Self-adaptive predefined time preset performance control method for piezoelectric micro-positioning platform

Abstract

The invention discloses a self-adaptive predefined time preset performance control method for a piezoelectric micro-positioning platform, and the method comprises the steps: building a mathematical model according to the characteristics of the piezoelectric micro-positioning platform; then, a preset performance function is utilized to ensure that tracking errors are converged to any set set, and a constraint boundary adjustment mechanism is designed for the preset performance function to solve the problem of control output singularity during input saturation; thirdly, calculating a virtual error, and constructing a virtual control law, an auxiliary control law and a self-adaptive updating law in combination with a backstepping technology, a finite time sliding mode differentiator, an RBFNN and a self-adaptive method; finally, a performance controller is preset through the self-adaptive predefined time, the Lyapunov stability theory and the practical predefined time stability criterion are combined, appropriate design parameters are selected, and it is guaranteed that the closed-loop system is stable within the predefined time, and the tracking error is stabilized within the expected error range. The problems that in an existing control method, the hysteresis model building process is complex, and convergence time is long are solved.

Description

Technical Field

[0001] The present invention belongs to the technical field of tracking control of piezoelectric micro-positioning platforms, and more specifically, relates to an adaptive predefined time preset performance control method of a piezoelectric micro-positioning platform. Background Art

[0002] Piezoelectric micro-positioning platforms have the advantages of high resolution, fast response speed, and high tracking accuracy. They have been widely used in precision manufacturing fields such as atomic force microscopes, aerospace equipment, and ultra-precision machine tools. However, in practical applications, piezoelectric micro-positioning platforms are affected by many unfavorable factors. The main influencing factor is the inherent hysteresis nonlinearity of piezoelectric materials. How to eliminate the adverse effects of hysteresis nonlinearity to improve the tracking accuracy and dynamic performance of the platform has attracted great attention.

[0003] At present, hysteresis is mainly handled by the hysteresis inverse compensation method. This method requires finding an inverse model to offset the hysteresis nonlinearity, which requires first establishing a suitable hysteresis model. However, due to the influence of environmental factors, creep and frequency dependence, the process of establishing an accurate hysteresis model is very complicated.

[0004] It is worth noting that most of the control methods applied to piezoelectric micro-positioning platforms are discussed under asymptotically stable conditions, and theoretically require an infinitely long time to converge to the vicinity of the equilibrium point. Some scholars have adopted finite-time control methods to ensure that the system converges in finite time and improve the control performance. However, the convergence time of the finite-time control method depends on the initial conditions of the system. Fixed-time control can get rid of the constraints of initial conditions on the convergence time and is a more realistic control method. Although fixed-time control has obvious advantages over finite-time control, its upper bound on convergence time is affected by many control parameters, and it is difficult to obtain a clear upper bound on the convergence time. Therefore, it is of great application value to design a predefined time controller whose convergence time does not depend on the initial conditions and control parameters of the system.

[0005] In the practical application of piezoelectric micro-positioning platforms, it is often necessary to explicitly constrain the tracking error. Preset performance functions are an effective tool and have been widely used in the controller design of practical systems. A problem with traditional preset performance functions is that they cannot handle the peak tracking error caused by sudden disturbances and input saturation. Obviously, these factors will increase the tracking error. When the tracking error reaches or even exceeds the constraint boundary of the preset performance function, a larger control signal is usually required to return the tracking error to the constraint boundary. However, the drive module in the piezoelectric micro-positioning platform system may not be able to provide enough voltage, which makes it difficult to keep the tracking error within the constraint boundary, and even causes the controller output to become singular. Therefore, it is of practical value to design a preset performance function that takes into account the input saturation problem.

[0006] In summary, how to take the input saturation problem into consideration in the controller design of the piezoelectric micro-positioning platform, and thus design an effective control strategy so that the tracking error of the system falls within the desired error range within a predefined time, while avoiding complex modeling of hysteresis nonlinearity to achieve high-performance tracking effects, has practical engineering significance. Summary of the invention

[0007] The present invention proposes an adaptive predefined time preset performance control method for a piezoelectric micro-positioning platform to overcome the problems existing in the prior art. Note: (·) represents the first-order derivative of a physical quantity, (^) represents the estimated value of a physical quantity, and (~) represents the estimation error;

[0008] The specific implementation steps of the technical solution of the present invention are as follows:

[0009] Step 1: The piezoelectric micro-positioning platform can be regarded as a spring-damper-mass system with input hysteresis; considering the influence of interference and uncertainty in practical applications, the mathematical model of the piezoelectric micro-positioning platform can be established as follows:

[0010]

[0011] Where c is the control gain with unknown magnitude but known sign. Considering the actual system, assume that c>0 and there is an upper bound Make f(x) is an unknown continuous function; x=[x1,x2] T ∈R 2 , y∈R represent the state and output of the system respectively; u is the system input; ω(u) is the input with hysteresis information, where H is the hysteresis term; d is the external interference. Considering the actual situation, it is assumed that there is a constant Make

[0012] Use radial basis neural network (RBFNN) to approximate the unknown continuous function f(x); the expression of f(x) is as follows:

[0013] f(x)=θ *T φ(x)+ε (2)

[0014] Where ε is the approximation error of the neural network, and there is a positive constant Make φ(x)=[φ1(x),φ2(x),...,φ m (x)] is the Gaussian basis function vector, where φ j (x) = exp(-||xc j || 2 / (2b j2 )),j=1,...,m, m is the number of hidden layer nodes, b j ,c j are the Gaussian basis function width and center vector of the jth node in the hidden layer; θ * is the ideal weight of the neural network;

[0015] The mathematical model of formula (1) can be re-expressed as:

[0016]

[0017] where c0 and k c is the design constant, c1 is the constant that makes the equation c=c0+k c c1 is the unknown constant for which it holds true;

[0018] Considering the input saturation problem of piezoelectric positioning system in practical application, the input saturation is described as:

[0019]

[0020] where u r is the actual voltage driving the piezoelectric micro-positioning platform; u h and u l are the upper and lower bounds of the saturation boundary, respectively.

[0021] Step 2: Using the preset performance function can ensure that the tracking error converges to an arbitrary set; introducing the constraint boundary adjustment mechanism into the preset performance function avoids the problem of controlling the output singularity when the input is saturated;

[0022] The expression of the prescribed performance function with the constraint boundary adjustment mechanism is:

[0023]

[0024] where υ ∞ <1 is a positive design constant that affects the tracking error range after the system stabilizes; e is a natural constant; T Γ ,c Γ is a positive design constant that affects the stability time and convergence speed of the system; t k is the boundary adjustment time, and t k (0) = 0;

[0025] The constraint boundary adjustment mechanism is:

[0026]

[0027] Where K is the number of times the boundary adjustment mechanism is triggered; △u e =|uu r|; In addition, Γ(t) also has the following properties: 1) For t>t k , Γ(t) is a bounded and time-differentiable function, and bounded; 2)Γ(0)=1, and when t≥(Τ Γ +t k ) when Γ(t)=υ ∞ ;

[0028] Define the system tracking error as e1, and its expression is as follows:

[0029] e1=x1-y d (7)

[0030] Where x1 is the actual output displacement of the piezoelectric micro-positioning platform; y d is the reference signal;

[0031] Since the initial state of the piezoelectric micro-positioning platform is unpredictable and variable, it is necessary to normalize the error. The Softsign function is used for normalization to relax the restrictions on the initial state of the system. The expression of the Softsign function is as follows:

[0032]

[0033] in is a normal constant set; the properties of this function are: 1) the range of Ψ(e1) is (-1,1), and Ψ(0)=0; 2) by It can be obtained that Ψ(e1) is monotonically increasing and differentiable; 3) The inverse function of Ψ(e1) is Ψ -1 (Γ(0))=∞.

[0034] Step 3: Based on the mathematical model of the piezoelectric micro-positioning platform in step 1 and the preset performance function in step 2, the virtual error is calculated, and the virtual control law, auxiliary control law and adaptive law are designed by combining the finite time sliding mode differentiator, backstepping method and adaptive method;

[0035] The virtual errors z1, z2 are designed as:

[0036]

[0037] Where x2 is the actual output speed of the piezoelectric micro-positioning platform, x 2d is the virtual control law;

[0038] Design of virtual control law based on backstepping method 2d for:

[0039]

[0040] in T c and η are positive design parameters, and 0<η<1;

[0041] Design auxiliary control law α c for:

[0042]

[0043] in, and They are θ * , estimated values ​​of c1 and H; μ 22 is the output of the finite-time sliding mode differentiator, which is:

[0044]

[0045] where x 2d is the input of the differentiator; μ 21 ,μ 22 is the state of the differentiator; Ξ1>0, Ξ2>0 are the designed differentiator parameters; Where δ2 is a bounded differential error, there exists a positive constant Make

[0046] Design based on adaptive technology and The adaptive law is:

[0047]

[0048] where τ1, τ2, τ3, k1, k2 and k3 are positive design parameters.

[0049] Step 4: Based on the practical predefined time stability lemma and Lyapunov stability theory, the following adaptive predefined time preset performance controller is designed:

[0050]

[0051] in is a designed normal number; is the time-varying disturbance △ H The upper bound of H =Hl H , l H is a constant (it can be the average value of the hysteresis term H, or an unknown constant).

[0052] The relevant parameters are selected according to the following principles:

[0053] 1) Preset performance function and normalization parameter υ ∞ , T Γ, c Γ and The selection of υ should meet the requirements of the tracking error range of the piezoelectric micro-positioning platform in practical applications; the parameters are: ∞ =0.2, T Γ =0.0025, c Γ =1,

[0054] 2) The selection of the parameters Ξ1 and Ξ2 of the finite time sliding mode differentiator should ensure that the output signal of the differentiator is effective, and the value should not be too large; the parameters are selected as Ξ1 = 0.05, Ξ2 = 0.001:

[0055] 3) Adaptive law, virtual control law and final control law parameters τ1, τ2, τ3, k1, k2, k3, η, T c , k c , c0, The selection of must ensure that the stability of the closed-loop system, the system convergence speed and the adjustment effect of the adaptive law can meet the requirements of practical applications; the parameters are selected as follows: τ1 = 0.05, τ2 = 1, τ3 = 1.8, k1 = 0.02, k2 = 1000, k3 = 5, η = 0.025, T c =0.025, k c =500, c0=9.6×10 7 ,

[0056] The beneficial effects of the present invention are:

[0057] Aiming at the tracking control problem of a piezoelectric micro-positioning platform, the present invention proposes an adaptive predefined time preset performance control method of a piezoelectric micro-positioning platform to ensure the tracking performance of the piezoelectric micro-positioning platform. The method has the following three advantages: First, the hysteresis term in the system is regarded as a time-varying parameter, and an adaptive method called "variable condensation" is used to process the time-varying hysteresis term, which simplifies the design process of the controller, avoids the complex description of hysteresis nonlinearity, and effectively eliminates the adverse effects of hysteresis nonlinearity on the tracking performance of the system. Second, the predefined time controller proposed by the present invention has better transient performance than the existing finite / fixed time convergence piezoelectric micro-positioning platform control strategy. The upper limit of the convergence time can be set flexibly and is independent of the initial conditions and parameters of the system. Third, a new constraint boundary adjustment mechanism is introduced in the preset performance function. The constraint boundary can be actively adjusted when the input is saturated, avoiding the problem of control output singularity that may exist in the traditional preset performance function when the input is saturated, and can ensure that the tracking error enters the expected error range within a predetermined time.

[0058] The method of the present invention takes into account the input saturation problem existing in the practical application of the piezoelectric micro-positioning platform system, and provides a feasible and effective control scheme for the practical engineering application of the piezoelectric micro-positioning platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 A control block diagram of the method for controlling the performance of the piezoelectric micro-positioning platform according to the present invention at an adaptive predefined time preset time;

[0060] Figure 2 It is a schematic diagram of an experimental device of a piezoelectric micro-positioning platform experimental system of the present invention;

[0061] Figure 3 It is an experimental principle block diagram of the piezoelectric micro-positioning platform experimental system of the present invention;

[0062] Figure 4 The tracking frequency conversion signal y of the present invention d1 Comparison diagram of the reference trajectory and actual displacement of the piezoelectric micro-positioning platform at ;

[0063] Figure 5 The tracking frequency conversion signal y of the present invention d1 The error curve between the reference trajectory and the actual displacement of the piezoelectric micro-positioning platform at ;

[0064] Figure 6 The tracking complex harmonic signal y of the present invention d2 Comparison diagram of the reference trajectory and actual displacement of the piezoelectric micro-positioning platform at ;

[0065] Figure 7 The tracking complex harmonic signal y of the present invention d2 The error curve between the reference trajectory and the actual displacement of the piezoelectric micro-positioning platform at ;

[0066] Figure 8 The tracking complex harmonic signal y of the present invention d2 The control signal curve when

[0067] Fig. 9 The tracking complex harmonic signal y of the present invention d2 Time adaptive parameters The adjustment curve of

[0068] Fig.10 The tracking complex harmonic signal y of the present invention d2 Time adaptive parameters The adjustment curve of

[0069] Fig.11 The tracking complex harmonic signal y of the present invention d2 Time adaptive parameters ( The L2 norm of the adjustment curve; DETAILED DESCRIPTION

[0070] The present invention is described in more detail below with reference to the accompanying drawings and the effectiveness of the present technical solution is further verified through an implementation case.

[0071] The control block diagram of the method for controlling the performance of a piezoelectric micro-positioning platform according to the present embodiment is shown in FIG. Figure 1 The specific steps are:

[0072] Step 1: The piezoelectric micro-positioning platform can be regarded as a spring-damper-mass system with input hysteresis; considering the influence of interference and uncertainty in practical applications, the piezoelectric micro-positioning platform can be modeled, and the state space equation model can be established as follows:

[0073]

[0074] Where c is the control gain with unknown magnitude but known sign. Considering the actual system, assume that c>0 and there is an upper bound Make f(x) is an unknown continuous function; x=[x1,x2] T ∈R 2 , y∈R represent the state and output of the system respectively; u is the system input; ω(u) is the input with hysteresis information, where H is the hysteresis term; d is the external interference. Considering the actual situation, it is assumed that there is a constant Make

[0075] Use radial basis neural network to approximate the unknown continuous function f(x); the expression of f(x) is as follows:

[0076] f(x)=θ *T φ(x)+ε (2)

[0077] Where ε is the approximation error of the neural network, and there is a positive constant Make φ(x)=[φ1(x),φ2(x),...,φ m (x)] is the Gaussian basis function vector, where φ j (x) = exp(-||xc j || 2 / (2b j 2 )),j=1,…,m, m is the number of hidden layer nodes, b j ,c j are the Gaussian basis function width and center vector of the jth node in the hidden layer; θ *is the ideal weight of the neural network;

[0078] Combining equation (2) with equation (1), the model of the piezoelectric micro-positioning platform can be re-expressed as:

[0079]

[0080] where c0 and k c is the design constant, c1 is the constant that makes the equation c=c0+k c c1 is the unknown constant for which it holds true;

[0081] Considering the input saturation problem of piezoelectric positioning system in practical application, input saturation can be described as:

[0082]

[0083] where u r is the actual voltage driving the piezoelectric micro-positioning platform; u h and u l are the upper and lower bounds of the saturation boundary, respectively.

[0084] Step 2: Using the preset performance function can ensure that the tracking error converges to an arbitrary set. In order to solve the singularity problem caused by the platform input saturation when the preset performance function is used, a new constraint boundary adjustment mechanism is introduced into the preset performance function to avoid the problem of controlling the output singularity when the input is saturated.

[0085] The expression of the prescribed performance function with the constraint boundary adjustment mechanism is:

[0086]

[0087] where υ ∞ <1 is a designable positive constant that affects the tracking error range after the system stabilizes; e is a natural constant; T Γ ,c Γ is a designable positive constant that affects the system's stability time and convergence speed; t k is the boundary adjustment time, and t k (0) = 0;

[0088] The constraint boundary adjustment mechanism is:

[0089]

[0090] Where K is the number of times the boundary adjustment mechanism is triggered; △u e =|uu r |; In addition, Γ(t) also has the following properties: 1) For t>t k , Γ(t) is a bounded and time-differentiable function, and bounded; 2)Γ(0)=1, and when t≥(Τ Γ +t k ) when Γ(t)=υ ∞ ;

[0091] Define the system tracking error as e1, and its expression is as follows:

[0092] e1=x1-y d (7)

[0093] Where x1 is the actual output displacement of the piezoelectric micro-positioning platform; y d is the reference signal;

[0094] Since the initial state of the piezoelectric micro-positioning platform is unpredictable and variable, it is necessary to normalize the error. The Softsign function is used for normalization to relax the restrictions on the initial state of the system. The expression of the Softsign function is as follows:

[0095]

[0096] in is a normal constant set; the properties of this function are: 1) the range of Ψ(e1) is (-1,1), and Ψ(0)=0; 2) by It can be obtained that Ψ(e1) is monotonically increasing and differentiable; 3) The inverse function of Ψ(e1) is Ψ -1 (Γ(0))=∞.

[0097] Step 3: Based on the mathematical model of the piezoelectric micro-positioning platform in step 1 and the preset performance function in step 2, the virtual error is calculated, and the virtual control law, auxiliary control law and adaptive law are designed by combining the finite time sliding mode differentiator, backstepping method and adaptive method;

[0098] The virtual errors z1, z2 are designed as:

[0099]

[0100] Where x2 is the actual output speed of the piezoelectric micro-positioning platform, x 2d is the virtual control law;

[0101] Design of virtual control law based on backstepping method 2d for:

[0102]

[0103] in T c and η are designable positive numbers, and 0<η<1;

[0104] Design auxiliary control law α c for:

[0105]

[0106] in, and They are θ * , estimated values ​​of c1 and H; μ 22 is the output of the finite-time sliding mode differentiator, which is:

[0107]

[0108] where x 2d is the input of the differentiator; μ 21 ,μ 22 is the state of the differentiator; Ξ1>0, Ξ2>0 are the designed differentiator parameters; Where δ2 is a bounded differential error, there exists a positive constant Make

[0109] Designable based on adaptive technology and The adaptive law is:

[0110]

[0111] where τ1, τ2, τ3, k1, k2 and k3 are positive design parameters.

[0112] Step 4: Based on the practical predefined time stability lemma and Lyapunov stability theory, the following adaptive predefined time preset performance controller is designed:

[0113]

[0114] in is a designed normal number; is the time-varying disturbance △ H The upper bound of H =Hl H , l H is a constant (it can be the average value of the hysteresis term H or an unknown constant).

[0115] Consider the Lyapunov function of the entire system as:

[0116]

[0117] in, is the estimation error. Taking the derivative of V and combining equations (9), (10), (11), (13) and (14) and transforming some terms through Young’s inequality, we can obtain:

[0118]

[0119] In order to facilitate the subsequent demonstration of the stability of the control method of the present invention, it is necessary to introduce the following lemma:

[0120] Lemma 1: For an uncertain system If there exists a continuous scalar function V(x) that satisfies the following inequality:

[0121]

[0122] System It is the actual predetermined time stability. Where 0<η<1, T c ,σ is a positive number. Its convergence domain is:

[0123]

[0124] The upper bound of the convergence time of the system is

[0125] Lemma 2: For κ i ,i=1,...,n, there exists a constant g i >0 and g i ≥κ i , so that the following inequality holds:

[0126]

[0127] Among them, ο>1,0<μ<1,

[0128] Lemma 3: For z i ∈R,i=1,...,n, there exists p>0, so that the following inequality holds:

[0129]

[0130] According to Lemma 1, the following inequality holds:

[0131]

[0132] Among them, ξ1,ξ2,ξ3 can be calculated according to Lemma 2. Further, we can get:

[0133]

[0134] in,

[0135] By selecting appropriate parameters such as k1, k2, k3 and τ1, τ2, τ3, η, the inequality Λ1≤-pηT can be obtained. c ,Λ2≤-5 η / 2 pηT c Established.

[0136] The main results of the present invention can be summarized as the following theorem:

[0137] Theorem 1: For a piezoelectric micro-positioning platform system (1), by designing a preset performance function (5), a constraint boundary adjustment mechanism (6), a virtual control law (10), a control law (14) and an adaptive law (13), and selecting appropriate design parameters, the adaptive predefined time preset performance control method constructed by the present invention can ensure that the signal of the closed-loop system is stable within a predefined time, and the tracking error can converge to a preset error range within the predefined time.

[0138] The proof of Theorem 1 is as follows:

[0139] For formula (22) Item, perform Taylor expansion, and rearrange to get:

[0140]

[0141] According to Lemma 3, formula (23) can be organized as:

[0142]

[0143] According to Lemma 1 and Equation (23), V contains all variables at the predefined time T max It converges to a bounded neighborhood near the origin, and the expression of convergence time is:

[0144] The beneficial effects of the present invention are verified by using the following specific implementation cases.

[0145] Implementation Cases:

[0146] The adaptive predefined time preset performance control designed by the present invention is applied to Figure 2 The piezoelectric micro-positioning platform system shown is used to perform trajectory tracking experiments.

[0147] The piezoelectric micro-positioning platform system mainly consists of five parts: an anti-vibration workbench (model: SPFO-B-30-12, Wuhan Xinlianxing Precision Electromechanical Co., Ltd.), a piezoelectric micro-positioning platform (model: MPT-2MRL102A, Suzhou Maikeron Automation Co., Ltd.), an integrated positioning controller (model: PPC-2CR0510, Suzhou Maikeron Automation Co., Ltd.), a data acquisition card (model: PCI-1716-AE, Advantech Co., Ltd. (China)), and a host computer. The experimental schematic diagram is as shown in Figure 3 shown. The proposed method is implemented in Matlab / Simulink and is compiled in real-time through the External mode of Desktop real-time. A / D and D / A conversions are performed through the data acquisition card, and the sampling frequency is set to 10 kHz. The output signal of the host computer is input to the drive module after D / A conversion. After being amplified by the drive module, a control signal is output to drive the piezoelectric micro-positioning platform. The sensor can obtain the output displacement of the platform, which is transmitted to the host computer after A / D conversion for further processing.

[0148] To verify the effectiveness of the control method of the present invention, the following reference trajectories are selected: y d1 = 18sin(2πft + 1.5π) + 18 (f = 1 (t ≤ 2), 5 (2 < t ≤ 2.4), 10 (2.4 < t ≤ 2.7), 20 (2.7 < t ≤ 2.8), 50 (2.8 < t ≤ 2.9), 100 (2.9 < t ≤ 3)); complex harmonic y d2 = 6(sin(3p + 1.5π) + sin(5p + 1.5π) + sin(8p + 1.5π)) + 18.

[0149] The parameters of the designed controller are selected as follows:

[0150] The parameters υ ∞ , T Γ , c Γ and of the preset performance function and the normalization function should meet the requirements for the tracking error range of the piezoelectric micro-positioning platform during actual application. The parameters are set as υ ∞ = 0.2, T Γ = 0.0025, c Γ = 1, The selection of the parameters Ξ1 and Ξ2 of the finite-time sliding mode differentiator should ensure the effectiveness of the output signal of the differentiator, and the values should not be too large. The parameters are set as Ξ1 = 0.05 and Ξ2 = 0.001. The parameters τ1, τ2, τ3, k1, k2, k3, η, T c , k c , c0, The selection of must ensure that the stability of the closed-loop system, the system convergence speed and the adjustment effect of the adaptive law can meet the requirements of practical applications. The parameters are set to τ1 = 0.05, τ2 = 1, τ3 = 1.8, k1 = 0.02, k2 = 1000, k3 = 5, η = 0.025, T c =0.025, k c =500, c0=9.6×10 7 , The initial values ​​of the parameters are all set to 0.

[0151] The experimental results are as follows Figure 4-Figure 11 As shown. Figure 4 and Figure 6 It can be seen that the adaptive predefined design preset performance control method described in the present invention can make the actual displacement of the piezoelectric micro-positioning platform accurately track the reference trajectory y d1 ,y d2 . Figure 5 and Figure 7 are the tracking reference trajectories y d1 and d2 From the tracking error curve between the reference trajectory and the actual displacement of the platform, it can be seen that under the control method of the present invention, the output displacement of the piezoelectric micro-positioning platform can effectively track sinusoidal trajectories of different frequencies and complex harmonic trajectories. Figure 8 The method of the present invention tracks y d2 It can be seen from the control input signal at that time that no control signal peak occurs during the entire control stage, which reflects the feasibility of this method in practical applications. Figure 9-11 It is the adaptive parameter adjustment curve of the method of the present invention. Figure 4-Figure 11 From the experimental results, it can be concluded that under the control method of the present invention, thanks to the application of the predefined time technology, the tracking error can converge to the preset error range within the predefined time, which reflects the excellent transient steady-state performance of the control method proposed in the present invention.

Claims

1. An adaptive predefined time preset performance control method for a piezoelectric micro-positioning platform, characterized in that: The steps of this method are as follows: Step 1: The piezoelectric micro-positioning platform can be regarded as a spring-damper-mass system with input hysteresis; considering the influence of interference and uncertainty in practical applications, the mathematical model of the piezoelectric micro-positioning platform can be established as follows: where c>0 and there is an upper bound Make f(x) is an unknown continuous function; x=[x1,x2] T ∈R 2 , y∈R represent the state and output of the system respectively; u is the system input; ω(u) is the input with hysteresis information, where H is the hysteresis term; d is the external interference, and there is a constant Make Use radial basis neural network to approximate the unknown continuous function f(x); the expression of f(x) is as follows: f(x)=θ *T f(x)+ε (2) Where ε is the approximation error of the neural network, and there is a positive constant Make f(x)=[f1(x),f2(x),...,f m (x)] is the Gaussian basis function vector, where f j (x) = exp(-||xc j || 2 / (2b j 2 )),j=1,...,m, m is the number of hidden layer nodes, b j ,c j are the Gaussian basis function width and center vector of the jth node in the hidden layer; θ * is the ideal weight of the neural network; The mathematical model of formula (1) can be re-expressed as: where c0 and k c is the design constant, c1 is the constant that makes the equation c=c0+k c c1 is the unknown constant for which it holds true; Considering the input saturation problem of piezoelectric positioning system in practical application, the input saturation is described as: where u r is the actual voltage driving the piezoelectric micro-positioning platform; u h and u l are the upper and lower bounds of the saturation boundary, respectively; Step 2: Using the preset performance function can ensure that the tracking error converges to an arbitrary set; introducing the constraint boundary adjustment mechanism into the preset performance function avoids the problem of controlling the output singularity when the input is saturated; The expression of the prescribed performance function with the constraint boundary adjustment mechanism is: where υ ∞ <1 is a positive design constant, which affects the tracking error range after the system stabilizes; e is a natural constant, T Γ ,c Γ is a positive design constant that affects the stability time and convergence speed of the system; t k is the boundary adjustment time, and t k (0) = 0; The constraint boundary adjustment mechanism is: Where K is the number of times the boundary adjustment mechanism is triggered; △u e =|uu r |; In addition, Γ(t) also has the following properties: 1) For t>t k , Γ(t) is a bounded and time-differentiable function, and bounded; 2)Γ(0)=1, and when t≥(Τ Γ +t k ) when Γ(t)=υ ∞ ; Define the system tracking error as e1, and its expression is as follows: e1=x1-y d (7) Where x1 is the actual output displacement of the piezoelectric micro-positioning platform; y d is the reference signal; The Softsign function is used to normalize the error. The expression of the Softsign function is as follows: in is a normal constant set; the properties of this function are: 1) the range of Ψ(e1) is (-1,1), and Ψ(0)=0; 2) by It can be obtained that Ψ(e1) is monotonically increasing and differentiable; 3) The inverse function of Ψ(e1) is Ψ -1 (Γ(0)) = ∞; Step 3: Based on the mathematical model of the piezoelectric micro-positioning platform in step 1 and the preset performance function in step 2, the virtual error is calculated, and the virtual control law, auxiliary control law and adaptive law are designed by combining the finite time sliding mode differentiator, backstepping method and adaptive method; The virtual errors z1, z2 are designed as: Where x2 is the actual output speed of the piezoelectric micro-positioning platform, x 2d is the virtual control law; Design of virtual control law based on backstepping method 2d for: in T c and η are positive design parameters, and 0<η<1; Design auxiliary control law α c for: in, and They are θ * , estimated values ​​of c1 and H; μ 22 is the output of the finite-time sliding mode differentiator, which is: where x 2d is the input of the differentiator; μ 21 ,μ 22 is the state of the differentiator; Ξ1>0, Ξ2>0 are the designed differentiator parameters; Where δ2 is a bounded differential error, there exists a positive constant Make Design based on adaptive technology and The adaptive law is: Where τ1, τ2, τ3, k1, k2 and k3 are positive design parameters; Step 4: Based on the practical predefined time stability lemma and Lyapunov stability theory, the following adaptive predefined time preset performance controller is designed: in is a designed normal number; is the time-varying disturbance △ H The upper bound of H =Hl H , l H is a constant, the average value of the hysteresis term H, or an unknown constant.

2. The method for controlling the performance of a piezoelectric micro-positioning platform according to claim 1, characterized in that: The relevant design parameters are selected according to the following principles: Preset performance function and normalized function parameter υ ∞ , T Γ , c Γ and The selection should meet the requirements of the tracking error range and convergence speed of the piezoelectric micro-positioning platform in practical applications. The parameter is set to υ ∞ =0.2, T Γ =0.0025, c Γ =1, The selection of the parameters Ξ1 and Ξ2 of the finite time sliding mode differentiator should ensure that the output signal of the differentiator is effective, and the value should not be too large. The parameters are set to Ξ1 = 0.05 and Ξ2 = 0.001; Adaptive law, virtual control law and final control law parameters τ1, τ2, τ3, k1, k2, k3, η, T c , k c , c0, The selection of must ensure that the stability of the closed-loop system, the system convergence speed and the adjustment effect of the adaptive law can meet the requirements of practical applications. The parameters are set to τ1 = 0.05, τ2 = 1, τ3 = 1.8, k1 = 0.02, k2 = 1000, k3 = 5, η = 0.025, T c =0.025, k c =500, c0=9.6×10 7 ,

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