t))Ce2)η(t)=0.

[0066] Let Y = GK, from the above we can get in

[0067] Σ(d(t))=Ξ(d(t))+Ξ 42 (d(t))+Ξ 43 (d(t))+(a1e1+a2e8) T GHS -1 G T H T (a1e1+a2e8),

[0068]

[0069] like

[0070] Σ(0)<0,Σ(d)<0,-d 2 Γ+Σ(0)<0, (11)

[0071] but From Schur's complement lemma, we can see that if (8)-(10) hold, then (11) holds.

[0072] Compared with the prior art, the beneficial technical effects of the above technical solution adopted by the present invention are as follows: (1) When designing the controller, the parameter changes of the controller components are taken into account and modeled as additive disturbances; (2) An augmented Lyapunov functional containing error system state, error system lag state, error system single integral, double integral state and fan function information is constructed; (3) In estimating When the integral term in is used, the integral inequality, inverted convex combination and other methods are used to obtain The sufficient conditions for negative definiteness are obtained, and the Schur complement lemma and second-order convex combination are used to express it in the form of linear matrix inequality, which can be easily solved using the toolkit in Matlab. (4) Based on the linear matrix inequality, the calculation method of the controller gain matrix is ​​given, so that when there is an additive disturbance in the controller, the error system can still be asymptotically stable, that is, the master-slave synchronization of the chaotic Lur'e system is achieved. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1 It is a flow chart of the present invention.

[0074] Figure 2 It is the phase trajectory of the main system when γ=0 in the specific implementation case of the present invention.

[0075] Figure 3 It is the state of the error system when γ=0 in the specific implementation case of the present invention.

[0076] Figure 4 It is the phase trajectory of the main system when γ=0.1 in the specific implementation case of the present invention.

[0077] Figure 5 It is the state of the error system when γ=0.1 in the specific implementation case of the present invention. DETAILED DESCRIPTION

[0078] In order to better understand the technical content of the present invention, specific embodiments are given and described as follows in conjunction with the accompanying drawings.

[0079] Various aspects of the invention are described herein with reference to the accompanying drawings, in which many illustrative embodiments are shown. The embodiments of the invention are not limited to those described in the accompanying drawings. It should be understood that the invention is implemented by any of the various concepts and embodiments described above, as well as the concepts and embodiments described in detail below, because the concepts and embodiments disclosed in the invention are not limited to any implementation. In addition, some aspects disclosed in the invention may be used alone or in any appropriate combination with other aspects disclosed in the invention.

[0080] refer to Figure 1 The present invention proposes a non-fragile master-slave synchronization control method for a chaotic Lur'e system based on linear matrix inequality, comprising the following steps:

[0081] S1. Establish the master-slave system of the chaotic Lur'e system;

[0082] S2, construct a hysteresis non-fragile controller for chaotic Lur'e system;

[0083] S3, the delayed non-fragile controller acts on the slave system to establish the error system of the chaotic Lur'e system;

[0084] S4, Augmented Lyapunov Function for constructing state information and fan function information;

[0085] S5. Find the derivative of the Lyapunov functional along the error system, and based on the linear matrix inequality, derive the calculation method of the controller gain matrix to stabilize the error system and achieve master-slave system synchronization.

[0086] Step S1 establishes the following master-slave system of the chaotic Lur'e system as follows:

[0087] Main system:

[0088]

[0089] From the system

[0090]

[0091] where x(t),y(t)∈R n are the state vectors of the master system and the slave system respectively, p(t),q(t)∈R n are the output vectors of the master system and the slave system respectively, A, A1, W, D, C are constant matrices with appropriate dimensions, and the lag function d(t) satisfies f(·) satisfies the fan-shaped condition, that is,

[0092]

[0093] Among them, f i (·) represents the i-th component of f(·), are constants, u∈R,v∈R.

[0094] The hysteresis non-fragile controller in step S2 is as follows:

[0095] U:u(t)=(K+ΔK(t))(p(td(t))-q(td(t))) (4)

[0096] Where K is the controller gain matrix, ΔK(t) = HF(t)E, H, E is a constant matrix of preset dimensions, and F(t) satisfies F T (t)F(t)≤I, I is the identity matrix.

[0097] In step S3, the error system of the chaotic Lur'e system is established, and e(t)=x(t)-y(t). The error system is as follows:

[0098]

[0099] g(De(t))=f(Dx(t))-f(Dy(t)), from (3) we can get

[0100]

[0101] D i Represents the i-th row vector of matrix D.

[0102] In step S4, an augmented Lyapunov function of state information and fan function information is constructed as follows:

[0103] V(t)=V1(t)+V2(t)+V3(t)+V4(t) (7)

[0104] In the formula,

[0105]

[0106] P i ,R i ,(i=1,2)Z a ,Z b , Q is a constant matrix with appropriate dimension, δ ki (k=1,2,3,4,i=1,2,…,m) is a positive scalar.

[0107] Step S5: Derivative V(t) along the error system, and use the generalized free weight matrix integral inequality, inverted convex combination and other methods to find The single integral term is estimated and the calculation method of the controller gain is given.

[0108] η1(t)=col(e(t) e(td(t)) e(td) v1(t) v2(t) v3(t) v4(t)),

[0109]

[0110] eta(t)=col(eta1(t) eta2(t) eta3(t)),

[0111]

[0112] e i =(0 n,(i-1)n I n 0 n,(10-i)n+3m ),i=1,2,…,10,e j =(0 m,10n+(j-11)m I m 0 m,(13-j)m ),j=11,12,13,

[0113]

[0114] e d =col(e1 e2 e3 e4 e5 e6 e7 e8 e 10 ).

[0115] For given constants d, μ, a1, a2, the error system (5) is asymptotically stable if there exists a symmetric positive definite matrix P i ,R i (i=1,2),Z a ,Z b ,Q, positive definite diagonal matrix Θ i (i=1,2,3,4), F bi ,F di (i=1,2),H b1 ,H d1 , symmetric matrices T1, T2, and a matrix M of appropriate dimension ij (i,j=1,2),S kj (k=1,2,3),F,G,Y, the positive definite matrix S>0 makes the following linear matrix inequalities (8)-(10) hold, and the corresponding controller gain matrix is ​​K=G -1 Y,

[0116]

[0117]

[0118] Ξ(d(t))=Ξ1(d(t))+Ξ2+Ξ3+Ξ 41 (d(t))+Ξ5(d(t))+Ξ6(d(t))+Ξ7+(ECe2) T SECe2,

[0119] Γ1=diag{-Z a2 -3Z a2 -Z b -3Z b -5Z b},

[0120] Γ2=diag{-Z a1 -3Z a1 -Z b -3Z b -5Z b},

[0121]

[0122] Ξ3=sym((e 11 -L1De1) T Θ1De8+(L2De1-e 11 ) T Θ2De8)+sym((e 12 -L1De3) T Θ3De9+(L2De3-e 12 ) T Θ4De9),

[0123]

[0124] Ξ7(d(t))=sym{(a1e1+a2e8) T G(-e8+Ae1+A1e2+We 11 -YCe2)},

[0125] Υ1=(e1 T e3 T (d(t)e4+(dd(t))e5) T e 10 T de 13 T ) T ,

[0126]

[0127]

[0128] V i (t) is derived along the error system (5) to obtain

[0129]

[0130] right Estimate the integral term in . If T1 and T2 are symmetric matrices, the following formula holds.

[0131]

[0132] By the generalized free weight matrix integral inequality, we get

[0133]

[0134]

[0135] On the other hand, the inverted convex combination inequality gives

[0136]

[0137] For a positive definite diagonal matrix F bi ,F di (i=1,2),H b1 ,H d1 The following inequality holds

[0138] p i (s) = 2(g(De(s)) - L1De(s)) T (d(t)F bi +(dd(t))F di )(L2De(s)-g(De(s))≥0,

[0139] q i (s1,s2)=2((g(De(s1))-g(De(s2)))-L1D(e(s1)-e(s2))) T (d(t)H b1 +(dd(t))H d1 )

[0140] ×(L2D(e(s1)-e(s2))-(g(De(s1))-g(De(s2)))≥0,

[0141] For any matrix F, G with appropriate dimensions, the following equation holds true:

[0142] 2(e d η(t)) T F(d 2 (t)e6+(dd(t)) 2 e7(t)+(dd(t))d(t)e4-e 10 )η(t)=0,

[0143] 2η T (t)(a1e1+a2e8) T G(-e8+Ae1+A1e2+We8-(K+ΔK(t))Ce2)η(t)=0.

[0144] Let Y = GK, from the above we can get in

[0145] Σ(d(t))=Ξ(d(t))+Ξ 42(d(t))+Ξ 43 (d(t))+(a1e1+a2e8) T GHS -1 G T H T (a1e1+a2e8),

[0146]

[0147] like

[0148] Σ(0)<0,Σ(d)<0,-d 2 Γ+Σ(0)<0, (11)

[0149] but According to Schur's complement lemma, if (8)-(10) hold, then (11) holds.

[0150] Provide an implementation case of the present invention below:

[0151]

[0152] in, It is expressed in the form of (1)

[0153] α=9,β=14.28, L1=0,L2=1.

[0154] Remember H T =[0.1 0.1 0.1], E=1, F(t)=sin t, d=0.5, μ=0.3, take a1=1, a2=0.5,

[0155] x(0) = [-0.2 -0.33 0.2] T ,y(0)=[0.5 0.1 -0.66] T ,d(t)=0.25+0.25sin(1.2t),

[0156] γ=0, at this time, the phase trajectory of the main system is as follows Figure 2 As shown. Solving (8)-(10) we get K = [-3.9641 -0.37062.9719] T , the state trajectory of the corresponding error system is as follows Figure 3 As shown. γ=0.1, the phase trajectory of the main system is as follows Figure 4 As shown, solving (8)-(10) yields K = [-3.8318 -0.2575 2.6749] T , the state trajectory of the corresponding error system is as follows Figure 5It can be seen from the simulation results that, in the case of additive disturbance in the controller gain, the chaotic Lur'e system can still achieve master-slave synchronization by using the controller designed by the present invention.

[0157] Although the present invention has been described above with preferred embodiments, it is not intended to limit the present invention. A person skilled in the art of the present invention may make various modifications and improvements without departing from the spirit and scope of the present invention. Therefore, the scope of protection of the present invention shall be determined by the claims.

Claims

1. A non-fragile master-slave synchronization control method for chaotic Lur'e system based on linear matrix inequality, characterized in that: The following steps are involved: S1. Establish the master-slave system of the chaotic Lur'e system; S2, construct a hysteresis non-fragile controller for chaotic Lur'e system; S3, the delayed non-fragile controller acts on the slave system to establish the error system of the chaotic Lur'e system; S4, Augmented Lyapunov Function for constructing state information and fan function information; S5. Find the derivative of the Lyapunov functional along the error system, and based on the linear matrix inequality, derive the calculation method of the controller gain matrix to stabilize the error system and achieve master-slave system synchronization.

2. The non-fragile master-slave synchronization control method of chaotic Lur'e system based on linear matrix inequality according to claim 1, characterized in that: In step S1, the master-slave system of the chaotic Lur'e system is established as follows: Main system: From the system Where x(t),y(t)∈R n are the state vectors of the master system and the slave system respectively, p(t),q(t)∈R n are the output vectors of the master system and the slave system respectively, A, A1, W, D, C are constant matrices with appropriate dimensions, and the lag function d(t) satisfies 0<d(t)<d, f(·) satisfies the fan-shaped condition, that is, Among them, f i (·) represents the i-th component of f(·), are constants, u∈R,v∈R.

3. The non-fragile master-slave synchronization control method of chaotic Lur'e system based on linear matrix inequality according to claim 1, characterized in that: The hysteresis non-fragile controller in step S2 is as follows: U:u(t)=(K+ΔK(t))(p(td(t))-q(td(t))) (4) Where K is the controller gain matrix, ΔK(t) = HF(t)E, H, E is a constant matrix of preset dimensions, and F(t) satisfies F T (t)F(t)≤I, I is the identity matrix.

4. The non-fragile master-slave synchronization control method of chaotic Lur'e system based on linear matrix inequality according to claim 3, characterized in that: In step S3, the error system of the chaotic Lur'e system is established, and e(t)=x(t)-y(t). The error system is as follows: Where g(De(t))=f(Dx(t))-f(Dy(t)), then D i Represents the i-th row vector of matrix D.

5. The non-fragile master-slave synchronization control method of chaotic Lur'e system based on linear matrix inequality according to claim 4, characterized in that: In step S4, an augmented Lyapunov function of state information and fan function information is constructed as follows: V(t)=V1(t)+V2(t)+V3(t)+V4(t) (7) In the formula, P i ,R i ,Z a ,Z b , Q is a constant symmetric positive definite matrix with a preset dimension, i = 1, 2; δ ki is a positive scalar, where k=1,2,3,4, i=1,2,…,m.

6. The non-fragile master-slave synchronization control method of chaotic Lur'e system based on linear matrix inequality according to claim 5, characterized in that: Step S5 specifically involves taking the derivative of V(t) along the error system and using methods such as generalized free weight matrix integral inequality and inverted convex combination to calculate The single integral term is estimated and the calculation method of the controller gain is obtained: η1(t)=col(e(t) e(td(t)) e(td) v1(t) v2(t) v3(t) v4(t)), η(t)=col(η1(t)η2(t)η3(t)), yes i =(0 n,(i-1)n I n 0 n,(10-i)n+3m ),i=1,2,…,10,e j =(0 m,10n+(j-11)m I m 0 m,(13-j)m ),j=11,12,13, e d =col(e1e2e3e4e5e6e7e8e 10 ), For the preset constants d, μ, a1, a2, the error system (5) is stable if there exists a symmetric positive definite matrix P i ,R i ,Z a ,Z b ,Q, where i = 1,2, positive definite diagonal matrix Θ i , where i = 1, 2, 3, 4, F bi ,F di , H b1 ,H d1 , i = 1, 2, symmetric matrices T1, T2, matrix M with preset dimensions ij , S kj ,F,G,Y;i,j=1,2,k=1,2,3,positive definite matrix S>0 makes the following linear matrix inequalities (8)-(10) hold, and the corresponding controller gain matrix is ​​K=G -1 Y, Ξ(d(t))=Ξ1(d(t))+Ξ2+Ξ3+Ξ 41 (d(t))+Ξ5(d(t))+Ξ6(d(t))+Ξ7+(ECe2) T SECe2, <h2 style=";text-align:left;direction:ltr">Γ1 = diag{-Z<h2 style=";text-align:left;direction:ltr"> a2 <h2 style=";text-align:left;direction:ltr"> -3Z<h2 style=";text-align:left;direction:ltr"> a2 <h2 style=";text-align:left;direction:ltr"> -Z<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr"> -3Z<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr"> -5Z<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr">}, <h2 style=";text-align:left;direction:ltr">Γ2=diag{-Z<h2 style=";text-align:left;direction:ltr"> a1 <h2 style=";text-align:left;direction:ltr"> -3Z<h2 style=";text-align:left;direction:ltr"> a1 <h2 style=";text-align:left;direction:ltr"> -Z<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr"> -3Z<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr"> -5Z<h2 style=";text-align:left;direction:ltr"> b <h2 style=";text-align:left;direction:ltr">}, V i (t) is derived along the error system (5) to obtain: right Estimate the integral term in. If T1 and T2 are symmetric matrices, the following formula holds; By the generalized free weight matrix integral inequality, we get The following formula is obtained from the inverted convex combination inequality: For a positive definite diagonal matrix F bi ,F di (i=1,2),H b1 ,H d1 The following inequality holds true i (s) = 2(g(De(s)) - L1De(s)) T (d(t)F bi +(dd(t))F di )(L2De(s)-g(De(s))≥0, q i (s1,s2)=2((g(De(s1))-g(De(s2)))-L1D(e(s1)-e(s2))) T (d(t)H b1 +(d-d(t))H d1 ) ×(L2D(e(s1)-e(s2))-(g(De(s1))-g(De(s2)))≥0 For any matrix F, G with a preset dimension, the following equation holds: 2(e) d η(t)) T F(d 2 (t)e6+(dd(t)) 2 e7(t)+(dd(t))d(t)e4-e 10 )η(t)=0, 2η T (t)(a1e1+a2e8) T G(-e8+Ae1+A1e2+We 11 -(K+ΔK(t))Ce2)η(t)=0, Let Y = GK, from the above we can get in, Σ(d(t))=Ξ(d(t))+Ξ 42 (d(t))+Ξ 43 (d(t))+(a1e1+a2e8) T GHS -1 G T H T (a1e1+a2e8), like Σ(0)<0,Σ(d)<0,-d 2 C+S(0)<0, (11) but From Schur's complement lemma, we can see that if (8)-(10) hold, then (11) holds.

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