CN106788086A - Consider the asynchronous machine command filtering finite time fuzzy control method of input saturation - Google Patents

Abstract

The invention discloses a kind of asynchronous machine command filtering finite time fuzzy control method for considering input saturation, the control method drives and nonlinear problem present in control system for motor, using the nonlinear function in fuzzy logic system approximation system, command filtering technology is introduced in traditional Backstepping design method, and by introducing compensation mechanism, reduce the error that filtering is produced, it is successfully overcome " calculating blast " problem in traditional Reverse Step Control caused by continuous derivation, and improve control accuracy.System tenacious tracking error is small under finite-time control, and dynamic response time is short, improves the convergence rate and interference rejection capability of system.The present invention is in the case where input saturation is considered, it is ensured that all of parameter of closed-loop system is all bounded, and can ensure that the tracking error of system can converge to one of origin fully small neighborhood within the limited time.

Description

Consider the asynchronous machine command filtering finite time fuzzy control method of input saturation

Technical field

The invention belongs to electric machine speed regulation control technology field, more particularly to a kind of asynchronous machine order for considering input saturation Filtering finite time fuzzy control method.

Background technology

In the technological innovation of industrial development, motor drives has extremely important status, because the type of drive in future The features such as must having stronger stability and more preferable control effect.Asynchronous machine is with its cheap, simple structure, high reliability And the advantage of durability, it is widely used in the industrial production.But the dynamic model of asynchronous machine is a high-order How the system of non-linear close coupling, realize it being a problem to effective control of asynchronous machine.To meet wanting for practical application Ask, it is proposed that the control strategy based on modern control theory such as fuzzy logic control, Backstepping control and sliding formwork control.

Backstepping is a kind of method for controlling uncertain, nonlinear system, in particular for control be unsatisfactory for The system of fixed condition.Backstepping is used in Induction Motor System, the high-order of asynchronous machine is made using virtual controlling variable Simple system, final output result can automatically be obtained by suitable Lyapunov equations.However, traditional contragradience Continuous derivation is carried out to virtual master function in control, easily causes " calculating blast " problem.

Finite time (Finite-time) is a kind of easy to use, effective control method.

From control system it is time-optimized from the perspective of, compole when being that makes the control method of closed-loop system finite time convergence control Excellent control method.Finite time stability refers to that the Phase Pathway of system is remained in a limited time interval In previously given boundary.Condition of the finite time stability to control system in itself is lower, with wider in current conditions It is general, more correspond to actual needs.In addition to the extremely excellent advantage of constringency performance, and Infinite Time control technology (exponential convergence is general Other asymptotic convergences) compare, when to being controlled with the system under uncertain parameter and external disturbance disturbed condition, have Systems stabilisation has faster convergence near origin between in limited time, and with more preferable robustness and anti-interference.

The content of the invention

It is an object of the invention to propose a kind of asynchronous machine command filtering finite time Fuzzy Control for considering input saturation Method processed, the method can overcome the influence of unknown parameters and load change, to realize significantly more efficient Position Tracking Control.

To achieve these goals, the present invention is adopted the following technical scheme that：

Consider the asynchronous machine command filtering finite time fuzzy control method of input saturation, comprise the following steps：

A. the dynamic mathematical models of asynchronous machine are set up

The dynamic mathematical models of asynchronous machine are represented by under synchronous rotary d-q coordinates：

Wherein,ω is angular speed, the L of asynchronous machine rotor_mIt is mutual inductance, n_pFor number of pole-pairs, J are used to rotate Amount, Θ are asynchronous machine rotor angle, T_LIt is load torque, ψ_dIt is rotor flux, i_qIt is q axles stator current, i_dIt is d axle stators Electric current, u_qIt is asynchronous machine q axles stator voltage, u_dIt is asynchronous machine d axles stator voltage, L_sIt is stator leakage inductance, R_sIt is asynchronous machine Stator equivalent resistance, L_rIt is rotor leakage inductance, R_rIt is asynchronous machine rotor equivalent resistance；

To simplify the dynamic mathematical models of asynchronous machine, new variable is defined：

Then the dynamic mathematical models of asynchronous machine are represented by：

B. according to command filtering technology and finite time principle, a kind of asynchronous machine order filter for considering input saturation is designed Ripple finite time fuzzy control method；Assuming that f (Z) is compacting Ω_ZIn be a continuous function, for arbitrary constant ε ＞ 0, always there is a fuzzy logic system Φ^TP (Z) meets：

In formula, input vectorQ is Indistinct Input dimension, R^qIt is real number vector set；

Φ=[Φ₁,Φ₂,...,Φ_l]^T∈R^lIt is fuzzy weight vector, obscures nodes l ＞ 1, R^lIt is real number vector set；

P (Z)=[p₁(Z),p₂(Z),...,p_l(Z)]^T∈R^lIt is basis function vector；

Generally choose basic function p_w(Z) it is following Gaussian function：

Wherein, μ_w=[μ_w1,...,μ_wq]^TIt is the center of Gaussian function distribution curves, and η_wIt is then its width；

Defining finite time command filtering device is：

Wherein,It is the output signal of command filtering device, α_uIt is the input of command filtering device Signal, v_uIt is the tracking error signal after compensation, u=1,2,3, constant R₁＞ 0, constant R₂＞ 0；If command filtering device is defeated Enter signal alpha_uFor all of t >=0 so thatAndSet up, wherein, ρ₁And ρ₂It is normal number；SimultaneouslyCan then draw, to arbitrary constant κ ＞ 0 so thatWithIt is bounded；So in a limited time for v₁WithTo have and be set up with lower inequality：

Wherein, constant0 is all higher than, and depending on the design parameter of second order differential equation, constant、It is all higher than 0；

u_qAnd u_dThe asymmetric saturation nonlinearity input of Induction Motor-Driven system is represented, due to u_qAnd u_dFundamental characteristics It is identical, define u acute pyogenic infection of finger tip u for statement is convenient_qAnd u_d；According to the characteristic of u, u can be described as：

In formula, sat (w) represents input saturation function, u_minAnd u_maxRespectively the minimum value of stator input voltages known to table and Maximum, u_max＞ 0 and u_min＜ 0 is unknown input saturation constant, and w is the input signal of saturation nonlinearity；

Using g (w) come approximate saturation function, it is defined as：

Sat (w) is expressed as u=sat (w)=g (w)+d (w)；

Wherein, d (w) is a bounded function,

Defining system tracking error variable is：

Define x_1dIt is desired rate signal, x_4dIt is desired rotor flux signal；Dummy pilot signal α₁,α₂,α₃For The input signal of command filtering device；x_1,c,x_2,c,x_3,cIt is the output signal of command filtering device；

C. the tracking error signal after definition command filtering compensation is：v₁=z₁-ξ₁；Choose Liapunov control function It is as follows：To V₁Derivation can be obtained：

Choose dummy pilot signal：

Wherein, constant k₁＞ 0, constant s₁＞ 0, the span of constant γ is：0 ＜ γ ＜ 1；

Definition compensation error

Wherein, constant l₁＞ 0；

Formula (4) is rewritten as according to formula (5), formula (6)：

D. nonlinear terms are occurred in that after step c, using finite-time control method, and is approached by fuzzy logic system Nonlinear function, it comprises the following steps：

D.1 according to the differential equationTo z₂Derivation obtains error dynamics equation：Definition Command filtering compensation after tracking error signal be：Lyapunov functions are chosen simultaneously：To V₁ Derivation is obtained：

Can not possibly be infinitely great due to being loaded in real system, it is assumed that 0≤T_L≤ d, constant d ＞ 0；

Can be obtained according to Young inequality：Wherein, ε₁It is arbitrarily small normal number；Then：

Wherein,

Obtained by almighty approaching theorem, for arbitrarily small positive number ε₂>=0 has fuzzy logic systemSo thatWherein δ₂(Z₂) approximate error is represented, and meet inequality | δ₂(Z₂)|≤ε₂, so as to have：

Wherein, | | Φ₂| | it is Φ₂Norm, constant h₂＞ 0；

Choose dummy pilot signal：

Wherein, constant k₂＞ 0, constant s₂＞ 0；WithIt is respectively unknown constant θ₁With the estimate of J；

Definition compensation error

Wherein, constant l₂＞ 0；

According to formula (10), formula (11) and formula (12), formula (9) is rewritten as：

D.2 according to the differential equationTo z₂Derivation obtains error dynamics side Journey：Tracking error signal after definition command filtering compensation is：v₃=z₃, simultaneous selection Lyapunov functions：To V₃Derivation is obtained：

Wherein,

Equally, by almighty approaching theorem, for arbitrarily small positive number ε₃, there is fuzzy logic system in ＞ 0Make Wherein, δ₃(Z₃) approximate error is represented, and meet | δ₃(Z₃)|≤ε₃, so as to have：

Wherein, | | Φ₃| | it is Φ₃Norm, constant h₃＞ 0；

Build true control rate u_qFor：

Wherein, constant k₃＞ 0, constant s₃＞ 0；It is unknown constant θ₂Estimate；

Formula (15) and formula (16) are substituted into formula (14) to obtain：

D.3 according to the differential equationTo z₄Derivation can obtain error dynamics equation：Definition life The tracking error signal after filtering compensation is made to be：v₄=z₄-ξ₄, simultaneous selection Lyapunov functions：To V₄Ask Leading to obtain：

Choose dummy pilot signal α₃For：

Wherein, constant k₄＞ 0, constant s₄＞ 0；

Definition compensation error

Wherein, constant l₄＞ 0；

Formula (19) and formula (20) are substituted into formula (18), can be obtained：

D.4 according to the differential equationTo z₅Derivation can obtain error dynamics side Journey：Tracking error signal after definition command filtering compensation is：v₅=z₅, simultaneous selection Lyapunov functions：To V₅Derivation can be obtained：

Wherein,Z₅=[x₂,x₃,x₄,x₅]^T；

For smooth function f₅(Z₅), equally, by almighty approaching theorem, for arbitrarily small positive number ε₅＞ 0, is present fuzzy Flogic systemSo thatWherein, δ₅(Z₅) approximate error is represented, and meet | δ₅ (Z₅)|≤ε₅；So as to have：

Wherein, | | Φ₅| | it is Φ₅Norm, constant h₅＞ 0；

Build true control rate u_dFor：

Wherein, constant k₅＞ 0, constant s₅＞ 0；

Formula (22) is rewritten as according to formula (23) and formula (24)：

E. the asynchronous machine command filtering finite time fuzzy controller of the consideration input saturation to setting up carries out stability Analysis

Definition It is θ₁Estimate, wherein,

Definition It is θ₂Estimate, wherein, θ₂=max (| | Φ₃||²,||Φ₅||²)；

Definition It is the estimate of J；Formula (25) is rewritten：

Choose liapunov functionThen

Formula (26) is substituted into：

Select corresponding adaptive law：

Wherein, constant r₁＞ 0, constant m₁＞ 0, constant r₂＞ 0, constant m₂＞ 0, constant r₃＞ 0, constant m₃＞ 0；

According to formula (28), formula (29) and formula (30), formula (27) is rewritten as：

Meanwhile, by Young inequalityCan obtain：

Obtained by Young inequality again：

Wherein, e=1,2,4；

With reference to formula (32), formula (33), formula (34) and formula (35),It is represented by：

IfIt is available：

IfIt is available：

Therefore, with reference to inequality (35), (36), obtain：

Wherein, g=1,2；

Similarly,

Therefore, by above-mentioned inequality, can be written as：

In formula,

Using finite time by v_eA minizone is constrained in, because of z_e=v_e+ξ_e, e=1,2,4；ξ need to be proved_eIt is available to have Limit time-constrain is so as to obtain tracking signal z_eFinite time can also be used to constrain in the small neighbourhood of origin；

Choose compensation system liapunov function be：It is available：

And becauseCan obtain：

Wherein, k₀=2min (k₁,k₂,k₄),Select suitable l_e,Realized with ρMake ξ_eThe bounded in finite time, e=1,2,4.

The invention has the advantages that：

(1) present invention is directed to nonlinear problem present in motor driving and control system, in traditional Backstepping design side Command filtering technology is introduced in method, by introducing compensation mechanism, the error that filtering is produced is reduced；The present invention utilizes fuzzy logic Nonlinear function in system approximation system, command filtering contragradience technology is combined with fuzzy self-adaption method and constructs mould Self-adaptive fuzzy controller；The present invention makes the Phase Pathway of system remain at previously given boundary using the method for finite time In limit；The present invention is in the case where input saturation is considered, it is ensured that the tracking error of system can be received within the limited time In holding back one of origin fully small neighborhood；The controller simple structure for designing of the invention is easy, it is convenient to realize, design conjunction Reason, has faster response speed, stronger antijamming capability and more preferable tracking effect compared with traditional controller.

(2) input signal that the present invention needs be it is readily available in Practical Project can rotating speed measured directly and electric current letter Number amount, fuzzy finite time algorithm can be realized by software programming in itself, and can save the parameter of asynchronous machine is set Put, it is easy to which asynchronous machine is directly controlled, it is reduces cost, safe and reliable, have broad application prospects.

(3) present invention need not change the parameter of controller according to the difference of asynchronous machine, and it is right to be realized in principle The stable speed regulating control of the asynchronous machine of all models and power, reduces the measurement to non-synchronous motor parameter in control process, Beneficial to the quick response for realizing asynchronous motor speed regulation.

Brief description of the drawings

Fig. 1 is by considering asynchronous machine command filtering finite time fuzzy controller, the coordinate of input saturation in the present invention The schematic diagram of the composite controlled object of conversion and SVPWM inverters, rotation speed detection unit and current detecting unit composition；

Fig. 2 is to turn after the asynchronous machine command filtering finite time fuzzy Control that input saturation is considered in the present invention The tracking analogous diagram of subangle and rotor angle setting value；

Fig. 3 is to turn after the asynchronous machine command filtering finite time fuzzy Control that input saturation is considered in the present invention The tracking error analogous diagram of subangle and rotor angle setting value；

Fig. 4 is different after the asynchronous machine command filtering finite time fuzzy Control that input saturation is considered in the present invention Asynchronous motor q axle stator voltage analogous diagrams after step motor q axles stator voltage and consideration input saturation；

Fig. 5 is different after the asynchronous machine command filtering finite time fuzzy Control that input saturation is considered in the present invention Asynchronous motor d axle stator voltage analogous diagrams after step motor d axles stator voltage and consideration input saturation.

Specific embodiment

Below in conjunction with the accompanying drawings and specific embodiment is described in further detail to the present invention：

With reference to shown in Fig. 1, it is considered to be input into the asynchronous machine command filtering finite time fuzzy control method of saturation, it is related to Part mainly include consider input saturation asynchronous machine command filtering finite time fuzzy controller 1, coordinate transformation unit 2nd, SVPWM inverters 3 and rotation speed detection unit 4 and current detecting unit 5.

Wherein, rotation speed detection unit 4 and current detecting unit 5 mainly for detection of asynchronous motor current value and rotating speed Correlated variables, is used as input, by the asynchronous machine order for considering input saturation by the electric current and speed variable of actual measurement Filtering finite time fuzzy controller 1 carries out voltage control, is ultimately converted to the rotating speed of three-phase electric control asynchronous motor, in order to One significantly more efficient controller of design, the dynamic mathematical models for setting up asynchronous machine are very necessary.

Consider the asynchronous machine command filtering finite time fuzzy control method of input saturation, comprise the following steps：

A. the dynamic mathematical models of asynchronous machine are set up

The dynamic mathematical models of asynchronous machine are represented by under synchronous rotary d-q coordinates：

Wherein,ω is angular speed, the L of asynchronous machine rotor_mIt is mutual inductance, n_pFor number of pole-pairs, J are used to rotate Amount, Θ are asynchronous machine rotor angle, T_LIt is load torque, ψ_dIt is rotor flux, i_qIt is q axles stator current, i_dIt is d axle stators Electric current, u_qIt is asynchronous machine q axles stator voltage, u_dIt is asynchronous machine d axles stator voltage, L_sIt is stator leakage inductance, R_sIt is asynchronous machine Stator equivalent resistance, L_rIt is rotor leakage inductance, R_rIt is asynchronous machine rotor equivalent resistance；

To simplify the dynamic mathematical models of asynchronous machine, new variable is defined：

Then the dynamic mathematical models of asynchronous machine are represented by：

B. according to command filtering technology and finite time principle, a kind of asynchronous machine order filter for considering input saturation is designed Ripple finite time fuzzy control method；Assuming that f (Z) is compacting Ω_ZIn be a continuous function, for arbitrary constant ε ＞ 0, always there is a fuzzy logic system Φ^TP (Z) meets：

In formula, input vectorQ is Indistinct Input dimension, R^qIt is real number vector set；

Φ=[Φ₁,Φ₂,...,Φ_l]^T∈R^lIt is fuzzy weight vector, obscures nodes l ＞ 1, R^lIt is real number vector set；

P (Z)=[p₁(Z),p₂(Z),...,p_l(Z)]^T∈R^lIt is basis function vector；

Generally choose basic function p_w(Z) it is following Gaussian function：

Wherein, μ_w=[μ_w1,...,μ_wq]^TIt is the center of Gaussian function distribution curves, and η_wIt is then its width；

Defining finite time command filtering device is：

Wherein,It is the output signal of command filtering device, α_uIt is the input of command filtering device Signal, v_uIt is the tracking error signal after compensation, u=1,2,3, constant R₁＞ 0, constant R₂＞ 0；If command filtering device is defeated Enter signal alpha_uFor all of t >=0 so thatAndSet up, wherein, ρ₁And ρ₂It is normal number；SimultaneouslyCan then draw, to arbitrary constant κ ＞ 0 so thatWith It is bounded；So in a limited time for v₁WithTo have and be set up with lower inequality：

Wherein, constant0 is all higher than, and depending on the design parameter of second order differential equation, constant、It is all higher than 0；

u_qAnd u_dThe asymmetric saturation nonlinearity input of Induction Motor-Driven system is represented, due to u_qAnd u_dFundamental characteristics It is identical, define u acute pyogenic infection of finger tip u for statement is convenient_qAnd u_d；According to the characteristic of u, u can be described as：

In formula, sat (w) represents input saturation function, u_minAnd u_maxRespectively the minimum value of stator input voltages known to table and Maximum, u_max＞ 0 and u_min＜ 0 is unknown input saturation constant, and w is the input signal of saturation nonlinearity；

Using g (w) come approximate saturation function, it is defined as：

Sat (w) is expressed as u=sat (w)=g (w)+d (w)；

Wherein, d (w) is a bounded function,

Defining system tracking error variable is：

Define x_1dIt is desired rate signal, x_4dIt is desired rotor flux signal；Dummy pilot signal α₁,α₂,α₃For The input signal of command filtering device；x_1,c,x_2,c,x_3,cIt is the output signal of command filtering device；

C. the tracking error signal after definition command filtering compensation is：v₁=z₁-ξ₁；Choose Liapunov control function It is as follows：To V₁Derivation can be obtained：

Choose dummy pilot signal：

Wherein, constant k₁＞ 0, constant s₁＞ 0, the span of constant γ is：0 ＜ γ ＜ 1；

Definition compensation error

Wherein, constant l₁＞ 0；

Formula (4) is rewritten as according to formula (5), formula (6)：

D. nonlinear terms are occurred in that after step c, using finite-time control method, and is approached by fuzzy logic system Nonlinear function, it comprises the following steps：

D.1 according to the differential equationTo z₂Derivation obtains error dynamics equation：Definition Command filtering compensation after tracking error signal be：v₂=z₂-ξ₂, while choosing Lyapunov functions：To V₁Ask Lead：

Can not possibly be infinitely great due to being loaded in real system, it is assumed that 0≤T_L≤ d, constant d ＞ 0；

Can be obtained according to Young inequality：Wherein, ε₁It is arbitrarily small normal number；Then：

Wherein,

Obtained by almighty approaching theorem, for arbitrarily small positive number ε₂>=0 has fuzzy logic systemSo thatWherein δ₂(Z₂) approximate error is represented, and meet inequality | δ₂(Z₂)|≤ε₂, so as to have：

Wherein, | | Φ₂| | it is Φ₂Norm, constant h₂＞ 0；

Choose dummy pilot signal：

Wherein, constant k₂＞ 0, constant s₂＞ 0；WithIt is respectively unknown constant θ₁With the estimate of J；

Definition compensation error

Wherein, constant l₂＞ 0；

According to formula (10), formula (11) and formula (12), formula (9) is rewritten as：

D.2 according to the differential equationTo z₂Derivation obtains error dynamics side Journey：Tracking error signal after definition command filtering compensation is：v₃=z₃, simultaneous selection Lyapunov functions：To V₃Derivation is obtained：

Wherein,

Equally, by almighty approaching theorem, for arbitrarily small positive number ε₃, there is fuzzy logic system in ＞ 0Make Wherein, δ₃(Z₃) approximate error is represented, and meet | δ₃(Z₃)|≤ε₃, so as to have：

Wherein, | | Φ₃| | it is Φ₃Norm, constant h₃＞ 0；

Build true control rate u_qFor：

Wherein, constant k₃＞ 0, constant s₃＞ 0；It is unknown constant θ₂Estimate；

Formula (15) and formula (16) are substituted into formula (14) to obtain：

D.3 according to the differential equationTo z₄Derivation can obtain error dynamics equation：Definition life The tracking error signal after filtering compensation is made to be：v₄=z₄-ξ₄, simultaneous selection Lyapunov functions：To V₄Ask Leading to obtain：

Choose dummy pilot signal α₃For：

Wherein, constant k₄＞ 0, constant s₄＞ 0；

Definition compensation error

Wherein, constant l₄＞ 0；

Formula (19) and formula (20) are substituted into formula (18), can be obtained：

D.4 according to the differential equationTo z₅Derivation can obtain error dynamics side Journey：Tracking error signal after definition command filtering compensation is：v₅=z₅, simultaneous selection Lyapunov functions：To V₅Derivation can be obtained：

Wherein,Z₅=[x₂,x₃,x₄,x₅]^T；

For smooth function f₅(Z₅), equally, by almighty approaching theorem, for arbitrarily small positive number ε₅＞ 0, is present fuzzy Flogic systemSo thatWherein, δ₅(Z₅) approximate error is represented, and meet | δ₅ (Z₅)|≤ε₅；So as to have：

Wherein, | | Φ₅| | it is Φ₅Norm, constant h₅＞ 0；

Build true control rate u_dFor：

Wherein, constant k₅＞ 0, constant s₅＞ 0；

Formula (22) is rewritten as according to formula (23) and formula (24)：

E. the asynchronous machine command filtering finite time fuzzy controller of the consideration input saturation to setting up carries out stability Analysis

Definition It is θ₁Estimate, wherein, θ₁=max (| | Φ₂||²)；

Definition It is θ₂Estimate, wherein, θ₂=max (| | Φ₃||²,||Φ₅||²)；

Definition It is the estimate of J；Formula (25) is rewritten：

Choose liapunov functionThen

Formula (26) is substituted into：

Select corresponding adaptive law：

Wherein, constant r₁＞ 0, constant m₁＞ 0, constant r₂＞ 0, constant m₂＞ 0, constant r₃＞ 0, constant m₃＞ 0；

According to formula (28), formula (29) and formula (30), formula (27) is rewritten as：

Meanwhile, by Young inequalityCan obtain：

Obtained by Young inequality again：

Wherein, e=1,2,4；

With reference to formula (32), formula (33), formula (34) and formula (35),It is represented by：

IfIt is available：

IfIt is available：

Therefore, with reference to inequality (35), (36), obtain：

Wherein, g=1,2；

Similarly,

Therefore, by above-mentioned inequality, can be written as：

In formula,

Using finite time by v_eA minizone is constrained in, because of z_e=v_e+ξ_e, e=1,2,4；ξ need to be proved_eIt is available to have Limit time-constrain is so as to obtain tracking signal z_eFinite time can also be used to constrain in the small neighbourhood of origin；

Choose compensation system liapunov function be：It is available：

And becauseCan obtain：

Wherein, k₀=2min (k₁,k₂,k₄),Select suitable l_e,Realized with ρMake ξ_eThe bounded in finite time, e=1,2,4.

Imitated considering the asynchronous machine command filtering finite time fuzzy controller of input saturation under virtual environment Very, the feasibility of the asynchronous motor command filtering finite time fuzzy control method of proposed consideration input saturation is verified.

Motor and load parameter are：

J=0.0586kgm²,R_s=0.1 Ω, R_r=0.15 Ω, L_m=0.068H,

L_s=0.0699H, L_r=0.0699H, n_p=1, T_L=0.4.

Selection control rate parameter be：

k₁=k₂=k₃=k₄=k₅=37；s₁=s₂=s₃=s₄=s₅=0.1；

R₁=400, R₂=0.4；l₁=l₂=l₄=0.05；r₁=r₂=r₃=0.07；

h₂=h₃=h₅=100；m₁=m₂=m₃=0.08.

Input saturation constant：

Selection tracks signal：x_1d=sin (t), expects that rotor flux signal is：x_4d=1.

Load torque is：

Selection fuzzy membership function be：

Emulation is carried out on the premise of systematic parameter and nonlinear function are unknown, corresponding simulation result such as Fig. 2, figure 3rd, shown in Fig. 4 and Fig. 5.Wherein, Fig. 2 and Fig. 3 are respectively and consider that the asynchronous machine command filtering finite time of input saturation is obscured Rotor angle and rotor angle setting value tracking and rotor angle and rotor angle setting value tracking error after controller control Analogous diagram, shows that effect is preferable by simulation result, and tracking effect is preferable, fast response time.It is defeated that Fig. 4 and Fig. 5 are respectively consideration Enter after the asynchronous machine command filtering finite time fuzzy Control of saturation asynchronous motor q axles stator voltage and consider defeated Enter the voltage after saturation and asynchronous motor d axles stator voltage and consider the voltage analogous diagram after input saturation, by emulation Result shows that effect is preferable, fluctuate small, fast response time.The present invention overcomes Parameter uncertainties influence and reach Preferable control effect, realizes the quickly and stably response to rotating speed.

Certainly, described above is only presently preferred embodiments of the present invention, and the present invention is not limited to enumerate above-described embodiment, should When explanation, any those of ordinary skill in the art are all equivalent substitutes for being made, bright under the teaching of this specification Aobvious variant, all falls within the essential scope of this specification, ought to be subject to protection of the invention.

Claims (1)

1. the asynchronous machine command filtering finite time fuzzy control method of input saturation is considered, it is characterised in that including as follows Step：

A. the dynamic mathematical models of asynchronous machine are set up

The dynamic mathematical models of asynchronous machine are represented by under synchronous rotary d-q coordinates：

$\left\{\begin{array}{c}\frac{d\mathrm{\Θ}}{dt}=\mathrm{\ω}\\ \frac{d\mathrm{\ω}}{dt}=\frac{n_p{L}_m}{{\mathrm{JL}}_r}{i}_q{\mathrm{\ψ}}_d-\frac{T_L}{J}\\ \frac{{\mathrm{di}}_q}{dt}=-\frac{L_m^2{R}_r+L_r^2{R}_s}{{\mathrm{\σL}}_s{L}_r^2}{i}_q-\frac{L_m{n}_p}{{\mathrm{\σL}}_s{L}_r}{\mathrm{\ω\ψ}}_d-n_p{\mathrm{\ωi}}_d+\frac{L_m{R}_r}{L_r}\frac{i_q{i}_d}{{\mathrm{\ψ}}_d}+\frac{1}{{\mathrm{\σL}}_s}{u}_q\\ \frac{{\mathrm{d\ψ}}_d}{dt}=-\frac{R_r}{L_r}{\mathrm{\ψ}}_d+\frac{L_m{R}_r}{L_r}{i}_d\\ \frac{{\mathrm{di}}_d}{dt}=-\frac{L_m^2{R}_r+L_r^2{R}_s}{{\mathrm{\σL}}_s{L}_r^2}{i}_d-\frac{L_m{R}_r}{{\mathrm{\σL}}_s{L}_r^2}{\mathrm{\ψ}}_d+n_p{\mathrm{\ωi}}_q+\frac{L_m{R}_r}{L_r}\frac{i_q^i}{{\mathrm{\ψ}}_d}+\frac{1}{{\mathrm{\σL}}_s}{u}_d\end{array}\right.---\left(1\right)$

Wherein,ω is angular speed, the L of asynchronous machine rotor_mIt is mutual inductance, n_pFor number of pole-pairs, J are rotary inertia, Θ It is asynchronous machine rotor angle, T_LIt is load torque, ψ_dIt is rotor flux, i_qIt is q axles stator current, i_dIt is d axles stator current, u_q It is asynchronous machine q axles stator voltage, u_dIt is asynchronous machine d axles stator voltage, L_sIt is stator leakage inductance, R_sIt is asynchronous machine stator etc. Effect resistance, L_rIt is rotor leakage inductance, R_rIt is asynchronous machine rotor equivalent resistance；

To simplify the dynamic mathematical models of asynchronous machine, new variable is defined：

$\left\{\begin{array}{c}{x}_1=\mathrm{\Θ},x_2=\mathrm{\ω},x_3=i_q,x_4={\mathrm{\ψ}}_d,x_5=i_q\\ a_1=\frac{n_p{L}_m}{L_r}\\ b_1=-\frac{L_m^2{R}_r+L_r^2{R}_s}{{\mathrm{\σL}}_s{L}_r^2},b_2=-\frac{L_m{n}_p}{{\mathrm{\σL}}_s{L}_r},b_3=n_p,b_4=\frac{L_m{R}_r}{L_r},b_5=\frac{1}{{\mathrm{\σL}}_s}\\ c_1=-\frac{R_r}{L_r}\\ d_2=\frac{L_m{R}_r}{{\mathrm{\σL}}_s{L}_r^2}\end{array}\right.---\left(2\right)$

Then the dynamic mathematical models of asynchronous machine are represented by：

$\left\{\begin{array}{c}{\stackrel{\·}{x}}_1=x_2\\ {\stackrel{\·}{x}}_2=\frac{1}{J}{a}_1{x}_3{x}_4-\frac{T_L}{J}\\ {\stackrel{\·}{x}}_3=b_1{x}_3+b_2{x}_2{x}_4-b_3{x}_2{x}_5-b_4\frac{x_3{x}_5}{x_4}+b_5{u}_q\\ {\stackrel{\·}{x}}_4=c_1{x}_4+b_4{x}_5\\ {\stackrel{\·}{x}}_5=b_1{x}_5+d_2{x}_4+b_3{x}_2{x}_3+b_4\frac{x_3^2}{x_4}+b_5{u}_d\end{array}\right.---\left(3\right)$

B. according to command filtering technology and finite time principle, designing a kind of asynchronous machine command filtering for considering input saturation has Fuzzy control method between in limited time；Assuming that f (Z) is compacting Ω_ZIn be a continuous function, for arbitrary constant ε ＞ 0, always It is have a fuzzy logic system Φ^TP (Z) meets：

In formula, input vectorQ is Indistinct Input dimension, R^qIt is real number vector set；

Φ=[Φ₁,Φ₂,...,Φ_l]^T∈R^lIt is fuzzy weight vector, obscures nodes l ＞ 1, R^lIt is real number vector set；

P (Z)=[p₁(Z),p₂(Z),...,p_l(Z)]^T∈R^lIt is basis function vector；

Generally choose basic function p_w(Z) it is following Gaussian function：

$p_w\left(Z\right)=\exp \[\frac{-{(Z-{\mathrm{\μ}}_w)}^T(Z-{\mathrm{\μ}}_w)}{{\mathrm{\η}}_w^2}\],w=1,2,\mathrm{...},l;$

Wherein, μ_w=[μ_w1,...,μ_wq]^TIt is the center of Gaussian function distribution curves, and η_wIt is then its width；

Defining finite time command filtering device is：

Wherein,It is the output signal of command filtering device, α_uIt is the input signal of command filtering device, v_uIt is the tracking error signal after compensation, u=1,2,3, constant R₁＞ 0, constant R₂＞ 0；If the input signal of command filtering device α_uFor all of t >=0 so thatAndSet up, wherein, ρ₁And ρ₂It is normal number；SimultaneouslyCan then draw, to arbitrary constant κ ＞ 0 so that With It is bounded；So in a limited time for v₁WithTo have and be set up with lower inequality：

Wherein, constantMore than 0, and depending on the design parameter of second order differential equation, constantIt is all higher than 0；

u_qAnd u_dThe asymmetric saturation nonlinearity input of Induction Motor-Driven system is represented, due to u_qAnd u_dFundamental characteristics it is identical, U acute pyogenic infection of finger tip u is defined for statement is convenient_qAnd u_d；According to the characteristic of u, u can be described as：

$u=sat\left(w\right)=\left\{\begin{array}{c}{u}_{\max },w\&GreaterEqual;u_{\max }\\ w,u_{\min }<w<u_{\max }\\ u_{\min },w\&le;u_{\min }\end{array}\right.;$

In formula, sat (w) represents input saturation function, u_minAnd u_maxThe minimum value and maximum of stator input voltages known to difference table Value, u_max＞ 0 and u_min＜ 0 is unknown input saturation constant, and w is the input signal of saturation nonlinearity；

Using g (w) come approximate saturation function, it is defined as：

$g\left(w\right)=\left\{\begin{array}{c}{u}_{\max }*\frac{e^{w/u_{\max }}-e^{-w/u_{\max }}}{e^{w/u_{\max }}+e^{-w/u_{\max }}},w\&GreaterEqual;0\\ u_{\min }*\frac{e^{w/u_{\min }}-e^{-w/u_{\min }}}{e^{w/u_{\min }}+e^{-w/u_{\min }}},w\&le;0\end{array}\right.;$

Sat (w) is expressed as u=sat (w)=g (w)+d (w)；

Wherein, d (w) is a bounded function,

Defining system tracking error variable is：

Define x_1dIt is desired rate signal, x_4dIt is desired rotor flux signal；Dummy pilot signal α₁,α₂,α₃For order is filtered The input signal of ripple device；x_1,c,x_2,c,x_3,cIt is the output signal of command filtering device；

C. the tracking error signal after definition command filtering compensation is：v₁=z₁-ξ₁；Choose Liapunov control function as follows：To V₁Derivation can be obtained：

${\stackrel{\&CenterDot;}{V}}_1=v_1{\stackrel{\&CenterDot;}{v}}_1=v_1({\stackrel{\&CenterDot;}{z}}_1-{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_1)=v_1(z_2+x_{1,c}-{\mathrm{\&alpha;}}_1+{\mathrm{\&alpha;}}_1-{\stackrel{\&CenterDot;}{x}}_{1d}-{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_1)---\left(4\right)$

Choose dummy pilot signal：

Wherein, constant k₁＞ 0, constant s₁＞ 0, the span of constant γ is：0 ＜ γ ＜ 1；

Definition compensation error

Wherein, constant l₁＞ 0；

Formula (4) is rewritten as according to formula (5), formula (6)：

${\stackrel{\&CenterDot;}{V}}_1=-k_1{v_1}^2+v_1{v}_2-s_1{v}_1^{\mathrm{\&gamma;}+1}+v_1{l}_{1}sign\left({\mathrm{\&xi;}}_1\right)---\left(7\right)$

D. nonlinear terms are occurred in that after step c, using finite-time control method, and non-thread is approached by fuzzy logic system Property function, it comprises the following steps：

D.1 according to the differential equationTo z₂Derivation obtains error dynamics equation：Definition command Tracking error signal after filtering compensation is：v₂=z₂-ξ₂, while choosing Lyapunov functions：To V₁Derivation ：

Can not possibly be infinitely great due to being loaded in real system, it is assumed that 0≤T_L≤ d, constant d ＞ 0；

Can be obtained according to Young inequality：Wherein, ε₁It is arbitrarily small normal number；Then：

${\stackrel{\&CenterDot;}{V}}_2\&le;{\stackrel{\&CenterDot;}{V}}_1+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+v_2(x_3+f_2(Z_2)-J{\stackrel{\&CenterDot;}{x}}_{1,c}-J{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_2)---\left(9\right)$

Wherein,

Obtained by almighty approaching theorem, for arbitrarily small positive number ε₂>=0 has fuzzy logic systemSo thatWherein δ₂(Z₂) approximate error is represented, and meet inequality | δ₂(Z₂)|≤ε₂, so as to have：

$v_2{f}_2\left(Z_2\right)\&le;\frac{1}{2h_2^2}{v}_2^2\left|\right|{\mathrm{\&Phi;}}_2|{|}^2{P}_2^T\left(Z_2\right)P_2\left(Z_2\right)+\frac{1}{2}{h}_2^2+\frac{1}{2}{v}_2^2+\frac{1}{2}{\mathrm{\&epsiv;}}_2^2---\left(10\right)$

Wherein, | | Φ₂| | it is Φ₂Norm, constant h₂＞ 0；

Choose dummy pilot signal：

Wherein, constant k₂＞ 0, constant s₂＞ 0；WithIt is respectively unknown constant θ₁With the estimate of J；

Definition compensation error

Wherein, constant l₂＞ 0；

According to formula (10), formula (11) and formula (12), formula (9) is rewritten as：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_2\&le;\underset{i=1}{\overset{2}{\&Sigma;}}(-k_i{v}_i^2-s_i{v}_i^{\mathrm{\&gamma;}}+l_i{v}_{i}sign({\mathrm{\&xi;}}_i\left)\right)+\frac{1}{2h_2^2}{v}_2^2\left(\right|\left|{\mathrm{\&Phi;}}_2\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_1)P_2^T{P}_2\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+v_2{v}_3+v_2(\hat{J}-J){\stackrel{\&CenterDot;}{x}}_{1,c}\end{array}---\left(13\right)$

D.2 according to the differential equationTo z₂Derivation obtains error dynamics equation：Tracking error signal after definition command filtering compensation is：v₃=z₃, simultaneous selection Lyapunov functions：To V₃Derivation is obtained：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_3\&le;\underset{i=1}{\overset{2}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}}+l_i{v}_{i}sign({\mathrm{\&xi;}}_i\left)\right)+\frac{1}{2h_2^2}{v}_2^2\left(\right|\left|{\mathrm{\&Phi;}}_2\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_1)P_2^T{P}_2+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2\\ +\frac{1}{2}{h}_2^2+\frac{1}{2}{\mathrm{\&epsiv;}}_2^2+v_3\left(f_3\right(Z_3)+b_5{u}_q-{\stackrel{\&CenterDot;}{x}}_{2,c})+v_2(\hat{J}-J){\stackrel{\&CenterDot;}{x}}_{1,c}\end{array}---\left(14\right)$

Wherein,Z₃=[x₂,x₃,x₄,x₅]^T；

Equally, by almighty approaching theorem, for arbitrarily small positive number ε₃, there is fuzzy logic system in ＞ 0So thatWherein, δ₃(Z₃) approximate error is represented, and meet | δ₃(Z₃)|≤ε₃, so as to have：

$v_3{f}_3\left(Z_3\right)\&le;\frac{1}{2h_3^2}{v}_3^2\left|\right|{\mathrm{\&Phi;}}_3|{|}^2{P}_3^T\left(Z_3\right)P_3\left(Z_3\right)+\frac{1}{2}{h}_3^2+\frac{1}{2}{v}_3^2+\frac{1}{2}{\mathrm{\&epsiv;}}_3^2---\left(15\right)$

Wherein, | | Φ₃| | it is Φ₃Norm, constant h₃＞ 0；

Build true control rate u_qFor：

$u_q=\frac{1}{b_5}(-k_3{z}_3-\frac{1}{2}{v}_3-z_2-\frac{1}{2{h_3}^2}{v}_3{\widehat{\mathrm{\&theta;}}}_2{P}_3^T{P}_3+{\stackrel{\&CenterDot;}{x}}_{2,c}-s_3{v_3}^{\mathrm{\&gamma;}})---\left(16\right)$

Wherein, constant k₃＞ 0, constant s₃＞ 0；It is unknown constant θ₂Estimate；

Formula (15) and formula (16) are substituted into formula (14) to obtain：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_3\&le;\underset{i=1}{\overset{3}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+\frac{1}{2h_2^2}{v}_2^2\left(\right|\left|{\mathrm{\&Phi;}}_2\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_1)P_2^T{P}_2\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2h_3^2}{v}_3^2\left(\right|\left|{\mathrm{\&Phi;}}_3\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_2)P_3^T{P}_3+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+v_2(\hat{J}-J){\stackrel{\&CenterDot;}{x}}_{1,c}\end{array}---\left(17\right)$

D.3 according to the differential equationTo z₄Derivation can obtain error dynamics equation：Definition command is filtered Ripple compensation after tracking error signal be：v₄=z₄-ξ₄, simultaneous selection Lyapunov functions：To V₄Derivation can ：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_4={\stackrel{\&CenterDot;}{V}}_3+v_4{\stackrel{\&CenterDot;}{v}}_4={\stackrel{\&CenterDot;}{V}}_3+v_4(b_4{z}_5+b_4(x_{3,c}-{\mathrm{\&alpha;}}_3)+b_4{\mathrm{\&alpha;}}_3+c_1{x}_4-{\stackrel{\&CenterDot;}{x}}_{4d}-{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_4)\\ \&le;\underset{i=1}{\overset{3}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+\frac{1}{2h_2^2}{v}_2^2\left(\right|\left|{\mathrm{\&Phi;}}_2\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_1)P_2^T{P}_2\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2h_3^2}{v}_3^2\left(\right|\left|{\mathrm{\&Phi;}}_3\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_2)P_3^T{P}_3+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+v_2(\hat{J}-J){\stackrel{\&CenterDot;}{x}}_{1,c}\\ +v_4(b_4{z}_5+b_4(x_{3,c}-{\mathrm{\&alpha;}}_3)+b_4{\mathrm{\&alpha;}}_3+c_1{x}_4-{\stackrel{\&CenterDot;}{x}}_{4d}-{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_4)\end{array}---\left(18\right)$

Choose dummy pilot signal α₃For：

Wherein, constant k₄＞ 0, constant s₄＞ 0；

Definition compensation error

Wherein, constant l₄＞ 0；

Formula (19) and formula (20) are substituted into formula (18), can be obtained：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_4\&le;\underset{i=1}{\overset{4}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+b_4{v}_4{v}_5+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+v_4{l}_{4}sign\left({\mathrm{\&xi;}}_4\right)\\ +\frac{1}{2h_2^2}{v}_2^2\left(\right|\left|{\mathrm{\&Phi;}}_2\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_1)P_2^T{P}_2+\frac{1}{2h_3^2}{v}_3^2\left(\right|\left|{\mathrm{\&Phi;}}_3\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_2)P_3^T{P}_3\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+v_2(\hat{J}-J){\stackrel{\&CenterDot;}{x}}_{1,c}\end{array}---\left(21\right)$

D.4 according to the differential equationTo z₅Derivation can obtain error dynamics equation：Tracking error signal after definition command filtering compensation is：v₅=z₅, simultaneous selection Lyapunov functions：To V₅Derivation can be obtained：

${\stackrel{\&CenterDot;}{V}}_5={\stackrel{\&CenterDot;}{V}}_4+v_5{\stackrel{\&CenterDot;}{v}}_5={\stackrel{\&CenterDot;}{V}}_4+v_5\&lsqb;b_5{u}_d+f_5\left(Z_5\right)-{\stackrel{\&CenterDot;}{x}}_{3,c}\&rsqb;---\left(22\right)$

Wherein,Z₅=[x₂,x₃,x₄,x₅]^T；

For smooth function f₅(Z₅), equally, by almighty approaching theorem, for arbitrarily small positive number ε₅, there is fuzzy logic in ＞ 0 SystemSo thatWherein, δ₅(Z₅) approximate error is represented, and meet | δ₅(Z₅)| ≤ε₅；So as to have：

$v_5{f}_5\left(Z_5\right)\&le;\frac{1}{2h_5^2}{v}_5^2\left|\right|{\mathrm{\&Phi;}}_5|{|}^2{P}_5^T\left(Z_5\right)P_5\left(Z_5\right)+\frac{1}{2}{h}_5^2+\frac{1}{2}{v}_5^2+\frac{1}{2}{\mathrm{\&epsiv;}}_5^2---\left(23\right)$

Wherein, | | Φ₅| | it is Φ₅Norm, constant h₅＞ 0；

Build true control rate u_dFor：

$u_d=\frac{1}{b_5}(-k_5{z}_5-\frac{1}{2}{v}_5-b_4{z}_4-\frac{1}{2{h_5}^2}{v}_5{\widehat{\mathrm{\&theta;}}}_2{P}_5^T{P}_5+{\stackrel{\&CenterDot;}{x}}_{3,c}-s_5{v_5}^{\mathrm{\&gamma;}})---\left(24\right)$

Wherein, constant k₅＞ 0, constant s₅＞ 0；

Formula (22) is rewritten as according to formula (23) and formula (24)：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_5\&le;\underset{i=1}{\overset{5}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+l_4{v}_{4}sign\left({\mathrm{\&xi;}}_4\right)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2\\ +\frac{1}{2h_2^2}{v}_2^2\left(\right|\left|{\mathrm{\&Phi;}}_2\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_1)P_2^T{P}_2+\frac{1}{2h_3^2}{v}_3^2\left(\right|\left|{\mathrm{\&Phi;}}_3\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_2)P_3^T{P}_3+\frac{1}{2h_5^2}{v}_5^2\left(\right|\left|{\mathrm{\&Phi;}}_5\right|{|}^2-{\widehat{\mathrm{\&theta;}}}_2)P_5^T{P}_5\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)+v_2(\hat{J}-J){\stackrel{\&CenterDot;}{x}}_{1,c}\end{array}---\left(25\right)$

E. the asynchronous machine command filtering finite time fuzzy controller of the consideration input saturation to setting up carries out stability analysis

Definition It is θ₁Estimate, wherein,

Definition It is θ₂Estimate, wherein,

Definition It is the estimate of J；Formula (25) is rewritten：

$\begin{array}{c}{\stackrel{\&CenterDot;}{V}}_5\&le;\underset{i=1}{\overset{5}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+l_4{v}_{4}sign\left({\mathrm{\&xi;}}_4\right)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2-\tilde{J}{v}_2{\stackrel{\&CenterDot;}{x}}_{1,c}\\ +\frac{1}{2h_2^2}{v}_2^2{\widetilde{\mathrm{\&theta;}}}_1{P}_2^T{P}_2+\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2h_3^2}{v}_3^2{\widetilde{\mathrm{\&theta;}}}_2{P}_3^T{P}_3+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2h_5^2}{v}_5^2{\widetilde{\mathrm{\&theta;}}}_2{P}_5^T{P}_5+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)\end{array}---\left(26\right)$

Choose liapunov functionThen

Formula (26) is substituted into：

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\&le;\underset{i=1}{\overset{5}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+l_4{v}_{4}sign\left({\mathrm{\&xi;}}_4\right)\\ +\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)+\frac{1}{r_1}{\widetilde{\mathrm{\&theta;}}}_1\left(\frac{r_1}{2h_2^2}{v}_2^2{P}_2^T{P}_2-{\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}}_1\right)\\ +\frac{1}{r_2}{\widetilde{\mathrm{\&theta;}}}_2\left(\frac{r_2}{2h_3^2}{v}_3^2{P}_3^T{P}_3+\frac{r_2}{2h_5^2}{v}_5^2{P}_5^T{P}_5-{\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}}_2\right)-\frac{1}{r_3}\tilde{J}(\stackrel{\&CenterDot;}{\hat{J}}+r_3{v}_2{\stackrel{\&CenterDot;}{x}}_{1,c})\end{array}---\left(27\right)$

Select corresponding adaptive law：

${\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}}_1=\frac{r_1}{2h_2^2}{v}_2^2{P}_2^T{P}_2-m_1{\widehat{\mathrm{\&theta;}}}_1---\left(28\right)$

${\stackrel{\&CenterDot;}{\widehat{\mathrm{\&theta;}}}}_2=\frac{r_2}{2h_3^2}{v}_3^2{P}_3^T{P}_3+\frac{r_2}{2h_5^2}{v}_5^2{P}_5^T{P}_5-m_2{\widehat{\mathrm{\&theta;}}}_2---\left(29\right)$

$\stackrel{\&CenterDot;}{\hat{J}}=-r_3{v}_2{\stackrel{\&CenterDot;}{x}}_{1,c}-m_3\hat{J}---\left(30\right)$

Wherein, constant r₁＞ 0, constant m₁＞ 0, constant r₂＞ 0, constant m₂＞ 0, constant r₃＞ 0, constant m₃＞ 0；

According to formula (28), formula (29) and formula (30), formula (27) is rewritten as：

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\&le;\underset{i=1}{\overset{5}{\&Sigma;}}(-k_i{v}_i^2-s_i{v_i}^{\mathrm{\&gamma;}+1})+l_1{v}_{1}sign\left({\mathrm{\&xi;}}_1\right)+l_2{v}_{2}sign\left({\mathrm{\&xi;}}_2\right)+v_4{l}_{4}sign\left({\mathrm{\&xi;}}_4\right)+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)+\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1{\widehat{\mathrm{\&theta;}}}_1}{r_1}+\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2{\widehat{\mathrm{\&theta;}}}_2}{r_2}+\frac{m_3\tilde{J}\hat{J}}{r_3}\end{array}---\left(31\right)$

Meanwhile, by Young inequalityCan obtain：

$\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1{\widehat{\mathrm{\&theta;}}}_1}{r_1}=\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1}{r_1}(-{\widetilde{\mathrm{\&theta;}}}_1+{\mathrm{\&theta;}}_1)=\frac{m_1}{r_1}(-{\widetilde{\mathrm{\&theta;}}}_1^2+{\mathrm{\&theta;}}_1{\widetilde{\mathrm{\&theta;}}}_1)\&le;\frac{m_1}{r_1}\left(-{\widetilde{\mathrm{\&theta;}}}_1^2+\frac{1}{4}{\widetilde{\mathrm{\&theta;}}}_1^2+{\mathrm{\&theta;}}_1^2\right)\&le;\frac{-3m_1}{4r_1}{\widetilde{\mathrm{\&theta;}}}_1^2+\frac{m_1}{r_1}{\mathrm{\&theta;}}_1^2---\left(32\right)$

$\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2{\widehat{\mathrm{\&theta;}}}_2}{r_2}=\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2}{r_2}(-{\widetilde{\mathrm{\&theta;}}}_2+{\mathrm{\&theta;}}_2)=\frac{m_2}{r_2}(-{\widetilde{\mathrm{\&theta;}}}_2^2+{\mathrm{\&theta;}}_2{\widetilde{\mathrm{\&theta;}}}_2)\&le;\frac{m_2}{r_2}\left(-{\widetilde{\mathrm{\&theta;}}}_2^2+\frac{1}{4}{\widetilde{\mathrm{\&theta;}}}_2^2+{\mathrm{\&theta;}}_2^2\right)\&le;\frac{-3m_2}{4r_2}{\widetilde{\mathrm{\&theta;}}}_2^2+\frac{m_2}{r_2}{\mathrm{\&theta;}}_2^2---\left(33\right)$

$\frac{m_3\tilde{J}\hat{J}}{r_3}=\frac{m_3\tilde{J}}{r_3}(-\tilde{J}+J)=\frac{m_3}{r_3}(-{\tilde{J}}^2+J\tilde{J})\&le;\frac{m_3}{r_3}\left(-{\tilde{J}}^2+\frac{1}{4}{\tilde{J}}^2+J^2\right)\&le;\frac{-3m_3}{4r_3}{\tilde{J}}^2+\frac{m_3}{r_3}{J}^2---\left(34\right)$

Obtained by Young inequality again：

$l_e{v}_{e}sign\left({\mathrm{\&xi;}}_e\right)\&le;\frac{l_e}{2}{v}_e^2+\frac{l_e}{2}{\&lsqb;sign\left({\mathrm{\&xi;}}_e\right)\&rsqb;}^2\&le;\frac{l_e}{2}{v}_e^2+\frac{l_e}{2}---\left(35\right)$

Wherein, e=1,2,4；

With reference to formula (32), formula (33), formula (34) and formula (35),It is represented by：

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\&le;-\underset{i=1}{\overset{5}{\&Sigma;}}\&lsqb;k_i{v_i}^2+s_i{v_i}^{\mathrm{\&gamma;}+1}\&rsqb;+\frac{l_1}{2}{v_1}^2+\frac{l_2}{2}{v_2}^2+\frac{l_4}{2}{v_4}^2+\frac{1}{2}{l}_1+\frac{1}{2}{l}_2+\frac{1}{2}{l}_4+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2\\ +\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)+\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1{\widehat{\mathrm{\&theta;}}}_1}{r_1}-{\left(\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{2r_1}\right)}^{\mathrm{\&gamma;}+1/2}+{\left(\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{2r_1}\right)}^{\mathrm{\&gamma;}+1/2}\\ +\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2{\widehat{\mathrm{\&theta;}}}_2}{r_2}-{\left(\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{2r_2}\right)}^{\mathrm{\&gamma;}+1/2}+{\left(\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{2r_2}\right)}^{\mathrm{\&gamma;}+1/2}+\frac{m_3\tilde{J}\hat{J}}{r_3}-{\left(\frac{m_3{\tilde{J}}^2}{2r_3}\right)}^{\mathrm{\&gamma;}+1/2}\\ \&le;-\underset{i=1}{\overset{5}{\&Sigma;}}\&lsqb;k_i{v_i}^2+s_i{v_i}^{\mathrm{\&gamma;}+1}\&rsqb;-{\left(\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{2r_1}\right)}^{\mathrm{\&gamma;}+1/2}-{\left(\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{2r_1}\right)}^{\mathrm{\&gamma;}+1/2}-{\left(\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{2r_2}\right)}^{\mathrm{\&gamma;}+1/2}-{\left(\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{2r_2}\right)}^{\mathrm{\&gamma;}+1/2}\\ -{\left(\frac{m_3{\tilde{J}}^2}{2r_3}\right)}^{\mathrm{\&gamma;}+1/2}-{\left(\frac{m_3{\tilde{J}}^2}{2r_3}\right)}^{\mathrm{\&gamma;}+1/2}+\frac{l_1}{2}{v_1}^2+\frac{l_2}{2}{v_2}^2+\frac{l_4}{2}{v_4}^2+\frac{1}{2}{l}_1+\frac{1}{2}{l}_2\\ +\frac{1}{2}{l}_4+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)-\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{4r_1}+{\left(\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{2r_1}\right)}^{\mathrm{\&gamma;}+1/2}\\ -\frac{m_1{\widetilde{\mathrm{\&theta;}}}_1^2}{2r_1}+\frac{m_1{\mathrm{\&theta;}}_1^2}{2r_1}-\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{4r_2}+{\left(\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{2r_2}\right)}^{\mathrm{\&gamma;}+1/2}-\frac{m_2{\widetilde{\mathrm{\&theta;}}}_2^2}{2r_2}+\frac{m_2{\mathrm{\&theta;}}_2^2}{2r_2}-\frac{m_3{\tilde{J}}^2}{4r_3}+{\left(\frac{m_3{\tilde{J}}^2}{2r_3}\right)}^{\mathrm{\&gamma;}+1/2}\\ -\frac{m_3{\tilde{J}}^2}{2r_3}+\frac{m_3{J}^2}{r_3}\end{array}---\left(36\right)$

IfIt is available：

${\left(\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2\right)}^{\mathrm{\&gamma;}+1/2}-\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2+\frac{m_g}{r_g}{\mathrm{\&theta;}}_g^2<\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2-\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2+\frac{m_g}{r_g}{\mathrm{\&theta;}}_g^2=\frac{m_g}{r_g}{\mathrm{\&theta;}}_g^2---\left(37\right)$

IfIt is available：

${\left(\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^{2g}\right)}^{\mathrm{\&gamma;}+1/2}{|}_{\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2\&le;1}<{\left(\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2\right)}^{\mathrm{\&gamma;}+1/2}{|}_{\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2>1}---\left(38\right)$

Therefore, with reference to inequality (35), (36), obtain：

${\left(\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2\right)}^{\mathrm{\&gamma;}+1/2}-\frac{m_g}{2r_g}{\widetilde{\mathrm{\&theta;}}}_g^2+\frac{m_g}{r_g}{\mathrm{\&theta;}}_g^2\&le;\frac{m_g}{r_g}{\mathrm{\&theta;}}_g^2---\left(39\right)$

Wherein, g=1,2；

Similarly,

Therefore, by above-mentioned inequality, can be written as：

$\begin{array}{c}\stackrel{\&CenterDot;}{V}\&le;-\underset{i=1}{\overset{5}{\&Sigma;}}{k}_i{v_i}^2-\frac{m_1}{4r_1}{\widetilde{\mathrm{\&theta;}}}_1^2-{\left(\frac{m_1}{2r_1}{\widetilde{\mathrm{\&theta;}}}_1^2\right)}^{\mathrm{\&gamma;}+1/2}-\frac{m_2}{4r_2}{\widetilde{\mathrm{\&theta;}}}_2^2-{\left(\frac{m_2}{2r_2}{\widetilde{\mathrm{\&theta;}}}_2^2\right)}^{\mathrm{\&gamma;}+1/2}-\frac{m_3}{4r_3}{\tilde{J}}^2-{\left(\frac{m_3}{2r_3}{\tilde{J}}^2\right)}^{\mathrm{\&gamma;}+1/2}\\ -\underset{i=1}{\overset{5}{\&Sigma;}}{s}_j{v_j}^{\mathrm{\&gamma;}+1}+\frac{l_1}{2}{v_1}^2+\frac{l_2}{2}{v_2}^2+\frac{l_4}{2}{v_4}^2+\frac{1}{2}{l}_1+\frac{1}{2}{l}_2+\frac{1}{2}{l}_4+\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)\\ +\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)+\frac{1}{2}{{\mathrm{\&epsiv;}}_1}^2{d}^2+\frac{m_1}{r_1}{\mathrm{\&theta;}}_1^2+\frac{m_2}{r_2}{\mathrm{\&theta;}}_2^2+\frac{m_3}{r_3}{J}^2\\ \&le;-aV-{\mathrm{bV}}^{\frac{\mathrm{\&gamma;}+1}{2}}+c\end{array}---\left(41\right)$

In formula,

$b=\min \{\left(s_0\right)\&CenterDot;2^{\frac{\mathrm{\&gamma;}+1}{2}},m_1,m_2,m_3\},o=\mathrm{1,2,3,4,5};$

$c=\frac{1}{2}{l}_1+\frac{1}{2}{l}_2+\frac{1}{2}{l}_4+\frac{1}{2}{\mathrm{\&epsiv;}}_1^2{d}^2+\frac{1}{2}(h_2^2+{\mathrm{\&epsiv;}}_2^2)+\frac{1}{2}(h_3^2+{\mathrm{\&epsiv;}}_3^2)+\frac{1}{2}(h_5^2+{\mathrm{\&epsiv;}}_5^2)+\frac{m_1}{r_1}{\mathrm{\&theta;}}_1^2+\frac{m_2}{r_2}{\mathrm{\&theta;}}_2^2+\frac{m_3}{r_3}{J}^2;$

Using finite time by v_eA minizone is constrained in, because of z_e=v_e+ξ_e, e=1,2,4；ξ need to be proved_eUsing it is limited when Between constrain so as to obtain tracking signal z_eFinite time can also be used to constrain in the small neighbourhood of origin；

Choose compensation system liapunov function be：It is available：

$\begin{array}{c}\stackrel{\&CenterDot;}{\stackrel{\&OverBar;}{V}}={\mathrm{\&xi;}}_1{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_1+{\mathrm{\&xi;}}_2{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_2+{\mathrm{\&xi;}}_4{\stackrel{\&CenterDot;}{\mathrm{\&xi;}}}_4\\ =-k_1{\mathrm{\&xi;}}_1^2+{\mathrm{\&xi;}}_2{\mathrm{\&xi;}}_1+{\mathrm{\&xi;}}_1(x_{1,c}-{\mathrm{\&alpha;}}_1)-{\mathrm{\&xi;}}_1{l}_{1}sign\left({\mathrm{\&xi;}}_1\right)+\frac{{\mathrm{\&xi;}}_2}{J}\&lsqb;-k_2{\mathrm{\&xi;}}_2-{\mathrm{\&xi;}}_1+(x_{2,c}-{\mathrm{\&alpha;}}_2)-l_{2}sign\left({\mathrm{\&xi;}}_2\right)\&rsqb;\\ -k_4{\mathrm{\&xi;}}_4^2+b_4{\mathrm{\&xi;}}_4(x_{3,c}-{\mathrm{\&alpha;}}_3)-{\mathrm{\&xi;}}_4{l}_{4}sign\left({\mathrm{\&xi;}}_4\right)\\ =-k_1{\mathrm{\&xi;}}_1^2-k_2{\mathrm{\&xi;}}_2^2-k_4{\mathrm{\&xi;}}_4^2-{\mathrm{\&xi;}}_1{l}_{1}sign\left({\mathrm{\&xi;}}_1\right)-{\mathrm{\&xi;}}_2{l}_{2}sign\left({\mathrm{\&xi;}}_2\right)-{\mathrm{\&xi;}}_4{l}_{4}sign\left({\mathrm{\&xi;}}_4\right)\\ +{\mathrm{\&xi;}}_1(x_{1,c}-{\mathrm{\&alpha;}}_1)+\frac{{\mathrm{\&xi;}}_2}{J}(x_{2,c}-{\mathrm{\&alpha;}}_2)+b_4{\mathrm{\&xi;}}_4(x_{3,c}-{\mathrm{\&alpha;}}_3)\\ =-k_1{\mathrm{\&xi;}}_1^2-k_2{\mathrm{\&xi;}}_2^2-k_4{\mathrm{\&xi;}}_4^2-l_1\left|{\mathrm{\&xi;}}_1\right|-l_2\left|{\mathrm{\&xi;}}_2\right|-l_4\left|{\mathrm{\&xi;}}_4\right|+{\mathrm{\&xi;}}_1(x_{1,c}-{\mathrm{\&alpha;}}_1)+\frac{{\mathrm{\&xi;}}_2}{J}(x_{2,c}-{\mathrm{\&alpha;}}_2)+b_4{\mathrm{\&xi;}}_4(x_{3,c}-{\mathrm{\&alpha;}}_3)\end{array}---\left(42\right)$

And becauseCan obtain：

Wherein, k₀=2min (k₁,k₂,k₄),Select suitable l_e,Realized with ρMake ξ_eThe bounded in finite time, e=1,2,4.