CN106773694B - Precision Piezoelectric location platform adaptively exports feedback inverse control method - Google Patents

Abstract

The present invention is that a kind of Precision Piezoelectric location platform adaptively exports feedback inverse control method, devises an achievable adaptive output feedback Adverse control scheme neural network based for the first time for a kind of hysteresis nonlinearity system.Its main feature is that the high gain state observer of design comes the state of estimating system, and the indeterminate in processing system and environmental disturbances；By being tested in Precision Piezoelectric location platform using proposed control program, wherein Precision Piezoelectric location platform be regarded as one only systematic output be the third-order system that can be measured；By adjusting the primary condition of state observer and the adaptive law of unknown parameter, the arbitrarily small L for following error may be implemented_∞Norm.

Description

Precision Piezoelectric location platform adaptively exports feedback inverse control method

Technical field

It is that a kind of Precision Piezoelectric location platform adaptively exports feedback Adverse control the invention belongs to Precision Machining manufacturing field Method.

Background technique

In the prior art, the actuator based on intellectual material is widely used in miniature and nano-scale systems, metal Cutting system and other ultrahigh precision positioning systems.However, existing in actuator based on intellectual material inevitable The disadvantage is that hysteresis is non-linear.Because lag can not differential and multivalue, when a control system for lag lack mend When repaying, it may show undesirable attribute, such as vibrate or even unstable.

Usually one is the inversion models of construction magnetic hysteresis for the method for processing lag, and are placed on actuator as compensator In control system before；Another method is that robust adaptive scheme is designed in the case where not establishing hysteresis inversion model to subtract The influence of light magnetic hysteresis.For first method, because magnetic hysteresis is usually unknown, hysteresis model and its parsing error between Expression formula is difficult to set up.In the second approach, unknown magnetic hysteresis is divided into linear and nonlinear part, wherein non-linear Divide usually as interference.The reason of being designed here it is second method.However, since the non-linear partial of lag may be nothing Boundary, this meaning observer error may not establish the inverse situation of magnetic hysteresis when output signal can obtain without the upper bound Under, second method is possible to can be invalid.Up to the present, the very unobtainable output feedback ontrol that there is magnetic hysteresis against compensator Scheme.

In Precision Piezoelectric location platform control system, since velocity and acceleration is generally inconvenient to measure, one kind can be real Existing robust adaptive output feedback ontrol algorithm still can not obtain.

Summary of the invention

The purpose of the present invention is that can survey for output, there are the Precision Piezoelectric location platforms of magnetic hysteresis input, using based on sight The Precision Piezoelectric position that output feedback adaptive dynamic surface control, PI model inversion and the radial base neural net of survey device combine is flat Platform adaptively exports feedback inverse control method, it is ensured that the arbitrarily small L of its tracking error_∞It is all in norm and closed-loop system Signal is half global ultimately uniform boundary, overcomes counter " differential explosion " problem pushed away in control program, simplified control device knot Structure reduces calculation amount, real-time control of being more convenient for.

Realize the object of the invention the technical solution adopted is that: a kind of Precision Piezoelectric location platform adaptively export that feedback is inverse to be controlled Method processed, characterized in that it includes the following contents:

1) Precision Piezoelectric location platform mathematical model

In view of a kind of nonlinear system expression formula that hysteresis is added are as follows:

Y=x₁, i=0,1 ..., n-1 (1)

Wherein,It is state vector；Unknown smooth linear function, d_iIt (t) is external disturbance, b₀It is unknown constant parameter, w ∈ R is unknown hysteresis phenomenon, is indicated are as follows:

W (u)=P (u (t)) (2)

U is the input signal of actuator, and P is lag operator,

For systematic (1), it is assumed hereinafter that be required:

A1: interference d_i(t), i=1, n meets:

Wherein,It is some unknown normal numbers；A2: under design treatment, desired trajectory y_rIt is smooth, and y_rIt (0) is energy It accesses；For all t >=0,It is compacted known to belonging to one；A3:b₀Symbol be it is known, do not lose Generality, for convenience, it is assumed that b₀> 0；

2) Prandtl-Ishlinskii (PI) model and its inverse

Mitigate using the PI model suitable for describing hysteresis model piezo actuator, and using its corresponding model inversion The influence of hysteresis,

W (t)=P [u] (t) (4)

Wherein P [u] (t) is defined as:

Wherein r is threshold value, and p (r) is that given density function meets p (r) > 0,In, it rises for convenience See,It is the constant determined by density function p (r), Λ indicates the upper limit of integral, enables f_r: R → R, by (6) formula Definition:

f_r(u, w)=max (u-r, min (u+r, w)) (6)

In turn, play operator F_r[u] (t) meets:

Wherein, 0≤i≤N-1,0=t₀< t₁< ... < t_N=t_EIt is [0, t_E] on one segmentation so that function u exists (t_i,t_i+1] on each subinterval be it is dull, i.e., it is non-increasing or it is non-subtract,

In order to compensate for hysteresis nonlinearity w (u) in formula (1), the inverse of PI model is constructed:

Wherein, ο indicates compensating operator；P^-1[] (t) is the inverse compensating operator of PI model,

WhereinIt is constant, expression is upper limit of integral in formula (9), and,

Since in practice, hysteresis can not obtain, this means that density function p (r) is needed in measurement data On the basis of obtain, inversion model is based on density function estimationOn the basis of construct, useAs P [u] (t) estimated value, therefore, by by compensatory theory be applied to P [] (t) and:

Wherein,γ (r), δ (r) be P [] (t) andIt is initially loaded curve, u_dBe by The control signal of design,

In view of formula (11) and inequality F_r[u_d](t)+E_r[u_d] (t)=u_d(t), it obtains:

W (t)=φ ' (Λ) u_d+d_b(t) (12)

Wherein φ ' (Λ) normal number, E_r() is the stop operator of PI model, due to | E_r() | < Λ,Bounded and satisfaction:

|d_b(t)|≤D (13)

Wherein, D normal number obtains analytical error e (t) expression formula from formula (11) to formula (12) are as follows:

Formula (12) are substituted into formula (1), are obtained:

Wherein, b_ΛIt is normal number and satisfaction:

b_Λ=b₀φ′(Λ) (16)

3) radial basis function neural network (RBFNNs) approaches the unknown

Lemma 1 is followed, is compacted using the linear radial basis function neural network (RBFNNs) of a weight properties come approximate In a continuous function,

Lemma 1: for any given real continuous function f, RBFNNs is a universal approximator, f: Ω_ξ→ R its In,ξ is the input of neural network, and q is input dimension.For arbitrary ε_m> 0 passes through selection σ and ζ appropriate_k ∈R^q, k=1 ..., N, later, there are a RBFNN to make:

F (ξ)=ψ^T(ξ)θ^*+ε (17)

Wherein θ^*It is θ=[θ₁,…,θ_N]∈R^NBest initial weights vector, and is defined as:

Wherein, Y (ξ)=ψ^TThat (ξ) θ is indicated is the output of RBFNNs, ψ (ξ)=[ψ₁(ξ),…,ψ_N(ξ)]∈R^NIt is basic Functional vector, it is generally the case that so-called Gaussian function generally presses following form as basic function:

Wherein, σ > 0, k=1 ..., N,It is constant value vector, the referred to as center of basic function；σ is real number, referred to as The width of basic function, ε are approximate errors, and are met:

ε=f (ξ)-θ^*Tψ(ξ) (20)

Using lemma 1 and formula (17), RBFNNs obtains (21) as the unknown continuous function for approaching device and coming in approximate expression (17) Formula:

Wherein, ε_i, i=1 ..., N are any normal numbers, indicate neural network approximate error, and,

WhereinIt is state variable x₁,…,x_iEstimated value, and can introduce in the formula (40),

Formula (21) are substituted into formula (15), are obtained:

Systematic (1) is expressed as following state space form:

Wherein, [0 ... 0, b b=_Λ]^T∈Rⁿ,e₁=[1,0 ..., 0]^T,

B=D_b+ε+d (25)

Wherein, d=[d₁(t),…,d_n(t)]^T, ε=[ε₁,…,ε_n]^T,D_b=[0 ..., 0, d_b(t)]^T∈RⁿIt is class interference ,

Wherein,It is defined in formula (19), For a kind of hysteresis nonlinearity system, the target of control is to establish a kind of output feedback dynamic surface control based on adaptive neural network net Scheme processed, so that the L with tracking error_∞Norm is consistent, and output signal y can good track reference signal y_r, and closed-loop system All signals be all uniformly bounded；

4) the adaptive dynamic surface based on observer is against design of Compensator

1. high-gain Kalman Filter observer

Formula (24) is changed into (27) formula:

It enables

A₀=A-qe₁ ^T (28)

Wherein, q=[q₁,…,q_n], A is made by selection vector q appropriate₀For He Weici matrix,

Construction high-gain Kalman filter carrys out the state variable x in estimator (27),

Wherein k >=1 is positive design parameter, e_nWhat is represented belongs to RⁿThe n rank coordinate vector of form, and,

Φ=diag 1, k ..., k^n-1} (32)

From formula (29) to formula (32), estimated state vector is as follows,

Further, evaluated error is defined,

Then, it obtains,

Wherein, ε₁It is the first item of ε, B is defined in formula (25)；

Lemma 2: enabling high-gain Kalman filter be defined by formula (29)-formula (31) and following second order function,

V_ε:=ε^TPε (36)

Wherein,It is positive definite matrixAnd meet:

Wherein, A₀It is defined, is enabled by formula (28):

Wherein, | | B | |_maxBe | | B | | maximum value.For arbitrary k >=1, formula (36) derivation is obtained:

Due to the b in formula (33)_ΛAnd θ^*It is unknown,It can not obtain, therefore actual state estimation is:

Wherein,WithIt is b_ΛAnd θ^*Estimated value,

The improved high-gain Kalman filter is for handling bounded B in the formula of being defined on (25), by suitable When design parameter k >=1 and formula (32) of the selecting type (29) into formula (31) defined in matrix Φ, observation error ε can be made It is arbitrarily small；

2. dynamic surface inverse controller designs

The design of controller includes substitute variable, control law and adaptive law, wherein τ₂,…,τ_nWhen being low-pass filter Between constant, l_i, i=1 ..., n and γ_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bIt is positive design constant,

Substitute variable: S₁=y-y_r (T.1)

S_i=v_0,i-z_i, i=2 ..., n (T.2)

Wherein, z_iThere is following formula generation,

Wherein,

And

Control law:

Adaptive law:

Invention a kind of Precision Piezoelectric location platform adaptively export feed back inverse control method the advantages of be embodied in:

1) the high gain state observer designed carrys out the state of estimating system, and in processing system and environmental disturbances not Determine item, therefore, compared to STATE FEEDBACK CONTROL algorithm, the output of only control system is in the case of requirement can obtain, Itd is proposed control algolithm is set to be more suitable for practical application；

2) pass through and tested in Precision Piezoelectric location platform using proposed control program, wherein Precision Piezoelectric Location platform be regarded as one only it is systematic output be the third-order system that can be measured；

3) it by adjusting the primary condition of state observer and the adaptive law of unknown parameter, may be implemented to follow error Arbitrarily small L_∞Norm, overcomes counter " differential explosion " problem pushed away in control program, and simplified control device structure is reduced and calculated Amount, real-time control of being more convenient for.

4) its methodological science is reasonable, strong applicability, and effect is good.

Detailed description of the invention

Fig. 1 is PI model (a), PI inversion model (b) and compensation result (c) schematic diagram；

Fig. 2 is inverse compensation scheme schematic diagram；

Fig. 3 is that Precision Piezoelectric location platform of the invention adaptively exports feedback inverse control method schematic diagram；

Fig. 4 is practical Precision Piezoelectric location platform control system architecture schematic diagram；

Fig. 5 is the General Principle schematic diagram of Precision Piezoelectric location platform；

Fig. 6 input-output between piezoelectric ceramic actuator and PI model responds comparison schematic diagram；

Fig. 7 is modeling error schematic diagram；

Fig. 8 is emulation and tests against compensation result (3 μm) schematic diagram；

Fig. 9 is emulation and tests against compensation result (5 μm) schematic diagram；

Figure 10 is that actual displacement exports y and desired trajectory y_rSchematic diagram；

Figure 11 is to have Hysteresis compensation and tracking error schematic diagram when without Hysteresis compensation；

Figure 12 is to have Hysteresis compensation and the control V diagram without using Hysteresis compensation；

Figure 13 is the dynamic surface method proposed and the tracking error schematic diagram of the anti-pushing manipulation of tradition；

Figure 14 is the dynamic surface method proposed and the displacement diagram of the anti-pushing manipulation of tradition；

Figure 15 is the dynamic surface method proposed and the control V diagram of the anti-pushing manipulation of tradition.

In Fig. 1, ordinate indicates magnetic hysteresis output, and abscissa indicates magnetic hysteresis input；In Fig. 5, k_ampIt is fixed gain；R₀It is The equivalent internal resistance of driving circuit；v_hIt is the voltage generated due to hysteresis effect, H and T_emRepresent piezoelectric effect；C_AIndicate all pressures The summation of electroceramics capacitor；Q andThat respectively represent is all charges in PCA and the electric current for thus flowing through circuit, q_cIt is storage In linear capacitance C_AIn charge.q_pIndicate the conduction charge generated due to piezoelectric effect from mechanical side, v_AIndicate conduction electricity Pressure, for mechanical part, m, b_sAnd k_sRespectively indicate quality, the rigidity of damped coefficient and movement mechanism；Abscissa indicates magnetic in figure Stagnant input, ordinate indicate magnetic hysteresis output；Fig. 4 abscissa indicates the time, and ordinate indicates output y to objective function y_rTracking Performance；Fig. 6 abscissa indicates the time, and ordinate indicates displacement；Fig. 7 abscissa indicates time, ordinate modeling error (%)；Figure 8 and Fig. 9 abscissa indicates expectation displacement, and ordinate indicates actual displacement；Figure 10 abscissa indicates the time, and ordinate indicates position It moves；Figure 11 abscissa indicates the time, and ordinate indicates tracking error；Figure 12 abscissa indicates the time, and ordinate indicates control electricity Pressure；Figure 13 abscissa indicates the time, and ordinate indicates tracking error；Figure 14 abscissa indicates the time, and ordinate indicates displacement；Figure 15 abscissas indicate the time, and ordinate indicates control voltage.

Specific embodiment

Below with drawings and examples, the invention will be further described.

Precision Piezoelectric location platform of the invention adaptively exports feedback inverse control method, including the following contents:

1) Precision Piezoelectric location platform mathematical model

In view of a kind of nonlinear system expression formula that hysteresis is added are as follows:

Y=x₁, i=0,1 ..., n-1 (1)

Wherein,It is state vector；Unknown smooth linear function, d_iIt (t) is external disturbance, b₀Unknown constant parameter, w ∈ R unknown hysteresis phenomenon indicate are as follows:

W (u)=P (u (t)) (2)

U is the input signal of actuator, and P is lag operator,

For systematic (1), it is assumed hereinafter that be required:

A1: interference d_i(t), i=1, n meets:

Wherein,It is some unknown normal numbers；A2: under design treatment, desired trajectory y_rIt is smooth, and y_rIt (0) is energy It accesses；For all t >=0,It is compacted known to belonging to one；A3:b₀Symbol be it is known, do not lose Generality, for convenience, it is assumed that b₀> 0；

2) Prandtl-Ishlinskii (PI) model and its inverse

Mitigate using the PI model suitable for describing hysteresis model piezo actuator, and using its corresponding model inversion The influence of hysteresis,

W (t)=P [u] (t) (4)

Wherein P [u] (t) is defined as:

Wherein r is threshold value, and p (r) is that given density function meets p (r) > 0,In, it rises for convenience See,It is the constant determined by density function p (r), Λ indicates the upper limit of integral, enables f_r: R → R, by (6) formula Definition:

f_r(u, w)=max (u-r, min (u+r, w)) (6)

In turn, play operator F_r[u] (t) meets:

Wherein, t_i< t≤t_i+1, 0≤i≤N-1,0=t₀< t₁< ... < t_N=t_EIt is [0, t_E] on one segmentation, with Make function u in (t_i,t_i+1] on each subinterval be it is dull, i.e., it is non-increasing or it is non-subtract,

In order to compensate for hysteresis nonlinearity w (u) in formula (1), the inverse of PI model is constructed:

Wherein, ο indicates compensating operator；P^-1[] (t) is the inverse compensating operator of PI model,

WhereinIt is constant, expression is upper limit of integral in formula (9), and,

Since in practice, hysteresis can not obtain, this means that density function p (r) is needed in measurement data On the basis of obtain, inversion model is based on density function estimationOn the basis of construct, useAs P [u] (t) estimated value, therefore, by by compensatory theory be applied to P [] (t) and:

Wherein,γ (r), δ (r) be P [] (t) andIt is initially loaded curve, u_dBe by The control signal of design,

In view of formula (11) and inequality F_r[u_d](t)+E_r[u_d] (t)=u_d(t), it obtains:

W (t)=φ ' (Λ) u_d+d_b(t) (12)

Wherein φ ' (Λ) normal number, E_r() is the stop operator of PI model, due to | E_r() | < Λ,Bounded and satisfaction:

|d_b(t)|≤D (13)

Wherein, D normal number obtains analytical error e (t) expression formula from formula (11) to formula (12) are as follows:

Formula (12) are substituted into formula (1), are obtained:

Wherein, b_ΛIt is normal number and satisfaction:

b_Λ=b₀φ′(Λ) (16)

3) radial basis function neural network (RBFNNs) approaches the unknown

Lemma 1 is followed, is compacted using the linear radial basis function neural network (RBFNNs) of a weight properties come approximate In a continuous function,

Lemma 1: for any given real continuous function f, RBFNNs is a universal approximator, f: Ω_ξ→ R its In,ξ is the input of neural network, and q is input dimension.For arbitrary ε_m> 0 passes through selection σ and ζ appropriate_k ∈R^q, k=1 ..., N, later, there are a RBFNN to make:

F (ξ)=ψ^T(ξ)θ^*+ε (17)

Wherein θ^*It is θ=[θ₁,…,θ_N]∈R^NBest initial weights vector, and is defined as:

Wherein, Y (ξ)=ψ^TThat (ξ) θ is indicated is the output of RBFNNs, ψ (ξ)=[ψ₁(ξ),…,ψ_N(ξ)]∈R^NIt is basic Functional vector, it is generally the case that so-called Gaussian function generally presses following form as basic function:

Wherein, σ > 0, k=1 ..., N, ζ_k∈RⁿIt is constant value vector, the referred to as center of basic function.σ is real number, referred to as base The width of this function, ε are approximate errors, and are met:

ε=f (ξ)-θ^*Tψ(ξ) (20)

Using lemma 1 and formula (17), RBFNNs obtains (21) as the unknown continuous function for approaching device and coming in approximate expression (17) Formula:

Wherein, ε_i, i=1 ..., N are any normal numbers, indicate neural network approximate error, and,

WhereinIt is state variable x₁,…,x_iEstimated value, and can introduce in the formula (40),

Formula (21) are substituted into formula (15), obtain (23) formula:

Systematic (1) is expressed as following state space form:

Wherein, [0 ... 0, b b=_Λ]^T∈Rⁿ,e₁=[1,0 ..., 0]^T,

B=D_b+ε+d (25)

Wherein, d=[d₁(t),…,d_n(t)]^T, ε=[ε₁,…,ε_n]^T,D_b=[0 ..., 0, d_b(t)]^T∈RⁿIt is class interference ,

Wherein,It is defined in formula (19), For a kind of hysteresis nonlinearity system, the target of control is to establish a kind of output feedback dynamic surface control based on adaptive neural network net Scheme processed, so that the L with tracking error_∞Norm is consistent, and output signal y can good track reference signal y_r, and closed-loop system All signals be all uniformly bounded；

4) the adaptive dynamic surface based on observer is against design of Compensator

1. high-gain Kalman Filter observer

Formula (24) is changed into (27) formula:

It enables

A₀=A-qe₁ ^T (28)

Wherein, q=[q₁,…,q_n], A is made by selection vector q appropriate₀For He Weici matrix,

Construction high-gain Kalman filter carrys out the state variable x in estimator (27),

Wherein k >=1 is positive design parameter, e_nWhat is represented belongs to RⁿThe n rank coordinate vector of form, and,

Φ=diag 1, k ..., k^n-1} (32)

From formula (29)-formula (32), estimated state vector is as follows,

Further, evaluated error is defined,

Then, formula (34) derivation is obtained,

Wherein, ε₁It is the first time value of ε, B is defined in formula (25)；

Lemma 2: enabling high-gain Kalman filter be defined by formula (29)-formula (31) and following second order function,

V_ε:=ε^TPε (36)

Wherein,It is positive definite matrixAnd meet:

Wherein, A₀It is defined, is enabled by formula (28):

Wherein, | | B | |_maxBe | | B | | maximum value.For arbitrary k >=1, formula (36) derivation is obtained:

Due to the b in formula (33)_ΛAnd θ^*It is unknown,It can not obtain, therefore actual state estimation is:

Wherein,WithIt is b_ΛWithEstimated value,

The improved high-gain Kalman filter is for handling bounded B in the formula of being defined on (25), by suitable When selecting type (29)-formula (31) in design parameter k >=1 and formula (32) defined in matrix Φ, can make observation error ε appoint It anticipates small；

2. dynamic surface inverse controller designs

The design of controller includes substitute variable, control law and adaptive law, wherein τ₂,…,τ_nWhen being low-pass filter Between constant, l_i, i=1 ..., n and γ_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bIt is positive design constant,

Substitute variable: S₁=y-y_r (T.1)

S_i=v_0,i-z_i, i=2 ..., n (T.2)

Wherein, z_iThere is following formula generation,

Wherein,

And

Control law:

Adaptive law:

1, stability indicator is analyzed

In this part, it will stability and performance evaluation to the adaptive output feedback DSIC scheme proposed into Row discusses, to analyze the L of stability and tracking error_∞Performance.

For stability analysis of control system, the following liapunov function of definition:

Wherein,V_εIt is to close In the quadratic function of high-gain Kalman filtering observation error ε, provided in lemma 2.

Theorem 1: this closed-loop system is considered comprising there is magnetic hysteresis nonlinear systems with delay described in formula (4) (1), the unknown parameter adaptive law in (T.9)-(T.11), control law (T8) related with A1-A3 is assumed.Then for any Given positive number p, if V (0) meets V (0)≤p in formula (41),

A) by proper choice of design parameter k, l₁,…,l_n, timeconstantτ₂,…,τ_n, adaptation law coefficient γ_θ,σ_θ, γ_ζ,σ_ζ,γ_b,σ_b, all signal uniform boundes of closed-loop system, and can be arbitrarily small.

B) tracking error S₁L_∞Performance is available and arbitrarily small, and convolution (T.5) and formula (41) can obtain formula (42):

WhereinC₁It is normal Number, and meet

2, the experimental study of Precision Piezoelectric location platform

A. experimental provision

In order to show the validity of proposed control program, the experiment for carrying out Precision Piezoelectric location platform shown in Fig. 4 is ground Study carefully.The element of control system is as follows:

Piezoelectric ceramic actuator: the piezoelectric ceramics for having used Physik Instrument company to produce in experiment Actuator P-753.31 C.It provides a 38 μ m peak output displacements, and for actuator, voltage range is 0-100V.

Capacitance sensor: the dynamic respond in order to measure actuator, having used a sensitivity is 2.632V/ μm of collection At capacitance type sensor.

Voltage amplifier: the voltage amplifier (LVPZT, E-505) that fixed gain is 10 has been used to execute as piezoelectricity The excitation voltage of device.

Data collection system: the dSPACE control panel with 16 analog-to-digitals, D-A converter is used to obtain piezoelectricity essence The displacement of close location platform, it is to be measured to obtain by capacitance sensor.

B, the model of Precision Piezoelectric location platform

For the modeling result of the Precision Piezoelectric location platform accumulated, the General Principle figure of Precision Piezoelectric location platform Model can be indicated that it is by piezoelectric ceramic actuator (PCA) by Fig. 5) and mechanical part form.In Fig. 5, k_ampIt is fixed increasing Benefit；R₀It is the equivalent internal resistance of driving circuit；v_hIt is the voltage generated due to hysteresis effect.H and T_emRepresent piezoelectric effect；C_AIt indicates The summation of all piezoelectric ceramics capacitors；Q andWhat is respectively represented is all charges in PCA and the electric current for thus flowing through circuit.q_c It is stored in linear capacitance C_AIn charge.q_pIndicate the conduction charge generated due to piezoelectric effect from mechanical side, v_AIt indicates to pass Conduction pressure.For mechanical part, m, b_sAnd k_sRespectively indicate quality, the rigidity of damped coefficient and movement mechanism.Based on above-mentioned piezoelectricity The third-order model of the schematic model of precision positions platform, definition description Precision Piezoelectric location platform is as follows:

Wherein, w=φ ' (Λ) u provided in formula (12)_d+d_b(t) be piezo actuator output, and

C. about magnetic hysteresis modeling and its experiment of inverse compensator construction

For the ease of the parameter identification to model described in formula (5), corresponding discrete expression is as follows in formula (5):

In order to obtain optimal play operator p_i, i=1,2,3 ..., n, to describe piezoelectric ceramic actuator P- in Fig. 4 Hysteresis in 753.31, in conjunction with following constraint double optimization:

min{[CΛ-d]^T[CΛ-d]} (45)

Wherein C constant, d are sinusoidal signals, and Λ (i) meets

Λ(i)≥0,i∈{1,2,3,…,n} (46)

By using experimental data, p is recognized with the Least-squares minimization tool box in MATLAB_i.And these data are Based on Precision Piezoelectric location platform control system practical in Fig. 4, the sinusoidal input signal d's of the amplitude reduction designed by one Under the conditions of obtain.Since modeling error is inevitable, we can only obtain p_iEstimation parameter r_i= [0,0.1,1.7834,3.4669,5.1503,6.8338,8.5172,10.2007,11.8841,13.5676,15.2510, 21.9848,32.0855].Fig. 6 shows the input and output between piezoelectric ceramic actuator (dotted line) and PI model (solid line) The comparison of response.Modeling error shown in Fig. 7 is defined as follows:

Wherein, x (t) and w (t) respectively indicates the output of piezoelectric ceramic actuator and PI model.Comparison result and modeling miss It is relatively good that difference shows that PI model coincide with experimental data really (less than 1%).

It is inverse using the parsing in the identified parameters and formula (10) of above-mentioned PI modelIt can be expressed as The Numerical Implementation of discrete expression is as follows:

AndCause This,Threshold value and weight computing are as follows: In order to show formula (48) validity that inverse compensator is established in turns the code in MATLAB/SIMULINK through dSPACE module shown in Fig. 4 It is changed to real-time code, is thus tested.One input signal u_d(t)=B₁Sin (2 π ft), wherein B₁=3 μm, 5 μm, f= 1Hz is applied in compensator, and output is applied to piezoelectric ceramic actuator by power amplifier (LVPZT, E-505) In.

And then the dynamic respond of actuator is obtained by the sensor monitoring measurement in Fig. 4, then download to dSPACE mould In block.In order to make comparisons to simulation result with experimental result, the emulation of inverse compensator is also to carry out in MATLAB/Simulink. Fig. 8 is illustrated when magnetic hysteresis is against compensator in application drawing 2, and simulation result is compared with experimental result.From imitating in Fig. 8 and Fig. 9 True result can be seen that between expectation displacement and output displacement there are a perfect linear input/output relation, which imply The effect of magnetic hysteresis is entirely eliminated.However, there are an inverse compensation errors, in the experimental result of Fig. 8 and Fig. 9 clearly It shows.In fact, compensation error may be as caused by modeling error and environmental disturbances.From Fig. 8 (B₁=3 μm) experiment As a result from the point of view of, output and input between be still stagnant ring relationship, this shows that magnetic hysteresis cannot completely eliminate.Therefore, using being mentioned Output feedback adaptive control program out reduces compensation error.

Now, x is enabled₁=x,Later, formula (43) can be expressed as follows:

Wherein, w indicates that the magnetic hysteresis in piezoelectric ceramic actuator shown in fig. 5 is non-linear.

D. the design procedure and experimental result of controller

Adaptive dynamic surface based on observer is against the part of design of Compensator, in this experiment, high-gain K- filtering Device is designed according to formula (29) to formula (31), wherein v (0)=ξ₀(0)=Ξ (0)=0, k=1.5, q=[3,2,1]^T.For nerve Network system ψ₃(ξ₃), we have selected 5 nodes with basic function center, ζ_j, j=1 ..., 5, size is followed successively by- 0.5,0,1,2,3,4,5,6, width η_j=1, j=1 ..., 7；Next, Ψ^T(ξ)=diag 0,0, ψ₃, wherein ψ₃=[ψ_3,1(ξ₁),…,ψ_3,7(ξ₁)].According to Section III part, dynamic surface error is S₁=x₁-y_r, S₂=v_0,2-z₂, S₃=v_0,3-z₃.Adaptive law isVirtual controlling rule It is selected asWherein Final control law is selected asIn the 1,2nd step Firstorder filter beWherein,It is corresponding to be, on The design parameter for stating adaptive law and control signal is selected as l₁=50, l₂=2, l₂=l₃=2, γ_ζ=5, σ_ζ=0.7, γ_θ= 2,σ_θ=0.9, γ_b=5, σ_b=0.9.The primary condition of adaptive law is selected as

In order to verify the validity of proposed control program, following two examination has been carried out in Precision Piezoelectric location platform It tests.In order to achieve the purpose that real-time control, on dSPACE control panel, improved adaptive output feedback control algorithm is turned Turn to the S function file for being 10kHz by the sample frequency of C language compiling.

A. multi-frequency track following is tested

It is in both cases to multi-frequency desired trajectory y in this experiment_r=3+2sin (2 π * 2t)+sin (2 π * 15t) Carry out motion tracking control: with and without the Hysteresis compensation constructed in formula (48).Experimental result is as shown in fig. 10-15.Figure 10 Illustrate piezoelectric ceramic actuator actual displacement y (solid line) and desired trajectory y_rThe displacement of (dotted line).As can be seen that realizing one A quite satisfactory tracking performance, tracking error and its small.Figure 11 illustrate with and without use Hysteresis compensation with Track error.Figure 11 clearly shows, compared with the case where not using Hysteresis compensation, be added its transient state of Hysteresis compensation and stable state with Track error is smaller.For example, if using error_max=max (| y-y_r|) indicate the maximum value of tracking error, do not using magnetic In the case where stagnant compensation, error_maxIt is 0.014 μm, and error in the case where magnetic hysteresis is added_maxOnly 0.0059 μm, be not have The half of Hysteresis compensation situation is added.Further, since good tracking performance and minimum tracking error, Figure 10 and Figure 11 are abundant Demonstrate the validity with the output feedback ontrol scheme of Hysteresis compensation device in formula (48) proposed.What Figure 12 was indicated is control The track of voltage processed, solid line expression joined Hysteresis compensation, and dotted line is indicated that Hysteresis compensation is not added.It should be noted that not having The control program of Hysteresis compensation is a kind of situation that " hysteresis inversion model estimation " is not included in Fig. 3.Then, u=u_dWith Precision Piezoelectric location platform system is in the case where no any compensation, by control signal u_dIt directly drives.

B. single-frequency track following and the anti-experimental study for pushing away control and comparing

In order to show the advantage of proposed dynamic surface control, we do dynamic surface control scheme and the anti-control program that pushes away Compare.What Figure 12-14 was indicated is that desired trajectory is y_rThe experimental result of=2sin (40 π t).Figure 12 shows that is proposed moves The tracking error of state face control program, steady-state error error_max=0.0158 μm, the anti-control method steady-state error that pushes away of tradition is error_max=0.0647 μm.Figure 14 and Figure 15 respectively shows displacement and control voltage under two methods.From these results It can be evident that, dynamic surface method shows better control performance than traditional anti-control methods that push away.

Specific embodiment is only the description of the invention, does not constitute the limitation to claims, ability Field technique personnel all fall in the scope of protection of the present invention without the equivalent substitute of creative work.

Claims (1)

1. a kind of Precision Piezoelectric location platform adaptively exports feedback inverse control method, characterized in that it includes the following contents:

1) Precision Piezoelectric location platform mathematical model

In view of a kind of nonlinear system expression formula that hysteresis is added are as follows:

Y=x₁, i=0,1 ..., n-1 (1)

Wherein,It is state vector；Unknown smooth linear function, d_i(t) It is external disturbance, b₀Unknown constant parameter, w ∈ R unknown hysteresis phenomenon output, indicates are as follows:

W (u)=P (u (t)) (2)

U is the input signal of actuator, and P is lag operator,

For systematic (1), it is assumed hereinafter that be required:

A1: external disturbance d_i(t), i=1 ..., n meet:

Wherein,It is some unknown normal numbers；A2: under design treatment, desired trajectory y_rIt is smooth, and y_rIt (0) is that can obtain It arrives；For all t >=0,It is compacted known to belonging to one；A3:b₀Symbol be it is known, do not lose one As property, for convenience, it is assumed that b₀> 0；

2) Prandtl-Ishlinskii (PI) model and its inverse

Mitigate magnetic hysteresis using the PI model suitable for describing hysteresis model piezo actuator, and using its corresponding model inversion The influence of phenomenon,

W (t)=P [u] (t) (4)

Wherein P [u] (t) is defined as:

Wherein r is threshold value, and p (r) is given density function, meets p (r) > 0,For convenience's sake,It is the constant determined by density function p (r), Λ indicates the upper limit of integral, enables f_r: R → R, it is fixed by (6) formula Justice:

f_r(u, w)=max (u-r, min (u+r, w)) (6)

In turn, play operator F_r[u] (t) meets:

Wherein, t_i< t≤t_i+1, 0≤i≤N-1,0=t₀< t₁< ... < t_N=t_EIt is [0, t_E] on one segmentation so that letter Number u is in (t_i,t_i+1] on each subinterval be it is dull, i.e., it is non-increasing or it is non-subtract,

In order to compensate for hysteresis nonlinearity w (u) in formula (1), the inverse of PI model is constructed:

Wherein,Indicate compensating operator；P^-1[] (t) is the inverse compensating operator of PI model,

WhereinIt is constant, expression is upper limit of integral in formula (9), and,

Since in practice, hysteresis can not obtain, this means that density function p (r) needs on the basis of measurement data On obtain, inversion model is based on density function estimationOn the basis of construct, useAs P's [u] (t) Estimated value, therefore, by by compensatory theory be applied to P [] (t) and:

Wherein,γ (r), δ (r) be P [] (t) andIt is initially loaded curve, u_dIt is designed Signal is controlled,

In view of formula (11) and inequality F_r[u_d](t)+E_r[u_d] (t)=u_d(t), it obtains:

W (t)=φ ' (Λ) u_d+d_b(t) (12)

Wherein φ ' (Λ) normal number, E_r() is the stop operator of PI model, due to | E_r() | < Λ,Bounded and satisfaction:

|d_b(t)|≤D (13)

Wherein, D normal number obtains analytical error e (t) expression formula from formula (11) to formula (12) are as follows:

Formula (12) are substituted into formula (1), are obtained:

Wherein, b_ΛIt is normal number and satisfaction:

b_Λ=b₀φ′(Λ) (16)

3) radial basis function neural network (RBFNNs) approaches the unknown

Lemma 1 is followed, using the linear radial basis function neural network (RBFNNs) of a weight properties come in approximate compact One continuous function,

Lemma 1: for any given real continuous function f, RBFNNs is a universal approximator, f: Ω_ξ→ R wherein,ξ is the input of neural network, and q is input dimension, for arbitrary ε_m> 0 passes through selection σ and ζ appropriate_k∈ R^q, k=1 ..., N, later, there are a RBFNN to make:

F (ξ)=ψ^T(ξ)θ^*+ε (17)

|ε|≤ε_m, wherein θ^*It is θ=[θ₁,…,θ_N]∈R^NBest initial weights vector, and is defined as:

Wherein, Y (ξ)=ψ^TThat (ξ) θ is indicated is the output of RBFNNs, ψ (ξ)=[ψ₁(ξ),…,ψ_N(ξ)]∈R^NIt is basic function Vector, it is generally the case that so-called Gaussian function generally presses following form as basic function:

Wherein, σ > 0, k=1 ..., N, ζ_k∈RⁿIt is constant value vector, the referred to as center of basic function；σ is real number, referred to as basic letter Several width, ε are approximate errors, and are met:

ε=f (ξ)-θ^*Tψ(ξ) (20)

Using lemma 1 and formula (17), RBFNNs obtains (21) formula as the unknown continuous function for approaching device and coming in approximate expression (17):

Wherein, ε_i, i=1 ..., N are any normal numbers, indicate neural network approximate error, and,

WhereinIt is state variable x₁,…,x_iEstimated value, and can introduce in the formula (40),

Formula (21) are substituted into formula (15), are obtained:

Systematic (1) is expressed as following state space form:

Wherein, [0 ... 0, b b=_Λ]^T∈Rⁿ,e₁=[1,0 ..., 0]^T,

B=D_b+ε+d (25)

Wherein, d=[d₁(t),…,d_n(t)]^T, ε=[ε₁,…,ε_n]^T,D_b=[0 ..., 0, d_b(t)]^T∈RⁿIt is class distracter,

Wherein,It is defined in formula (19), for A kind of hysteresis nonlinearity system, the target of control are to establish a kind of output feedback dynamic surface control side based on adaptive neural network net Case, so that the L with tracking error_∞Norm is consistent, and output signal y can good track reference signal y_r, and the institute of closed-loop system Signal is all uniformly bounded；

4) the adaptive dynamic surface based on observer is against design of Compensator

1. high-gain Kalman Filter observer

Formula (24) is changed into (27) formula:

It enables

A₀=A-qe₁ ^T (28)

Wherein, q=[q₁,…,q_n], A is made by selection vector q appropriate₀For He Weici matrix,

Construction high-gain Kalman filter carrys out the state variable x in estimator (27),

Wherein k >=1 is positive design parameter, e_nWhat is represented belongs to RⁿThe n rank coordinate vector of form, and,

Φ=diag 1, k ..., k^n-1} (32)

From formula (29) to formula (32), estimated state vector is as follows,

Further, observation error is defined,

Then, formula (34) derivation is obtained,

Wherein,It isFirst item, B is defined in formula (25)；

Lemma 2: enabling high-gain Kalman filter be defined by formula (29)-formula (31) and following second order function,

Wherein, It is positive definite matrixAnd meet:

Wherein, A₀It is defined, is enabled by formula (28):

Wherein, | | B | |_maxBe | | B | | maximum value arbitrary k >=1 obtains formula (36) derivation:

Due to the b in formula (33)_ΛAnd θ^*It is unknown,It can not obtain, therefore actual state estimation is:

Wherein,WithIt is b_ΛAnd θ^*Estimated value,

The improved high-gain Kalman filter is for handling bounded B in the formula of being defined on (25), by appropriate Matrix Φ defined in design parameter k >=1 and formula (32) of the selecting type (29) into formula (31), can make observation errorArbitrarily It is small；

2. dynamic surface inverse controller designs

The design of controller includes substitute variable, control law and adaptive law, wherein τ₂,…,τ_nBe low-pass filter time it is normal Number, l_i, i=1 ..., n and γ_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bIt is positive design constant,

Substitute variable: S₁=y-y_r (41)

S_i=v_0,i-z_i, i=2 ..., n (42)

Wherein, z_iThere is following formula generation,

Wherein,

And

Control law:

Adaptive law: