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

Self-adaptive output feedback inverse control method for piezoelectric precise position platform

Technical Field

The invention belongs to the field of precision machining and manufacturing, and relates to a self-adaptive output feedback inverse control method for a piezoelectric precision position platform.

Background

In the prior art, smart material based actuators are widely used in micro and nano-scale systems, metal cutting systems and other ultra-high precision positioning systems. However, an inevitable disadvantage in smart material based actuators is hysteresis nonlinearity. Because the hysteresis is non-differentiable and multivalued, a control system may exhibit undesirable properties, such as oscillations or even instability, when it lacks compensation for the hysteresis.

One way to deal with hysteresis is to construct an inverse model of the hysteresis and place it as a compensator in the control system before the actuator; another approach is to design a robust adaptation scheme to mitigate the effects of hysteresis without building an inverse model of hysteresis. For the first method, since hysteresis is generally unknown, an analytical error expression between the hysteresis model and its inverse is difficult to establish. In the second method, the unknown hysteresis is divided into linear and non-linear parts, where the non-linear part usually acts as a disturbance. This is why the second approach was devised. However, since the nonlinear part of the hysteresis may be unbounded, which means that the observer error may not have an upper bound, the second approach may not be effective without establishing the hysteresis inverse when the output signal is available. Up to now, it has been difficult to obtain an output feedback control scheme with a hysteresis inverse compensator.

In a piezoelectric precision position platform control system, a realizable robust adaptive output feedback control algorithm cannot be obtained due to the fact that the speed and the acceleration are usually inconvenient to measure.

Disclosure of Invention

The invention aims to provide a piezoelectric precision position platform with measurable output and hysteresis input, which adopts a piezoelectric precision position platform self-adaptive output feedback inverse control method based on output feedback self-adaptive dynamic surface control of an observer, PI model inverse and radial basis function neural network combination, and can ensure that the tracking error of the piezoelectric precision position platform is arbitrarily small_∞All signals in the norm and the closed-loop system are semi-global and finally are bounded, so that the problem of differential explosion in a reverse control scheme is solved, the structure of the controller is simplified, the calculated amount is reduced, and the real-time control is more convenient.

The technical scheme adopted for realizing the aim of the invention is as follows: a piezoelectric precise position platform self-adaptive output feedback inverse control method is characterized by comprising the following steps:

1) piezoelectric precision position platform mathematical model

Consider a class of nonlinear system expressions that add hysteresis:

y＝x₁,i＝0,1,…,n-1 (1)

wherein,is a state vector;unknown smooth linear function, d_i(t) is external interference, b₀Is an unknown constant parameter, w ∈ R is an unknown hysteresis phenomenon, and is expressed as:

w(u)＝P(u(t)) (2)

u is the input signal of the actuator, P is the hysteresis operator,

for system equation (1), the following assumptions are necessary:

a1 interference d_i(t), i ═ 1, ·, n, satisfying:

wherein,are some unknown normal numbers; a2-under the design process, the desired trajectory y_rIs smooth, and y is_r(0) Is obtainable; for all t ≧ 0,belonging to a known tight set; a3: b₀Is known without loss of generality, for convenience, let b be assumed₀＞0；

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

Adopts a PI model suitable for describing a hysteresis model in the piezoelectric actuator and adopts the corresponding model inverse to reduce the influence of hysteresis,

w(t)＝P[u](t) (4)

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

where r is a threshold value, p (r) is that a given density function satisfies p (r) > 0,in the above-mentioned publication, for the sake of convenience,is a constant determined by the density function p (r), Λ represents the upper limit of the integral, let f_rR → R, defined by the formula (6):

f_r(u,w)＝max(u-r,min(u+r,w)) (6)

furthermore, play operator F_r[u](t) satisfies:

wherein i is not less than 0 and not more than N-1, and t is not less than 0₀＜t₁＜…＜t_N＝t_EIs [0, t_E]Such that the function u is at (t)_i,t_i+1]Each subinterval above is monotonic, i.e. non-increasing or non-decreasing,

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

wherein omicron represents a compensation operator; p^-1[·](t) is the inverse compensation operator of the PI model,

whereinIs a constant, represents the upper limit of integration in the formula (9), and,

since in practice hysteresis is not available, which means that the density function p (r) needs to be obtained on the basis of measured data, the inverse model is based on estimating the density functionConstructed on the basis of, usingAs P [ u ]](t), and therefore, by applying the compensation theory to P [. cndot.](t) andobtaining:

wherein,gamma (r), delta (r) is P [ ·](t) andinitial loading curve u_dIs a control signal that is designed to be,

considering equation (11) and inequality F_r[u_d](t)+E_r[u_d](t)＝u_d(t), giving:

w(t)＝φ′(Λ)u_d+d_b(t) (12)

wherein phi' (Λ) Normal, E_r(. is) the stop operator of the PI model, since | E_r(·)|＜Λ，Is bounded and satisfies:

|d_b(t)|≤D (13)

wherein, D is a normal number, and the analysis error e (t) obtained from the formula (11) to the formula (12) is expressed as:

substituting formula (12) for formula (1) to obtain:

wherein, b_ΛIs a normal number and satisfies:

b_Λ＝b₀φ′(Λ) (16)

3) approximation of unknown terms by Radial Basis Function Neural Networks (RBFNNs)

Following lemma 1, a linear Radial Basis Function Neural Network (RBFNNs) of weighting attributes is used to approximate a continuous function in the compact set,

theorem 1 for any given continuous real function f, RBFNNs are global approximators, f: omega_ξ→ R wherein in the reaction mixture,ξ is the input to the neural network and q is the input dimension. For an arbitrary epsilon_m> 0, by appropriate selection of σ and ζ_k∈R^qK is 1, …, N, after which there is one RBFNN such that:

f(ξ)＝ψ^T(ξ)θ^*+ε (17)

wherein theta is^*Is θ ═ θ₁,…,θ_N]∈R^NThe optimal weight vector is defined as:

wherein Y (xi) ═ ψ^T(xi) theta denotes an output from RBFNNs, and ψ (xi) ═ ψ₁(ξ),…,ψ_N(ξ)]∈R^NIs a vector of basis functions, usually the so-called gaussian function is generally taken as a basis function in the form:

where σ > 0, k ═ 1, …, N,is a constant vector, called the center of the basis function; σ is a real number, called the width of the basis function, ε is the approximation error, and satisfies:

ε＝f(ξ)-θ^*Tψ(ξ) (20)

using theorem 1 and equation (17), RBFNNs as approximators to the unknown continuous function in equation (17), equation (21):

wherein epsilon_iN is any constant number, i is 1 …, represents the neural network approximation error, and,

whereinIs a state variable x₁,…,x_iAnd is introduced in equation (40),

substituting formula (21) for formula (15) to obtain:

system equation (1) is expressed in the following state space form:

wherein, b ═ 0, … 0, 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∈RⁿIs a term of interference-like type,

wherein,defined in equation (19), for a class of hysteretic nonlinear systems, the goal of the control is to establish an adaptive neural network-based output feedback dynamic surface control scheme that is compatible with the L of the tracking error_∞The norm is consistent, and the output signal y can well track the reference signal y_rAnd all signals of the closed loop system are consistently bounded;

4) observer-based adaptive dynamic surface inverse compensator design

High gain Kalman filter observer

Converting formula (24) to formula (27):

order to

A₀＝A-qe₁ ^T (28)

Wherein q is [ q ]₁,…,q_n]By appropriate selection of the vector q for A₀Is a matrix of a Huviz matrix,

a high gain kalman filter is constructed to estimate the state variable x in equation (27),

where k ≧ 1 is a positive design parameter, e_nIs represented by the formulaⁿA vector of coordinates of order n in form, and,

Φ＝diag{1,k,…,k^n-1} (32)

from equation (29) to equation (32), the state vector is estimated as follows,

further, the estimation error is defined,

then, the obtained product is processed by the following steps,

wherein epsilon₁Is a first term of epsilon, B being defined in formula (25);

2, leading: let the high-gain kalman filter be defined by equation (29) -equation (31) and a second order function,

V_ε:＝ε^TPε (36)

wherein,is a positive definite matrixAnd satisfies the following conditions:

wherein A is₀Defined by equation (28), let:

wherein | B | Y calculation_maxIs the maximum value of B. For any k ≧ 1, the derivation of equation (36) yields:

due to b in the formula (33)_ΛAnd theta^*It is not known that the user is,are not available, so the actual state estimate is:

wherein,andis b_ΛAnd theta^*Is determined by the estimated value of (c),

the improved high-gain Kalman filter is used for processing the bounded B term defined in the formula (25), and the observation error epsilon can be made arbitrarily small by properly selecting the design parameters k ≧ 1 in the formulas (29) to (31) and the matrix phi defined in the formula (32);

design of dynamic surface inverse controller

The design of the controller includes alternative variables, control laws and adaptation laws, where₂,…,τ_nIs the time constant of the low-pass filter,/_iI-1, …, n and y_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bIs a positive design constant that is constant for the design,

substitution variables: s₁＝y-y_r (T.1)

S_i＝v_0,i-z_i,i＝2,…,n (T.2)

Wherein z is_iThere is the following formula to produce,

wherein,

and is

Control law:

adaptive law:

the invention discloses a self-adaptive output feedback inverse control method of a piezoelectric precise position platform, which has the advantages that:

1) the designed high-gain state observer is used for estimating the state of the system and processing uncertain items in the system and environmental interference, so that compared with a state feedback control algorithm, the proposed control algorithm is more suitable for practical application only under the condition that the output of the control system is required to be obtained;

2) tests were performed on a piezo-electric precision position stage by applying the proposed control scheme, wherein the piezo-electric precision position stage can be considered as a three-order system where the output of the system alone is measurable;

3) by adjusting the initial conditions of the state observer and the adaptive law of unknown parameters, it is possible to achieve an arbitrarily small L following error_∞The norm solves the problem of differential explosion in a reverse control scheme, simplifies the structure of the controller, reduces the calculated amount and is more convenient for real-time control.

4) The method is scientific and reasonable, and has strong applicability and good effect.

Drawings

FIG. 1 is a diagram of PI model (a), PI inverse model (b) and compensation result (c);

FIG. 2 is a schematic diagram of an inverse compensation scheme;

FIG. 3 is a schematic diagram of the adaptive output feedback inversion control method for a piezoelectric precision position stage according to the present invention;

FIG. 4 is a schematic structural diagram of an actual piezoelectric precision position stage control system;

FIG. 5 is a general schematic diagram of a piezoelectric precision position stage;

FIG. 6 is a schematic diagram of input-output response comparison between a piezoceramic actuator and a PI model;

FIG. 7 is a schematic of modeling errors;

FIG. 8 is a schematic diagram of simulation and experimental inverse compensation results (3 μm);

FIG. 9 is a schematic diagram of simulation and experimental inverse compensation results (5 μm);

FIG. 10 shows the actual displacement output y and the desired trajectory y_rA schematic diagram;

FIG. 11 is a schematic diagram of tracking error with and without hysteresis compensation;

FIG. 12 is a schematic of control voltages with and without hysteresis compensation;

FIG. 13 is a schematic diagram of tracking errors of the proposed dynamic surface method and the conventional back-extrapolation method;

FIG. 14 is a displacement diagram of the proposed dynamic surface method and the conventional back-extrapolation method;

fig. 15 is a schematic diagram of control voltages of the proposed dynamic surface method and the conventional back-extrapolation method.

In fig. 1, the ordinate represents the hysteresis output and the abscissa represents the hysteresis input; in FIG. 5, k_ampIs a fixed gain; r₀Is the equivalent internal resistance of the drive circuit; v. of_hDue to the voltage, H and T, generated by hysteresis effects_emRepresents the piezoelectric effect; c_ARepresents the sum of all piezoelectric ceramic capacitances; q andrespectively, all charges in the PCA and the current flowing through the circuit, q_cIs stored in a linear capacitor C_AOf the charge of (1). q. q.s_pRepresenting the conduction charge, v, generated from the machine side due to the piezoelectric effect_ARepresenting the conduction voltage, m, b for the mechanical part_sAnd k_sRespectively representing the mass, the damping coefficient and the rigidity of the motion mechanism; the abscissa in the figure represents the hysteresis input,the ordinate represents the hysteresis output; FIG. 4 shows time on the abscissa and output y on the ordinate versus an objective function y_rTracking performance of (2); FIG. 6 shows time on the abscissa and displacement on the ordinate; FIG. 7 shows time in abscissa and modeling error (%) in ordinate; FIGS. 8 and 9 show the desired displacement on the abscissa and the actual displacement on the ordinate; FIG. 10 shows time on the abscissa and displacement on the ordinate; FIG. 11 shows time on the abscissa and tracking error on the ordinate; FIG. 12 shows time on the abscissa and control voltage on the ordinate; FIG. 13 shows time on the abscissa and tracking error on the ordinate; FIG. 14 shows time on the abscissa and displacement on the ordinate; fig. 15 shows time on the abscissa and control voltage on the ordinate.

Detailed Description

The invention is further illustrated by the following figures and examples.

The invention discloses a self-adaptive output feedback inverse control method of a piezoelectric precise position platform, which comprises the following steps:

1) piezoelectric precision position platform mathematical model

Consider a class of nonlinear system expressions that add hysteresis:

y＝x₁,i＝0,1,…,n-1 (1)

wherein,is a state vector;unknown smooth linear function, d_i(t) is external interference, b₀Unknown constant parameters, w ∈ R unknown hysteresis, expressed as:

w(u)＝P(u(t)) (2)

u is the input signal of the actuator, P is the hysteresis operator,

for system equation (1), the following assumptions are necessary:

a1 interference d_i(t), i ═ 1, ·, n, satisfying:

wherein,are some unknown normal numbers; a2-under the design process, the desired trajectory y_rIs smooth, and y is_r(0) Is obtainable; for all t ≧ 0,belonging to a known tight set; a3: b₀Is known without loss of generality, for convenience, let b be assumed₀＞0；

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

Adopts a PI model suitable for describing a hysteresis model in the piezoelectric actuator and adopts the corresponding model inverse to reduce the influence of hysteresis,

w(t)＝P[u](t) (4)

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

where r is a threshold value, p (r) is that a given density function satisfies p (r) > 0,in the above-mentioned publication, for the sake of convenience,is a constant determined by the density function p (r), Λ represents the upper limit of the integral, let f_rR → R, defined by the formula (6):

f_r(u,w)＝max(u-r,min(u+r,w)) (6)

furthermore, play operator F_r[u](t) satisfies:

wherein, t_i＜t≤t_i+1，0≤i≤N-1，0＝t₀＜t₁＜…＜t_N＝t_EIs [0, t_E]Such that the function u is at (t)_i,t_i+1]Each subinterval above is monotonic, i.e. non-increasing or non-decreasing,

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

wherein omicron represents a compensation operator; p^-1[·](t) is the inverse compensation operator of the PI model,

whereinIs a constant, represents the upper limit of integration in the formula (9), and,

since in practice hysteresis is not available, which means that the density function p (r) needs to be obtained on the basis of measured data, the inverse model is based on estimating the density functionConstructed on the basis of, usingAs P [ u ]](t), and therefore, by applying the compensation theory to P [. cndot.](t) andobtaining:

wherein,gamma (r), delta (r) is P [ ·](t) andinitial loading curve u_dIs a control signal that is designed to be,

considering equation (11) and inequality F_r[u_d](t)+E_r[u_d](t)＝u_d(t), giving:

w(t)＝φ′(Λ)u_d+d_b(t) (12)

where phi' (Λ) is a normal number, E_r(. is) the stop operator of the PI model, since | E_r(·)|＜Λ,Is bounded and satisfies:

|d_b(t)|≤D (13)

wherein, D is a normal number, and the analysis error e (t) obtained from the formula (11) to the formula (12) is expressed as:

substituting formula (12) for formula (1) to obtain:

wherein, b_ΛIs a normal number and satisfies:

b_Λ＝b₀φ′(Λ) (16)

3) approximation of unknown terms by Radial Basis Function Neural Networks (RBFNNs)

Following lemma 1, a linear Radial Basis Function Neural Network (RBFNNs) of weighting attributes is used to approximate a continuous function in the compact set,

theorem 1 for any given continuous real function f, RBFNNs are global approximators, f: omega_ξ→ R wherein in the reaction mixture,ξ is the input to the neural network and q is the input dimension. For an arbitrary epsilon_m> 0, by appropriate selection of σ and ζ_k∈R^qK is 1, …, N, after which there is one RBFNN such that:

f(ξ)＝ψ^T(ξ)θ^*+ε (17)

wherein theta is^*Is θ ═ θ₁,…,θ_N]∈R^NThe optimal weight vector is defined as:

wherein Y (xi) ═ ψ^T(xi) theta denotes an output from RBFNNs, and ψ (xi) ═ ψ₁(ξ),…,ψ_N(ξ)]∈R^NIs a vector of basis functions, usually the so-called gaussian function is generally taken as a basis function in the form:

wherein, sigma is more than 0, k is 1, …, N, zeta_k∈RⁿIs a constant vector called the center of the basis function. σ is a real number, called the width of the basis function, ε is the approximation error, and satisfies:

ε＝f(ξ)-θ^*Tψ(ξ) (20)

using theorem 1 and equation (17), RBFNNs as approximators to the unknown continuous function in equation (17), equation (21):

wherein epsilon_iN is any constant number, i is 1 …, represents the neural network approximation error, and,

whereinIs a state variable x₁,…,x_iAnd is introduced in equation (40),

substituting formula (21) for formula (15) to obtain formula (23):

system equation (1) is expressed in the following state space form:

wherein, b ═ 0, … 0, 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∈RⁿIs a term of interference-like type,

wherein,defined in equation (19), for a class of hysteretic nonlinear systems, the goal of the control is to establish an adaptive neural network-based output feedback dynamic surface control scheme that is compatible with the L of the tracking error_∞The norm is consistent, and the output signal y can well track the reference signal y_rAnd all signals of the closed loop system are consistently bounded;

4) observer-based adaptive dynamic surface inverse compensator design

High gain Kalman filter observer

Converting formula (24) to formula (27):

order to

A₀＝A-qe₁ ^T (28)

Wherein q is [ q ]₁,…,q_n]By appropriate selection of the vector q for A₀Is a matrix of a Huviz matrix,

a high gain kalman filter is constructed to estimate the state variable x in equation (27),

where k ≧ 1 is a positive design parameter, e_nIs represented by the formulaⁿA vector of coordinates of order n in form, and,

Φ＝diag{1,k,…,k^n-1} (32)

from equation (29) -equation (32), the state vector is estimated as follows,

further, the estimation error is defined,

then, the result of equation (34) is obtained,

wherein epsilon₁Is the first value of epsilon, B is defined in formula (25);

2, leading: let the high-gain kalman filter be defined by equation (29) -equation (31) and a second order function,

V_ε:＝ε^TPε (36)

wherein,is a positive definite matrixAnd satisfies the following conditions:

wherein A is₀Defined by equation (28), let:

wherein | B | Y calculation_maxIs the maximum value of B. For any k ≧ 1, pairEquation (36) is derived:

due to b in the formula (33)_ΛAnd theta^*It is not known that the user is,are not available, so the actual state estimate is:

wherein,andis b_ΛAndis determined by the estimated value of (c),

the improved high-gain Kalman filter is used for processing the bounded B term defined in the formula (25), and the observation error epsilon can be made arbitrarily small by properly selecting the design parameter k ≧ 1 in the formula (29) -formula (31) and the matrix phi defined in the formula (32);

design of dynamic surface inverse controller

The design of the controller includes alternative variables, control laws and adaptation laws, where₂,…,τ_nIs the time constant of the low-pass filter,/_iI-1, …, n and y_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bIs a positive design constant that is constant for the design,

substitution variables: s₁＝y-y_r (T.1)

S_i＝v_0,i-z_i,i＝2,…,n (T.2)

Wherein z is_iThere is the following formula to produce,

wherein,

and is

Control law:

adaptive law:

1. stability index analysis

In this section, a discussion will be made of stability and performance analysis of the proposed adaptive output feedback DSIC scheme to analyze the L of stability and tracking error_∞And (4) performance.

To control the system stability analysis, the following lyapunov function is defined:

wherein,V_εis a quadratic function on the high gain kalman filter observation error epsilon, which is given in lemma 2.

Theorem 1: considering such a closed loop system, which includes the unknown parameter adaptation law in the hysteresis nonlinear time lag system (1), (t.9) - (t.11) described by equation (4), the control law (T8) is related to the assumptions a 1-A3. Then, for any given positive number p, if V (0) in equation (41) satisfies V (0). ltoreq.p,

a) by appropriate selection of the design parameters k, l₁,…,l_nTime constant τ₂,…,τ_nAdaptive law parameter gamma_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bAll signals of the closed loop system are bounded and can be arbitrarily small.

b) Tracking error S₁L of_∞Performance can be obtained and is arbitrarily small, and formula (42) can be obtained by combining formula (T.5) with formula (41):

whereinC₁Is a normal number and satisfies

2. Experimental study of piezoelectric precision position platform

A. Experimental device

To demonstrate the effectiveness of the proposed control scheme, an experimental study of the piezoelectric precision position stage shown in fig. 4 was conducted. The components of the control system are as follows:

piezoelectric ceramic actuator: a piezoceramic actuator P-753.31C from Physik Instrument was used for the experiments. It provides a 38 μm peak output displacement, for an actuator, in the voltage range 0-100V.

Capacitive sensors: to measure the displacement response of the actuator, an integrated capacitive sensor with a sensitivity of 2.632V/μm was used.

Voltage amplifier: a voltage amplifier (LVPZT, E-505) with a fixed gain of 10 was used as the excitation voltage for the piezoelectric actuator.

Data acquisition system: a dSPACE control board with a 16-bit analog-to-digital, digital-to-analog converter was used to obtain the displacement of the piezoelectric precision position stage, which was measured by a capacitive sensor.

B. Model of piezoelectric precision position platform

From the accumulated modeling results of the piezo-electric precision position stage, a general schematic model of the piezo-electric precision position stage can be represented by fig. 5, which is composed of a piezo-electric ceramic actuator (PCA)) and a mechanical part. In FIG. 5, k_ampIs a fixed gain; r₀Is the equivalent internal resistance of the drive circuit; v. of_hIs the voltage due to hysteresis effects. H and T_emRepresents the piezoelectric effect; c_ARepresents the sum of all piezoelectric ceramic capacitances; q andrespectively, represent all charges in the PCA and thus the current flowing through the circuit. q. q.s_cIs stored in a linear capacitor C_AOf the charge of (1). q. q.s_pRepresenting the conduction charge, v, generated from the machine side due to the piezoelectric effect_ARepresenting the conduction voltage. For the mechanical part, m, b_sAnd k_sRespectively representing the mass, the damping coefficient and the stiffness of the moving mechanism. Based on the schematic model of the piezoelectric precise position platform, a third-order model describing the piezoelectric precise position platform is defined as follows:

wherein w ═ Φ' (Λ) u given in equation (12)_d+d_b(t) is the output of the piezoelectric actuator, and

C. experiments on hysteresis modeling and its inverse compensator construction

To facilitate parameter identification for the model described in equation (5), the corresponding discrete expression in equation (5) is as follows:

to obtain the optimal play operator p_iI-1, 2,3, …, n, to describe hysteresis in the piezoceramic actuator P-753.31 in fig. 4, in combination with constrained quadratic optimization as follows:

min{[CΛ-d]^T[CΛ-d]} (45)

wherein C constant, d is a sinusoidal signal, and Λ (i) satisfies

Λ(i)≥0,i∈{1,2,3,…,n} (46)

Identifying p with least squares optimization toolkit in MATLAB by using experimental data_i. These data were obtained under a designed amplitude reduced sinusoidal input signal d based on the actual piezoelectric precision position stage control system of figure 4. Since modeling errors are unavoidable, we can only obtain p_iIs estimated from 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 thatComparison of input-output response between piezoceramic actuators (dashed line) and PI model (solid line). The modeling error shown in FIG. 7 is defined as follows:

where x (t) and w (t) represent the outputs of the piezoceramic actuator and PI model, respectively. The comparison results and modeling errors (less than 1%) indicate that the PI model does match the experimental data well.

Using the identification parameter of the PI model and the analysis inverse of equation (10)The values that can be expressed as discrete expressions are implemented as follows:

and isTherefore, the temperature of the molten metal is controlled,the threshold and weight are calculated as follows: to demonstrate the effectiveness of the inverse compensator established in equation (48), experiments were conducted by converting the code in MATLAB/SIMULINK to real-time code via the dSPACE module shown in FIG. 4. An input signal u_d(t)＝B₁sin (2 π ft), where B₁3 μm,5 μm, and f 1Hz, applied to a compensator, and the output applied to a piezo ceramic via a power amplifier (LVPZT, E-505)The line driving device.

The displacement response of the actuator is then monitored and measured by the sensor of fig. 4 and downloaded to the dSPACE module. For comparison of the simulation results with the experimental results, the simulation of the inverse compensator was also performed in MATLAB/Simulink. Fig. 8 shows a comparison of simulation results with experimental results when the hysteresis inverse compensator of fig. 2 is applied. As can be seen from the simulation results in fig. 8 and 9, there is a perfect linear input-output relationship between the expected displacement and the output displacement, which implies that the effect of hysteresis is completely cancelled. However, there is an inverse compensation error, which is clearly shown in the experimental results of fig. 8 and 9. In fact, the compensation error may be caused by modeling errors and environmental disturbances. From FIG. 8 (B)₁3 μm) the input and output are still hysteretic, indicating that hysteresis cannot be completely eliminated. Therefore, the proposed output feedback adaptive control scheme is applied to reduce the compensation error.

Now, let x₁＝x,Then, formula (43) can be expressed as follows:

where w represents the hysteresis nonlinearity in the piezoceramic actuator shown in fig. 5.

D. Design procedure of controller and experimental results

Part of the observer-based adaptive dynamic inverse compensator design, in this experiment, the high-gain K-filter was designed according to equations (29) to (31), where v (0) ═ ξ₀(0)＝Ξ(0)＝0,k＝1.5,q＝[3,2,1]^T. For neural network system psi₃(ξ₃) We have chosen 5 nodes, ζ, with a basis function center_jJ is 1, …,5, and has the size of-0.5, 0,1,2,3,4,5,6, width eta_j＝1,j＝1,…,7；Then Ψ^T(ξ)＝diag{0,0,ψ₃Therein ψ₃＝[ψ_3,1(ξ₁),…,ψ_3,7(ξ₁)]. According to section III, the dynamic surface error is S₁＝x₁-y_r，S₂＝v_0,2-z₂，S₃＝v_0,3-z₃. Is adaptive to law ofThe virtual control law is selected asWherein The final control law is selected asThe first order filter in steps 1 and 2 isWherein,correspondingly, the design parameters of the adaptive law and control signal are 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_b0.9. The initial conditions of the adaptation law are selected as

To verify the effectiveness of the proposed control scheme, the following two tests were performed on a piezoelectric precision position stage. For the purpose of real-time control, the improved adaptive output feedback control algorithm is converted into an S function file with the sampling frequency of 10kHz and compiled by C language on a dSPACE control panel.

A. Multi-frequency trajectory tracking test

In this experiment, the trajectory y is expected for multiple frequencies in two cases_rMotion tracking control was performed at 3+2sin (2 pi × 2t) + sin (2 pi × 15 t): with and without hysteresis compensation as constructed in equation (48). The results of the experiment are shown in FIGS. 10-15. FIG. 10 illustrates the actual displacement y (solid line) and the desired trajectory y of the piezoceramic actuator_r(dotted line). It can be seen that a quite satisfactory tracking performance is achieved, the tracking error of which is extremely small. Fig. 11 shows tracking error with and without hysteresis compensation. Fig. 11 clearly shows that the transient and steady state tracking errors are smaller with the addition of hysteresis compensation than without the use of hysteresis compensation. For example, if error is used_max＝max(|y-y_r|) to represent the maximum value of the tracking error, error without hysteresis compensation_maxIs 0.014 μm, while the error is the case with the addition of hysteresis_maxOnly 0.0059 μm, is half of the case without hysteresis compensation. Furthermore, due to good tracking performance and minimal tracking error, fig. 10 and 11 fully verify the effectiveness of the proposed output feedback control scheme with the hysteresis compensator in equation (48). Fig. 12 shows a locus of the control voltage, the solid line shows that hysteresis compensation is added, and the dotted line shows that no hysteresis compensation is added. It should be noted that the control scheme without hysteresis compensation is a case where "hysteresis inverse model estimation" is not included in fig. 3. Then, u is equal to u_dAnd the piezoelectric precision position platform system is controlled by the control signal u without any compensation_dDirectly driven.

B. Experimental study of single-frequency trajectory tracking and back-stepping control comparison

To demonstrate the advantages of the proposed dynamic surface control, we compared the dynamic surface control scheme with the back-thrust control scheme. FIGS. 12-14 show the desired trajectory y_rExperimental results of 2sin (40 tt). FIG. 12 shows the proposedTracking error of dynamic surface control scheme, its steady state error_maxThe steady state error of the traditional reverse control method is error which is 0.0158 mu m_max0.0647 μm. Fig. 14 and 15 show the displacement and control voltage, respectively, under two methods. From these results, it is apparent that the dynamic surface method exhibits better control performance than the conventional back-thrust control method.

The description is given for the sake of illustration only, and is not to be construed as limiting the scope of the claims, which are intended to be covered by the appended claims without undue experimentation and equivalents thereof by those skilled in the art.

Claims (1)

1. A piezoelectric precise position platform self-adaptive output feedback inverse control method is characterized by comprising the following steps:

1) piezoelectric precision position platform mathematical model

Consider a class of nonlinear system expressions that add hysteresis:

y＝x₁,i＝0,1,…,n-1 (1)

wherein,is a state vector;unknown smooth linear function, d_i(t) is external interference, b₀Unknown constant parameters, w ∈ R unknown hysteresis output, expressed as:

w(u)＝P(u(t)) (2)

u is the input signal of the actuator, P is the hysteresis operator,

for system equation (1), the following assumptions are necessary:

a1 external disturbance d_i(t), i ═ 1, …, n, satisfying:

wherein,are some unknown normal numbers; a2-under the design process, the desired trajectory y_rIs smooth, and y is_r(0) Is obtainable; for all t ≧ 0,belonging to a known tight set; a3: b₀Is known without loss of generality, for convenience, let b be assumed₀＞0；

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

Adopts a PI model suitable for describing a hysteresis model in the piezoelectric actuator and adopts the corresponding model inverse to reduce the influence of hysteresis,

w(t)＝P[u](t) (4)

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

where r is a threshold value, p (r) is a given density function, p (r) > 0 is satisfied,for the sake of convenience in the art,is a constant determined by the density function p (r), Λ represents the upper limit of the integral, let f_rR → R, defined by the formula (6):

f_r(u,w)＝max(u-r,min(u+r,w)) (6)

furthermore, play operator F_r[u](t) satisfies:

wherein, t_i＜t≤t_i+1，0≤i≤N-1，0＝t₀＜t₁＜…＜t_N＝t_EIs [0, t_E]Such that the function u is at (t)_i,t_i+1]Each subinterval above is monotonic, i.e. non-increasing or non-decreasing,

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

wherein,representing a compensation operator; p^-1[·](t) is the inverse compensation operator of the PI model,

whereinIs a constant, represents the upper limit of integration in the formula (9), and,

since in practice hysteresis is not available, which means that the density function p (r) needs to be obtained on the basis of measured data, the inverse model is based on estimating the density functionConstructed on the basis of, usingAs P [ u ]](t), and therefore, by applying the compensation theory to P [. cndot.](t) andobtaining:

wherein,gamma (r), delta (r) is P [ ·](t) andinitial loading curve u_dIs a control signal that is designed to be,

considering equation (11) and inequality F_r[u_d](t)+E_r[u_d](t)＝u_d(t), giving:

w(t)＝φ′(Λ)u_d+d_b(t) (12)

where phi' (Λ) is a normal number, E_r(. is) the stop operator of the PI model, since | E_r(·)|＜Λ, Is bounded and satisfies:

|d_b(t)|≤D (13)

wherein, D is a normal number, and the analysis error e (t) obtained from the formula (11) to the formula (12) is expressed as:

substituting formula (12) for formula (1) to obtain:

wherein, b_ΛIs a normal number and satisfies:

b_Λ＝b₀φ′(Λ) (16)

3) approximation of unknown terms by Radial Basis Function Neural Networks (RBFNNs)

Following lemma 1, a linear Radial Basis Function Neural Network (RBFNNs) of weighting attributes is used to approximate a continuous function in the compact set,

theorem 1 for any given continuous real function f, RBFNNs are global approximators, f: omega_ξ→ R wherein in the reaction mixture,ξ is the input to the neural network and q is the input dimension for an arbitrary ε_m> 0, by appropriate selection of σ and ζ_k∈R^qK is 1, …, N, after which there is one RBFNN such that:

f(ξ)＝ψ^T(ξ)θ^*+ε (17)

|ε|≤ε_mwherein theta^*Is θ ═ θ₁,…,θ_N]∈R^NThe optimal weight vector is defined as:

wherein Y (xi) ═ ψ^T(xi) theta denotes an output from RBFNNs, and ψ (xi) ═ ψ₁(ξ),…,ψ_N(ξ)]∈R^NIs a vector of basis functions, usually the so-called gaussian function is generally taken as a basis function in the form:

wherein, sigma is more than 0, k is 1, …, N, zeta_k∈RⁿIs a constant vector, called the center of the basis function; σ is a real number, called the width of the basis function, ε is the approximation error, and satisfies:

ε＝f(ξ)-θ^*Tψ(ξ) (20)

using theorem 1 and equation (17), RBFNNs as approximators to the unknown continuous function in equation (17), equation (21):

wherein epsilon_iN is any constant number, i is 1 …, represents the neural network approximation error, and,

whereinIs a state variable x₁,…,x_iAnd is introduced in equation (40),

substituting formula (21) for formula (15) to obtain:

system equation (1) is expressed in the following state space form:

wherein, b ═ 0, … 0, 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∈RⁿIs a term of interference-like type,

wherein,defined in equation (19), for a class of hysteretic nonlinear systems, the goal of the control is to establish an adaptive neural network-based output feedback dynamic surface control scheme that is compatible with the L of the tracking error_∞The norm is consistent, and the output signal y can well track the reference signal y_rAnd all signals of the closed loop system are consistently bounded;

4) observer-based adaptive dynamic surface inverse compensator design

High gain Kalman filter observer

Converting formula (24) to formula (27):

order to

A₀＝A-qe₁ ^T (28)

Wherein q is [ q ]₁,…,q_n]By appropriate selection of the vector q for A₀Is a matrix of a Huviz matrix,

a high gain kalman filter is constructed to estimate the state variable x in equation (27),

where k ≧ 1 is a positive design parameter, e_nIs represented by the formulaⁿA vector of coordinates of order n in form, and,

Φ＝diag{1,k,…,k^n-1} (32)

from equation (29) to equation (32), the state vector is estimated as follows,

further, the error of observation is defined,

then, the result of equation (34) is obtained,

wherein,is thatB is as defined in formula (25);

2, leading: let the high-gain kalman filter be defined by equation (29) -equation (31) and a second order function,

wherein, is a positive definite matrixAnd satisfies the following conditions:

wherein A is₀Defined by equation (28), let:

wherein | B | Y calculation_maxIs the maximum value of B, for any k greater than or equal to 1, the derivation of equation (36) is obtained:

due to b in the formula (33)_ΛAnd theta^*It is not known that the user is,are not available, so the actual state estimate is:

wherein,andis b_ΛAnd theta^*Is determined by the estimated value of (c),

the improved high gain Kalman filter is used to process the bounded B term defined in equation (25), and the observation error can be made by properly selecting the design parameters k ≧ 1 in equations (29) through (31) and the matrix Φ defined in equation (32)Is arbitrarily small;

design of dynamic surface inverse controller

The design of the controller includes alternative variables, control laws and adaptation laws, where₂,…,τ_nIs the time constant of the low-pass filter,/_iI-1, …, n and y_θ,σ_θ,γ_ζ,σ_ζ,γ_b,σ_bIs a positive design constant that is constant for the design,

substitution variables: s₁＝y-y_r (41)

S_i＝v_0,i-z_i,i＝2,…,n (42)

Wherein z is_iThere is the following formula to produce,

wherein,

and is

Control law:

adaptive law: