CN107479370A

CN107479370A - A kind of four rotor wing unmanned aerial vehicle finite time self-adaptation control methods based on non-singular terminal sliding formwork

Info

Publication number: CN107479370A
Application number: CN201710532250.3A
Authority: CN
Inventors: 陈强; 叶艳; 胡如海
Original assignee: Zhejiang University of Technology ZJUT
Current assignee: Zhejiang University of Technology ZJUT
Priority date: 2017-07-03
Filing date: 2017-07-03
Publication date: 2017-12-15
Anticipated expiration: 2037-07-03
Also published as: CN107479370B

Abstract

A kind of four rotor wing unmanned aerial vehicle finite time self-adaptation control methods based on non-singular terminal sliding formwork, for the four rotor wing unmanned aerial vehicle systems with inertia uncertain factor and external disturbance.According to the dynamic system of four rotor wing unmanned aerial vehicles, using non-singular terminal sliding-mode control, in conjunction with Self Adaptive Control, a kind of four rotor wing unmanned aerial vehicle self-adaptation control methods based on non-singular terminal sliding formwork are designed.The design of non-singular terminal sliding formwork is to ensure the finite time convergence control characteristic of system, and avoids singularity problem existing for TSM control, effectively weakens buffeting problem.In addition, Self Adaptive Control is the inertia uncertainty and external disturbance for processing system.The invention provides one kind can eliminate sliding-mode surface singularity problem, and can effectively suppress with compensation system existing for inertia is uncertain and the control method of external disturbance, ensure the finite time convergence control characteristic of system.

Description

A kind of four rotor wing unmanned aerial vehicle finite times based on non-singular terminal sliding formwork are self-adaptive controlled Method processed

Technical field

The present invention relates to a kind of four rotor wing unmanned aerial vehicle finite time self-adaptation control methods based on non-singular terminal sliding formwork, Particular with inertia uncertain factor and four rotor wing unmanned aerial vehicle system control methods of external disturbance.

Background technology

The one kind of four rotor wing unmanned aerial vehicles as rotor craft, energy is controlled by the flight for controlling four rotor rotating speeds to realize Enough be conveniently accomplished take off with landing etc. action, be widely used in aeroplane photography, geologic prospect, rescue and relief work, environmental assessment Deng field.Due to four rotor wing unmanned aerial vehicle small volumes and in light weight, be in-flight vulnerable to external disturbance, how to realize to four rotors without Man-machine High Performance Motion Control has become a hot issue.For the control problem of four rotor wing unmanned aerial vehicles, exist a lot Control method, such as PID control, Active Disturbance Rejection Control, sliding formwork control etc..

Wherein sliding formwork control has been widely used for nonlinear system, and its advantage includes fast response time, easy to implement, right Uncertain robustness with external disturbance of system etc..Traditional sliding formwork control is contrasted, TSM control can realize finite time Convergence, but there is singular point in system, and its discontinuous switching characteristic in itself will cause the buffeting of system, to system in reality Application in the situation of border has very big obstruction.To solve this problem, non-singular terminal sliding formwork control is suggested, and this method is in reality The singularity problem of system is efficiently solved in the situation of border, and ensure that system finite time convergence control characteristic and stronger robust Property.

To with the probabilistic four rotor wing unmanned aerial vehicles dynamic system of inertia, external disturbance and Parameter uncertainties be present Property.The problems such as aerodynamic interference that external disturbance is brought, gyroscopic couple interference, Parameter Perturbation, can influence the flight control of four rotor wing unmanned aerial vehicles The sensitivity of system and stability.Therefore, adaptive method can be applied to estimate interference and systematic parameter on the basis of sliding formwork control, Design estimation rule causes system to have more preferable steady-state behaviour.

The content of the invention

In order to overcome external disturbance and inertia uncertain problem and terminal sliding mode existing for four rotor wing unmanned aerial vehicle systems The deficiency of the singularity problem of control, the invention provides a kind of four rotor wing unmanned aerial vehicle non-singular terminals with Self Adaptive Control Sliding-mode control, system singularity problem is eliminated, effectively inhibit chattering phenomenon, while carry out to being disturbed existing for system Suppress and compensate, ensure the finite time convergence control characteristic of system.

In order to solve the above-mentioned technical problem the technical scheme proposed is as follows：

A kind of four rotor wing unmanned aerial vehicle finite time self-adaptation control methods based on non-singular terminal sliding formwork, including following step Suddenly：

Step 1, the kinetic model of four rotor wing unmanned aerial vehicles is established, initializes system mode, sampling time and systematic parameter, Process is as follows：

1.1 hypothesis unmanned planes are that rigidity is full symmetric, and its center overlaps with body axis system origin, to four rotors without Man-machine system carries out force analysis, establishes coordinate system, one is the inertial coodinate system based on the earth, by reference axis X_E、Y_E、Z_EReally Fixed, another is the body axis system based on four rotor wing unmanned aerial vehicles, by reference axis X_B、Y_B、Z_BIt is determined that from body axis system to used The transfer matrix M of property coordinate system is:

Wherein, ψ, θ, φ are referred to as yaw angle, the angle of pitch, the roll angle of unmanned plane, represent that inertial coodinate system is each around its The anglec of rotation of reference axis；

1.2 according to newton Euler's formula analytic dynamics model, has under translation state：

Wherein x, y, z represents position of four rotor wing unmanned aerial vehicles under inertial coodinate system respectively, and m is the quality of unmanned plane, U_F Lift caused by four rotors is represented, mg is the gravity suffered by unmanned plane, and g is acceleration of gravity, U_FActed on the expression of mg sums Bonding force F on unmanned plane；

Formula (1) is substituted into formula (2) to obtain：

1.3 under body axis system, according to Euler's formula, has under rotary state：

Wherein τ_x、τ_y、τ_zEach axle power square of body axis system, I are represented respectively_xx、I_yy、I_zzEach axle of body axis system is represented respectively Rotary inertia, ω_x、ω_y、ω_zEach axle attitude angular velocity on body axis system is represented respectively, Machine is represented respectively Each axle posture angular acceleration on body coordinate system；

Obtained by formula (4)：

Four rotor wing unmanned aerial vehicles are to realize flight control by adjusting the rotating speed of rotor, its control moment and rotor lift with The rotating speed of rotor has direct relation, as shown in formula (6)：

Wherein L represents the barycenter of four rotor wing unmanned aerial vehicles to the distance of each rotor axis, k_FRepresent lift coefficient, k_MRepresent to turn round Moment coefficient, ω₁、ω₂、ω₃、ω₄The rotating speed of each rotor is represented respectively；

1.4, which consider under actual environments that outer bound pair system produces, disturbs, and establishes the kinetic model of four rotor wing unmanned aerial vehicles, four rotors without Man-machine to be typically in low-speed operations or floating state, attitude angle change is smaller, it is believed that As shown in formula (7)：

Wherein

d_x、d_y、d_z、d_φ、d_θ、d_ψThe external disturbance of representative model；

Formula (7) belongs to second order MIMO nonlinear systems, for ease of the design of controller, formula (7) be expressed as Lower form：

Wherein state variable X=(x, y, x, φ, θ, ψ)^T,It is right Angle matrix B (X)=diag { 1,1,1, b₁,b₂,b₃, input U=(U_x,U_y,U_z,τ_x,τ_y,τ_z)^T, external disturbance D (t)=(d_x, d_y,d_z,d_φ,d_θ,d_ψ)^T, and meet following condition：

||D(t)||_∞≤ρ (9)

Wherein | | D (t) | |_∞For D (t) Infinite Norm, ρ is a constant more than zero；

Step 2, calculating control system tracking error, designs non-singular terminal sliding-mode surface, and process is as follows：

2.1, which define system tracking error, is：

E=X-X_d (10)

Wherein X_dFor desired signal, X can be led_d=(x_d,y_d,z_d,φ_d,θ_d,ψ_d)^T, x_d、y_d、z_d、φ_d、θ_d、ψ_dRespectively pair The position answered and the desired value of attitude angle；

The first differential and second-order differential of formula (10) represent as follows：

2.2, which define non-singular terminal sliding-mode surface, is：

Wherein S=(s₁,s₂,s₃,s₄,s₅,s₆)^T, β^-1=diag { β₁ ^-1, β₂ ^-1, β₃ ^-1, β₄ ^-1, β₅ ^-1, β₆ ^-1It is positive definite square Battle array,s_i、β_i、e_iRespectively represent corresponding to x, y, z, ψ, θ, φ sliding variable, constant value coefficient, error first derivative, i=1,2,3,4,5,6, sign () are sign function, and p, q are Positive odd number, and 1<p/q<2；

Step 3, the dynamic system based on four rotor wing unmanned aerial vehicles, according to non-singular terminal sliding formwork, controller, process are designed It is as follows：

3.1 are based on formula (8), and non-singular terminal sliding mode controller is designed to：

U=U_eq+U_re (14)

Wherein, constant η>0,

3.2 design liapunov functions

Derivation is carried out to formula (13) to obtain

Wherein

Formula (14)~(16) are substituted into formula (18) and obtained

Derivation is carried out to formula (17) to obtain

Obtained by formula (20)

Formula (9) is substituted into formula (21) to obtain

Wherein η>0, β is positive definite matrix, and p, q are positive odd number,It is then obviousTherefore, it is determined thatSystem is stable；

Step 4, optimal controller, do not known using interference and inertia present in self-adaptation control method processing system Property, process is as follows：

4.1 are assumed on external disturbance D (t) boundary again：

Wherein, c, k₁、k₂It is unknown border, is not easy to obtain due to probabilistic labyrinth in actual control system Take；

4.2 are revised as control law based on formula (14)~(16)：

U₁=U_eq1+U_re1 (24)

Wherein, It is γ estimation；It is c, k respectively₁、k₂, ρ estimation；

Design estimates that the more new law of parameter is respectively：

Wherein, p₀＞ 0, p₁＞ 0, p₂＞ 0, ε₀＞ 0, ε₁＞ 0, ε₂＞ 0 is design parameter；

4.3 design liapunov functions

Wherein

Derivation is carried out to formula (32), then substitutes into formula (18) and obtains

(24)~(26) are substituted into formula (33) to obtain

Formula (28) is substituted into formula (34) to obtain

Obtained by formula (35)

Formula (27), (29)~(31) are substituted into formula (36) and obtained；

Formula (23) is substituted into

To meetThenThen haveWherein δ=ε₀c²+ε₁k₁ ²+ε₂k₂ ², Expression takes The wherein minterm of element；V_saReduction drive the track of closed-loop system to beTherefore the track of closed-loop system is most It is defined as eventually

The present invention is based on non-singular terminal sliding formwork and Self Adaptive Control, designs the non-singular terminal of four rotor wing unmanned aerial vehicle systems Sliding Mode Adaptive Control method, the finite time convergence control characteristic of system is realized, eliminate the nonsingular of TSM control and ask Topic, while the buffeting for weakening system influences, and improves the robustness of system.

The present invention technical concept be：For the dynamic system of four rotor wing unmanned aerial vehicles, non-singular terminal sliding formwork control is utilized Method processed, in conjunction with Self Adaptive Control, it is adaptive to design a kind of four rotor wing unmanned aerial vehicle finite times based on non-singular terminal sliding formwork Answer control method.The design of non-singular terminal sliding formwork is to ensure the finite time convergence control characteristic of system, and avoids end Singularity problem existing for holding sliding formwork control, effectively weakens buffeting problem.In addition, Self Adaptive Control is for processing system Inertia uncertainty and external disturbance.The invention provides one kind can eliminate sliding-mode surface singularity problem, and can effectively press down The control method of inertia uncertainty and external disturbance existing for system and compensation system, ensure that the finite time convergence control of system is special Property.

Advantage of the present invention is：Solve singularity problem, weaken and buffet, inertia is uncertain existing for suppression and compensation system And external disturbance, realize finite time convergence control.

Brief description of the drawings

Fig. 1 is the position tracking effect diagram of the present invention.

Fig. 2 is the attitude angle tracking effect schematic diagram of the present invention.

The positioner that Fig. 3 is the present invention inputs schematic diagram.

The posture angle controller that Fig. 4 is the present invention inputs schematic diagram.

The Position disturbance boundary parameter that Fig. 5 is the present invention estimates schematic diagram.

The Position disturbance that Fig. 6 is the present invention estimates border schematic diagram with it.

The attitude angle that Fig. 7 is the present invention disturbs border parameter Estimation schematic diagram.

The attitude angle interference that Fig. 8 is the present invention estimates that schematic diagram is estimated on border with it.

Fig. 9 is the system inertia uncertainty estimation schematic diagram of the present invention.

Figure 10 is the control flow schematic diagram of the present invention.

Embodiment

The present invention will be further described below in conjunction with the accompanying drawings.

Reference picture 1- Figure 10, a kind of four rotor wing unmanned aerial vehicle finite time Self Adaptive Control sides based on non-singular terminal sliding formwork Method, comprise the following steps：

Formula (1) is substituted into formula (2) to obtain：

Obtained by formula (4)：

1.4 consider that outer bound pair system produces interference under actual environment, establish the kinetic model of four rotor wing unmanned aerial vehicles, four rotations Wing unmanned plane is typically in low-speed operations or floating state, and attitude angle change is smaller, it is believed that Such as following formula (7)：

Wherein

||D(t)||_∞≤ρ (9)

2.1, which define system tracking error, is：

E=X-X_d (10)

2.2, which define non-singular terminal sliding-mode surface, is：

U=U_eq+U_re (14)

Wherein, constant η>0,

3.2 design liapunov functions

Derivation is carried out to formula (13) to obtain

Wherein

Formula (14)~(16) are substituted into formula (18) and obtained

Derivation is carried out to formula (17) to obtain

Obtained by formula (20)

Formula (9) is substituted into formula (21) to obtain

4.1 are assumed on external disturbance D (t) boundary again：

4.2 are revised as control law based on formula (14)~(16)：

U₁=U_eq1+U_re1 (24)

Design estimates that the more new law of parameter is respectively：

4.3 design liapunov functions

Wherein

(24)~(26) are substituted into formula (33) to obtain

Formula (28) is substituted into formula (34) to obtain

Obtained by formula (35)

Formula (27), (29)~(31) are substituted into formula (36) and obtained；

Formula (23) is substituted into

Can be by U_x、U_y、U_zRegard the input of positioner as, by τ_x、τ_y、τ_zRegard the input of posture angle controller as, according to Formula (24)~(27) provide the design of controller.

Position desired value x_d、y_d、z_dAnd attitude angle desired value ψ_dDirectly given by reference locus, attitude angle desired value φ_x、 θ_dDecoupled by position and attitude relation, it is as follows：

Wherein arcsin () is arcsin function, and arctan () is arctan function.

The feasibility of extracting method in order to verify, The present invention gives emulation knot of the control method on MATLAB platforms Fruit：

Parameter is given below：M=0.625kg, g=10 in formula (3)；I in formula (5)_xx=2.3 × 10^-3kg·m², I_yy= 2.4×10^-3kg·m², I_zz=2.6 × 10^-3kg·m²；K in formula (6)_F=2.103 × 10^-6N/(rad·s^-2), k_M=2.091 ×10^-8N/(rad·s^-2), L=0.1275m；X in formula (10)_d=1, y_d=1, z_d=1, ψ_d=0.5；β in formula (13)_i=1 (i =1,2,3,4,5,6), p=5, q=3；C=1, k in formula (23)₁=0.1, k₂=0.1；η=0.1 in formula (26)；Formula (29)~ (31) p in₀=0.1, p₀=0.1, p₀=0.1, ε₀=0.5, ε₁=0.5, ε₂=0.5；Interference signal is given as intensity as 0.01 White Gaussian noise.

Because four rotor wing unmanned aerial vehicle systems have six-freedom degree, so we track the dynamic of three positions and three attitude angles Step response.In order to further weaken the chattering phenomenon of system, we are replaced with saturation function sat () in formula (24)~(26) Sign function sign ()：

Wherein take μ=0.1.

From Fig. 1 and 2 as can be seen that system has good arrival performance, and equilbrium position is reached in finite time.From Fig. 3 and 4 is it can clearly be seen that system substantially weakens chattering phenomenon.From Fig. 5~9 as can be seen that the estimation rule of system finally becomes In stabilization, estimation parameter finally converges to constant.

In summary, non-singular terminal sliding formwork finite time self-adaptation control method obviously can efficiently solve nonsingular ask Topic and the influence for weakening buffeting, ensure system in finite time convergence control, and there is good robustness.

Described above is the excellent effect of optimization that one embodiment that the present invention provides is shown, it is clear that the present invention is not only Above-described embodiment is limited to, without departing from essence spirit of the present invention and the premise without departing from scope involved by substantive content of the present invention Under it can be made it is a variety of deformation be carried out.

Claims

A kind of 1. four rotor wing unmanned aerial vehicle finite time self-adaptation control methods based on non-singular terminal sliding formwork, it is characterised in that： Comprise the following steps：

Step 1, the kinetic model of four rotor wing unmanned aerial vehicles, initialization system mode, sampling time and systematic parameter, process are established It is as follows：

1.1 hypothesis unmanned planes are that rigidity is full symmetric, and its center overlaps with body axis system origin, to four rotor wing unmanned aerial vehicles System carries out force analysis, establishes coordinate system, one is the inertial coodinate system based on the earth, by reference axis X_E、Y_E、Z_EIt is determined that separately One is the body axis system based on four rotor wing unmanned aerial vehicles, by reference axis X_B、Y_B、Z_BIt is determined that from body axis system to inertial coordinate The transfer matrix M of system is:

<mrow> <mi>M</mi> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>+</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>+</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>sin</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&psi;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>+</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mo>-</mo> <mi>cos</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>+</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mi>sin</mi> <mi>&theta;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&theta;</mi> <mi>sin</mi> <mi>&phi;</mi> </mrow> </mtd> <mtd> <mrow> <mi>cos</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow>

Wherein, ψ, θ, φ are referred to as yaw angle, the angle of pitch, the roll angle of unmanned plane, represent inertial coodinate system around its each coordinate The anglec of rotation of axle；

1.2 according to newton Euler's formula analytic dynamics model, has under translation state：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>-</mo> <mi>m</mi> <mi>g</mi> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mi>M</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>U</mi> <mi>F</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mi>m</mi> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <mover> <mi>x</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>y</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> </mtd> </mtr> <mtr> <mtd> <mover> <mi>z</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mrow>

Wherein x, y, z represents position of four rotor wing unmanned aerial vehicles under inertial coodinate system respectively, and m is the quality of unmanned plane, U_FRepresent four Lift caused by individual rotor, mg are the gravity suffered by unmanned plane, and g is acceleration of gravity, U_FRepresent to act on nobody with mg sums Bonding force F on machine；

Formula (1) is substituted into formula (2) to obtain：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&phi;</mi> <mo>+</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> <mo>)</mo> </mrow> <mfrac> <msub> <mi>U</mi> <mi>F</mi> </msub> <mi>m</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&phi;</mi> <mo>+</mo> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&psi;</mi> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> <mo>)</mo> </mrow> <mfrac> <msub> <mi>U</mi> <mi>F</mi> </msub> <mi>m</mi> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>z</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <mo>-</mo> <mi>g</mi> <mo>+</mo> <mrow> <mo>(</mo> <mi>cos</mi> <mi>&theta;</mi> <mi>cos</mi> <mi>&phi;</mi> <mo>)</mo> </mrow> <mfrac> <msub> <mi>U</mi> <mi>F</mi> </msub> <mi>m</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mrow>

1.3 under body axis system, according to Euler's formula, has under rotary state：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&tau;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&tau;</mi> <mi>z</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>+</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>z</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>&times;</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&omega;</mi> <mi>z</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mrow>

Wherein τ_x、τ_y、τ_zEach axle power square of body axis system, I are represented respectively_xx、I_yy、I_zzRepresent that each axle of body axis system rotates respectively Inertia, ω_x、ω_y、ω_zEach axle attitude angular velocity on body axis system is represented respectively, Represent that body is sat respectively Mark each axle posture angular acceleration fastened；

Obtained by formula (4)：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>x</mi> </msub> <mo>=</mo> <msub> <mi>&omega;</mi> <mi>y</mi> </msub> <msub> <mi>&omega;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mrow> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mfrac> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>y</mi> </msub> <mo>=</mo> <msub> <mi>&omega;</mi> <mi>x</mi> </msub> <msub> <mi>&omega;</mi> <mi>z</mi> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mfrac> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mover> <mi>&omega;</mi> <mo>&CenterDot;</mo> </mover> <mi>z</mi> </msub> <mo>=</mo> <msub> <mi>&omega;</mi> <mi>x</mi> </msub> <msub> <mi>&omega;</mi> <mi>y</mi> </msub> <mrow> <mo>(</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mfrac> <mo>)</mo> </mrow> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mfrac> <msub> <mi>&tau;</mi> <mi>z</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mrow>

Four rotor wing unmanned aerial vehicles are to realize flight control, its control moment and rotor lift and rotor by adjusting the rotating speed of rotor Rotating speed have direct relation, as shown in formula (6)：

<mrow> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>U</mi> <mi>F</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&tau;</mi> <mi>y</mi> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>&tau;</mi> <mi>z</mi> </msub> </mtd> </mtr> </mtable> </mfenced> <mo>=</mo> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msub> <mi>k</mi> <mi>F</mi> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mi>F</mi> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mi>F</mi> </msub> </mtd> <mtd> <msub> <mi>k</mi> <mi>F</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>k</mi> <mi>F</mi> </msub> <mi>L</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mi>F</mi> </msub> <mi>L</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mi>F</mi> </msub> <mi>L</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mrow> <msub> <mi>k</mi> <mi>F</mi> </msub> <mi>L</mi> </mrow> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>k</mi> <mi>M</mi> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mi>M</mi> </msub> </mrow> </mtd> <mtd> <msub> <mi>k</mi> <mi>M</mi> </msub> </mtd> <mtd> <mrow> <mo>-</mo> <msub> <mi>k</mi> <mi>M</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mfenced open = "[" close = "]"> <mtable> <mtr> <mtd> <msubsup> <mi>&omega;</mi> <mn>1</mn> <mn>2</mn> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&omega;</mi> <mn>2</mn> <mn>2</mn> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&omega;</mi> <mn>3</mn> <mn>2</mn> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>&omega;</mi> <mn>4</mn> <mn>2</mn> </msubsup> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mrow>

Wherein L represents the barycenter of four rotor wing unmanned aerial vehicles to the distance of each rotor axis, k_FRepresent lift coefficient, k_MRepresent moment of torsion system Number, ω₁、ω₂、ω₃、ω₄The rotating speed of each rotor is represented respectively；

1.4 consider that outer bound pair system produces interference under actual environment, establishes the kinetic model of four rotor wing unmanned aerial vehicles, four rotor wing unmanned aerial vehicles Low-speed operations or floating state are typically in, attitude angle change is smaller, it is believed that Such as following formula (7)：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>x</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>U</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>d</mi> <mi>x</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>y</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>U</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>d</mi> <mi>y</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>z</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>U</mi> <mi>z</mi> </msub> <mo>+</mo> <msub> <mi>d</mi> <mi>z</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&phi;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>a</mi> <mn>1</mn> </msub> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mover> <mi>&psi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>b</mi> <mn>1</mn> </msub> <msub> <mi>&tau;</mi> <mi>x</mi> </msub> <mo>+</mo> <msub> <mi>d</mi> <mi>&phi;</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&theta;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mover> <mi>&psi;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <msub> <mi>&tau;</mi> <mi>y</mi> </msub> <mo>+</mo> <msub> <mi>d</mi> <mi>&theta;</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mover> <mi>&psi;</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>a</mi> <mn>3</mn> </msub> <mover> <mi>&phi;</mi> <mo>&CenterDot;</mo> </mover> <mover> <mi>&theta;</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msub> <mi>b</mi> <mn>3</mn> </msub> <msub> <mi>&tau;</mi> <mi>z</mi> </msub> <mo>+</mo> <msub> <mi>d</mi> <mi>&psi;</mi> </msub> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mrow>

Wherein

<mrow> <msub> <mi>U</mi> <mi>y</mi> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mo>-</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&phi;</mi> <mo>+</mo> <mi>sin</mi> <mi>&psi;</mi> <mi>sin</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> <mo>)</mo> </mrow> <mfrac> <msub> <mi>U</mi> <mi>F</mi> </msub> <mi>m</mi> </mfrac> <mo>,</mo> </mrow>

<mrow> <msub> <mi>U</mi> <mi>z</mi> </msub> <mo>=</mo> <mo>-</mo> <mi>g</mi> <mo>+</mo> <mrow> <mo>(</mo> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&theta;</mi> <mi>c</mi> <mi>o</mi> <mi>s</mi> <mi>&phi;</mi> <mo>)</mo> </mrow> <mfrac> <msub> <mi>U</mi> <mi>F</mi> </msub> <mi>m</mi> </mfrac> <mo>,</mo> </mrow>

<mrow> <msub> <mi>a</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mrow> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mfrac> <mo>,</mo> <msub> <mi>a</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mrow> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mfrac> <mo>,</mo> <msub> <mi>a</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mrow> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mfrac> <mo>,</mo> </mrow>

<mrow> <msub> <mi>b</mi> <mn>1</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mfrac> <mo>,</mo> <msub> <mi>b</mi> <mn>2</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>y</mi> <mi>y</mi> </mrow> </msub> </mfrac> <mo>,</mo> <msub> <mi>b</mi> <mn>3</mn> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <msub> <mi>I</mi> <mrow> <mi>z</mi> <mi>z</mi> </mrow> </msub> </mfrac> <mo>,</mo> </mrow>

d_x、d_y、d_z、d_φ、d_θ、d_ψThe external disturbance of representative model；

Formula (7) belongs to second order MIMO nonlinear systems, and for ease of the design of controller, formula (7) is expressed as shape Formula：

<mrow> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mi>U</mi> <mo>+</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi>Y</mi> <mo>=</mo> <mi>X</mi> </mrow> </mtd> </mtr> </mtable> </mfenced> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mrow>

Wherein state variable X=(x, y, x, φ, θ, ψ)^T, Diagonal matrix B (X)=diag { 1,1,1, b₁,b₂,b₃, input U=(U_x,U_y,U_z,τ_x,τ_y,τ_z)^T, external disturbance D (t)=(d_x, d_y,d_z,d_φ,d_θ,d_ψ)^T, and meet following condition：

||D(t)||_∞≤ρ (9)

Wherein | | D (t) | |_∞For D (t) Infinite Norm, ρ is a constant more than zero；

Step 2, calculating control system tracking error, designs non-singular terminal sliding-mode surface, and process is as follows：

2.1, which define system tracking error, is：

E=X-X_d (10)

Wherein X_dFor desired signal, X can be led_d=(x_d,y_d,z_d,φ_d,θ_d,ψ_d)^T, x_d、y_d、z_d、φ_d、θ_d、ψ_dIt is respectively corresponding Position and the desired value of attitude angle；

The first differential and second-order differential of formula (10) represent as follows：

<mrow> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mi>e</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mrow>

2.2, which define non-singular terminal sliding-mode surface, is：

<mrow> <mi>S</mi> <mo>=</mo> <mi>e</mi> <mo>+</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>sig</mi> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>,</mo> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mrow>

Wherein S=(s₁,s₂,s₃,s₄,s₅,s₆)^T, β^-1=diag { β₁ ^-1, β₂ ^-1, β₃ ^-1, β₄ ^-1, β₅ ^-1, β₆ ^-1It is positive definite matrix,s_i、β_i、e_iRespectively represent corresponding to x, y, z, ψ, θ, φ sliding variable, constant value coefficient, error first derivative, i=1,2,3,4,5,6, sign () are sign function, and p, q is just Odd number, and 1<p/q<2；

Step 3, the dynamic system based on four rotor wing unmanned aerial vehicles, according to non-singular terminal sliding formwork, controller is designed, process is such as Under：

3.1 are based on formula (8), and non-singular terminal sliding mode controller is designed to：

U=U_eq+U_re (14)

<mrow> <msub> <mi>U</mi> <mrow> <mi>e</mi> <mi>q</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>&beta;</mi> <mfrac> <mi>q</mi> <mi>p</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mn>2</mn> <mo>-</mo> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>U</mi> <mrow> <mi>r</mi> <mi>e</mi> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mfrac> <mn>1</mn> <mrow> <mi>B</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> </mrow> </mfrac> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <mi>&rho;</mi> <mo>+</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mrow>

Wherein, constant η>0,3.2 design Lee Ya Punuofu functions

<mrow> <mi>V</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msup> <mi>S</mi> <mi>T</mi> </msup> <mi>S</mi> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mrow>

Derivation is carried out to formula (13) to obtain

<mrow> <mover> <mi>S</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mi>U</mi> <mo>+</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mrow>

Wherein

Formula (14)~(16) are substituted into formula (18) and obtained

<mrow> <mover> <mi>S</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mi>&rho;</mi> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mrow>

Derivation is carried out to formula (17) to obtain

<mrow> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <mi>S</mi> <mo>=</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mi>&rho;</mi> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mrow>

Obtained by formula (20)

<mrow> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>&le;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>&lsqb;</mo> <mo>|</mo> <mo>|</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mi>&rho;</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mrow>

Formula (9) is substituted into formula (21) to obtain

<mrow> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mo>&le;</mo> <mo>-</mo> <mi>&eta;</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mrow>

Wherein η>0, β is positive definite matrix, and p, q are positive odd number,It is then obviousTherefore, it is determined thatSystem is stable；

Step 4, optimal controller, mistake uncertain using interference present in self-adaptation control method processing system and inertia Journey is as follows：

4.1 are assumed on external disturbance D (t) boundary again：

<mrow> <mo>|</mo> <mo>|</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>&le;</mo> <mi>c</mi> <mo>+</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>|</mo> <mo>|</mo> <mo>+</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>|</mo> <mo>|</mo> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mo>|</mo> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mrow>

Wherein, c, k₁、k₂It is unknown border, is not easy to obtain due to probabilistic labyrinth in actual control system；

4.2 are revised as control law based on formula (14)~(16)：

U₁=U_eq1+U_re1 (24)

<mrow> <msub> <mi>U</mi> <mrow> <mi>e</mi> <mi>q</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>&beta;</mi> <mfrac> <mi>q</mi> <mi>p</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mn>2</mn> <mo>-</mo> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>U</mi> <mrow> <mi>r</mi> <mi>e</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>-</mo> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mo>&lsqb;</mo> <mrow> <mo>(</mo> <mover> <mi>&rho;</mi> <mo>^</mo> </mover> <mo>+</mo> <mi>&eta;</mi> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mi>&rho;</mi> <mo>^</mo> </mover> <mo>=</mo> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>|</mo> <mo>|</mo> <mo>+</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>|</mo> <mo>|</mo> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mo>|</mo> <mo>|</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mrow>

Wherein, It is γ estimation；It is c, k respectively₁、k₂, ρ estimation；

Design estimates that the more new law of parameter is respectively：

<mrow> <mover> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <msup> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mn>3</mn> </msup> <mo>{</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>+</mo> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mover> <mi>&rho;</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>=</mo> <msub> <mi>p</mi> <mn>0</mn> </msub> <mo>{</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>{</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>|</mo> <mo>|</mo> <mi>X</mi> <mo>|</mo> <mo>|</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>=</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>{</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>|</mo> <mo>|</mo> <mover> <mi>X</mi> <mo>&CenterDot;</mo> </mover> <mo>|</mo> <mo>|</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>}</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mrow>

Wherein, p₀＞ 0, p₁＞ 0, p₂＞ 0, ε₀＞ 0, ε₁＞ 0, ε₂＞ 0 is design parameter；

4.3 design liapunov functions

<mrow> <msub> <mi>V</mi> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>&lsqb;</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <mi>S</mi> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>0</mn> </msub> </mfrac> <msup> <mover> <mi>c</mi> <mo>~</mo> </mover> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> <msup> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>2</mn> </msub> </mfrac> <msup> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msup> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <mn>2</mn> </msup> <mo>&rsqb;</mo> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mrow>

Wherein

Derivation is carried out to formula (32), then substitutes into formula (18) and obtains

<mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <mo>{</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>B</mi> <mrow> <mo>(</mo> <mi>X</mi> <mo>)</mo> </mrow> <mi>U</mi> <mo>+</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>&rsqb;</mo> <mo>}</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>0</mn> </msub> </mfrac> <mover> <mi>c</mi> <mo>~</mo> </mover> <mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mi>2</mi> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <msup> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mover> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mrow>

(24)~(26) are substituted into formula (33) to obtain

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <mo>{</mo> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msup> <mi>&gamma;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>+</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mi>f</mi> <mo>(</mo> <mi>X</mi> <mo>)</mo> <mo>-</mo> <msub> <mover> <mi>X</mi> <mo>&CenterDot;&CenterDot;</mo> </mover> <mi>d</mi> </msub> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>+</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mo>-</mo> <msup> <mi>&gamma;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mover> <mi>&rho;</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>+</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>0</mn> </msub> </mfrac> <mover> <mi>c</mi> <mo>~</mo> </mover> <mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mi>2</mi> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>+</mo> <mover> <mi>&gamma;</mi> <mo>~</mo> </mover> <msup> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mover> <mover> <mi>&gamma;</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mrow>

Formula (28) is substituted into formula (34) to obtain

<mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>S</mi> <mi>T</mi> </msup> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> <mo>&lsqb;</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>-</mo> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mover> <mi>&rho;</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mi>s</mi> <mi>i</mi> <mi>g</mi> <mi>n</mi> <mrow> <mo>(</mo> <mi>S</mi> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>0</mn> </msub> </mfrac> <mover> <mi>c</mi> <mo>~</mo> </mover> <mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>2</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mrow>

Obtained by formula (35)

<mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>&le;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>&lsqb;</mo> <mo>|</mo> <mo>|</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>-</mo> <mrow> <mo>(</mo> <mi>&eta;</mi> <mo>+</mo> <mover> <mi>&rho;</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>}</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>0</mn> </msub> </mfrac> <mover> <mi>c</mi> <mo>~</mo> </mover> <mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>2</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mrow>

Formula (27), (29)~(31) are substituted into formula (36) and obtained；

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>&le;</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>&lsqb;</mo> <mo>|</mo> <mo>|</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>-</mo> <mi>&eta;</mi> <mo>-</mo> <mrow> <mo>(</mo> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>|</mo> <mo>+</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>|</mo> <mo>|</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>|</mo> <mo>|</mo> <mo>)</mo> </mrow> <mo>&rsqb;</mo> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>0</mn> </msub> </mfrac> <mover> <mi>c</mi> <mo>~</mo> </mover> <mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>1</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <msub> <mi>p</mi> <mn>2</mn> </msub> </mfrac> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mover> <mi>k</mi> <mo>^</mo> </mover> <mo>&CenterDot;</mo> </mover> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>&lsqb;</mo> <mo>|</mo> <mo>|</mo> <mi>D</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>|</mo> <msub> <mo>|</mo> <mi>&infin;</mi> </msub> <mo>-</mo> <mi>&eta;</mi> <mo>-</mo> <mrow> <mo>(</mo> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>|</mo> <mo>+</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>|</mo> <mo>|</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>|</mo> <mo>|</mo> <mo>)</mo> </mrow> <mo>&rsqb;</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <mover> <mi>c</mi> <mo>~</mo> </mover> <mrow> <mo>(</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>0</mi> </msub> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>)</mo> </mrow> <mo>-</mo> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <mrow> <mo>(</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>|</mo> <mo>|</mo> <mi>x</mi> <mo>|</mo> <mo>|</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>1</mi> </msub> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>-</mo> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <mrow> <mo>(</mo> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <mrow> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>|</mo> <mo>|</mo> <mover> <mi>x</mi> <mo>&CenterDot;</mo> </mover> <mo>|</mo> <mo>|</mo> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mrow>

Formula (23) is substituted into

<mrow> <mtable> <mtr> <mtd> <mrow> <msub> <mover> <mi>V</mi> <mo>&CenterDot;</mo> </mover> <mrow> <mi>s</mi> <mi>a</mi> </mrow> </msub> <mo>&le;</mo> <mo>-</mo> <mi>&eta;</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mrow> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> </mrow> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <mover> <mi>c</mi> <mo>~</mo> </mover> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>1</mn> </msub> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <msub> <mover> <mi>k</mi> <mo>~</mo> </mover> <mn>2</mn> </msub> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>=</mo> <mo>-</mo> <mi>&eta;</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mrow> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> </mrow> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <msup> <mrow> <mo>(</mo> <mover> <mi>c</mi> <mo>^</mo> </mover> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>c</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>1</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <msup> <mrow> <mo>(</mo> <msub> <mover> <mi>k</mi> <mo>^</mo> </mover> <mn>2</mn> </msub> <mo>-</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> <msup> <msub> <mi>k</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <msup> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>&le;</mo> <mo>-</mo> <mi>&eta;</mi> <munderover> <mo>&Sigma;</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mn>6</mn> </munderover> <mo>{</mo> <msub> <mrow> <mo>(</mo> <msup> <mi>&beta;</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <msup> <mi>sig</mi> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>(</mo> <mover> <mi>e</mi> <mo>&CenterDot;</mo> </mover> <mo>)</mo> <mo>)</mo> </mrow> <mi>i</mi> </msub> <mo>|</mo> <msub> <mi>s</mi> <mi>i</mi> </msub> <mo>|</mo> <mo>}</mo> <mo>+</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mo>(</mo> <msub> <mi>&epsiv;</mi> <mn>0</mn> </msub> <msup> <mi>c</mi> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>1</mn> </msub> <msup> <msub> <mi>k</mi> <mn>1</mn> </msub> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>&epsiv;</mi> <mn>2</mn> </msub> <msup> <msub> <mi>k</mi> <mn>2</mn> </msub> <mn>2</mn> </msup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> <mo>-</mo> <mo>-</mo> <mo>-</mo> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mrow>

To meetThenThen haveWherein δ=ε₀c²+ε₁k₁ ²+ε₂k₂ ², Expression takes The wherein minterm of element；V_saReduction drive the track of closed-loop system to beTherefore the track of closed-loop system is most It is defined as eventually