An Enhanced Sliding Mode Predictive Control Method for Piezoelectric-Driven Vibration Isolation Systems Based on a Generalized Proportional-Integral Observer

Abstract

This invention proposes an enhanced sliding mode predictive control method based on a generalized proportional-integral observer for piezoelectric-driven vibration isolation systems to address the vibration suppression problem under variable mass load conditions. First, this invention considers the variable mass load in a piezoelectric-driven active-passive vibration isolator system under variable mass load conditions as an uncertainty in the system and expresses it as an unknown function, while vibration is considered as a disturbance in the system. Then, a state observer using a neural network is used to estimate the unknown system state, and the unknown function in the system is approximated by the neural network. A generalized proportional-integral observer is used to estimate and predict time-varying disturbances of mismatch and matching in the system. An enhanced sliding mode predictive method is employed to predict the sliding surface, and the optimal control sequence is solved by minimizing the loss function through rolling optimization, deriving the control signal applied to the system.

Description

Technical Field

[0001] This invention relates to the field of control engineering technology, and in particular to an enhanced sliding mode predictive control method for a piezoelectric-driven vibration isolation system based on a generalized proportional-integral observer. Background Technology

[0002] Currently, high-tech fields such as aerospace and optical instruments are booming, and the accuracy and reliability issues of precision instruments and equipment have received widespread attention from researchers both domestically and internationally. Current research indicates that vibration isolation technologies mainly include passive vibration isolation, active vibration isolation, semi-active vibration isolation, and hybrid active-passive vibration isolation. Among these, piezoelectric-driven active-passive vibration isolation systems offer high resolution, reliability, and controllability, and are widely used for micro-vibration suppression. However, for piezoelectric-driven active-passive vibration isolator systems under variable mass load conditions, system uncertainties and external disturbances coexist, and the design of a high-performance controller to protect the load from vibration remains a problem worthy of consideration. Therefore, this invention focuses on researching active vibration control methods for piezoelectric-driven vibration isolation systems under variable mass load conditions.

[0003] To ensure the performance of vibration isolation systems, the design of active control methods is of great significance. Currently, researchers have conducted studies on vibration suppression using robust control, adaptive control, and sliding mode control. Among these, observer-based sliding mode control is widely used to suppress external vibrations due to its robustness to uncertainties and disturbances. However, the accuracy of existing observers in estimating time-varying disturbances needs improvement. Furthermore, sliding mode control methods inevitably suffer from chattering. Therefore, improving the sliding mode control performance of complex uncertain systems remains a worthy research topic.

[0004] In sliding mode predictive control (SMD) methods, by fully utilizing system information to predict the sliding mode state and make it track the reference trajectory, and by deriving the control law through rolling time-domain optimization, control accuracy can be improved and chattering can be reduced. For piezoelectrically driven vibration isolation systems, system state information is difficult to obtain, and time-varying mismatch system uncertainties and disturbances continuously exist within the system. However, conventional SMD predictive control methods generally directly use the system state for controller design and only consider matching disturbances. Furthermore, traditional SMD predictive control methods typically assume the disturbance to be constant in the prediction time domain, which is conservative for handling time-varying disturbances.

[0005] Therefore, this invention designs an enhanced sliding mode predictive control based on a neural network state observer and a generalized proportional-integral observer, utilizing only the system's measured output, thus promoting the practical engineering application of the algorithm. Furthermore, the controller design considers mismatch uncertainties and disturbances, predicts time-varying disturbances using a generalized proportional-integral observer, and introduces a sliding mode state error integrator, improving the performance of sliding mode predictive control in piezoelectric-driven vibration isolation systems. Summary of the Invention

[0006] This invention mainly addresses the control challenges in piezoelectric-driven active and passive vibration isolation systems under variable load conditions. It fully considers the mismatch disturbances and uncertainties in the system and designs an enhanced sliding mode predictive control method based on a generalized proportional-integral observer according to the active and passive vibration isolation system model to solve the vibration suppression problem of the piezoelectric-driven active and passive vibration isolation system.

[0007] The technical solution adopted in this invention is as follows:

[0008] An enhanced sliding mode predictive control method based on a generalized proportional-integral observer for a piezoelectric-driven vibration isolation system is proposed, with the following specific steps:

[0009] Step 1: Describe the piezoelectric driven active and passive vibration isolation system under variable mass load conditions according to the dynamic model of the piezoelectric driven active and passive vibration isolation system. In this system, the variable mass load is considered as an uncertainty in the system and expressed as an unknown function, and the vibration is considered as a disturbance in the system. Both the mismatch disturbance and the uncertainty exist in the system.

[0010] Step 2: Use the state observer of the neural network to estimate the unknown system state. The unknown function in the system is approximated by the neural network.

[0011] Step 3: Use a generalized proportional-integral observer to estimate and predict time-varying disturbances of mismatch and matching in the system;

[0012] Step 4: Design a sliding surface capable of handling mismatched disturbances and uncertainties based on the estimated system state, disturbances, and approximation unknown functions, and predict the sliding surface; construct a loss function by combining the defined sliding surface approach law, the predicted sliding surface, and the sliding state error integrator; solve for the optimal control sequence by minimizing the loss function through rolling optimization, and derive the control signal applied to the system.

[0013] The specific process of step 1 is as follows:

[0014] Based on the dynamic model of the piezoelectric-driven active and passive vibration isolation system, the piezoelectric-driven active and passive vibration isolator system under variable mass load conditions is described as follows:

[0015]

[0016] Where x i For system state and n is the system order. u and y are the system input and output, b is the control gain, and h(u) is the hysteresis of the piezoelectric ceramic. The unknown function g... i (·) represents the system uncertainty caused by the variable mass load, d i,i=1,…,n represents disturbances caused by external vibrations; where g1(·),…,g n-1 (·) and d1,…,d n-1 This refers to the mismatch, uncertainty, and disturbance in the system. Given a function with parameters a0, ..., a n-1 Related to and known physical properties of the system.

[0017] Based on the Euler method, the discrete-time model of system (1) is derived as follows:

[0018]

[0019] Where T s The sampling period is The system state is unknown. The estimated value.

[0020] The specific process of step 2 is as follows:

[0021] A neural network is used to approximate the unknown function in the piezoelectric-driven active and passive vibration isolation system (2). The neural network is described as follows:

[0022]

[0023] Where ξ=[ξ1,ξ2,…,ξ q ] T ∈R q For the system input vector, Let ξ be the weight vector, δ(ξ) be the Gaussian active function vector, ε be the approximation error, and the optimal weights be ε(ξ).

[0024] By applying a neural network to approximate the unknown function, system (2) is rewritten in the following form:

[0025]

[0026] in This represents the composite disturbance of the system. u(k) + h(u(k)) + d n (k), b0 is the controller gain, ε i ,i=1,2,…,n represent the approximation error.

[0027] To simplify the explanation, the function Writing δ i (k), i=1,…,n, the system (4) is rewritten in the following form:

[0028]

[0029] Define the composite perturbation at time k as D(k) = [D1(k), ..., Dn (k)] T The approximation error at time k is ε(k) = [ε1(k),…,ε n (k)] T The rate of change of the composite perturbation is uniformly bounded. Furthermore, the rate of change of ε(k) is bounded, defined as follows: For piezoelectric active and passive vibration isolation systems, the function The rate of change is uniformly bounded, and the norm of the function is defined as follows:

[0030] The state observer for the discrete system (4) is as follows:

[0031]

[0032] in The estimates of the state and the disturbance are respectively, B = [0,…,0,T]. s b0], And l1,…,l n These are the design parameters in the state observer;

[0033] The state estimation error is:

[0034]

[0035] in and These represent the state, weight, and disturbance estimation errors, respectively.

[0036]

[0037] The specific process of step 3 is as follows:

[0038] First, define D i (k)=D i,0 (k), and its higher-order differences Where τ represents the order of the maximum difference; assume D i,τ+1 (k) = 0, and further, define and The derivation is as follows:

[0039]

[0040] in

[0041] The generalized proportional-integral observer is used to estimate and predict the aforementioned time-varying disturbances of mismatch and matching in the system. The generalized proportional-integral observer is as follows:

[0042]

[0043] in Indicates D i,0 (k),…,D i,τ The estimate of (k), These are the design parameters in the observer.

[0044] The expression for the estimated error is as follows:

[0045]

[0046] The matrix form of the estimation error is summarized as follows:

[0047]

[0048] in

[0049] By combining equations (7) and (8), the p-step forward perturbation preview is derived as follows:

[0050]

[0051] in yes The estimated value, and G = [1,0,…,0].

[0052] Consider the Lyapunov function V o (k)=V1(k)+V2(k), where Both matrices P1 and P2 are positive definite.

[0053] definition And γ>0. The calculation of V1(k) is as follows:

[0054] If the eigenvalues ​​of matrix A satisfy Then for any positive definite matrix Q1, there exists an equation satisfying Q1=-((1+γ)A T The positive definite matrix P1 is P1 of P1A-P1). Then formula (12) can be written as:

[0055]

[0056] Where λ min (Q1) represents the smallest eigenvalue of matrix Q1;

[0057] Define matrix The calculation for V2(k) is as follows:

[0058]

[0059] If the eigenvalues ​​of matrix A satisfy Then for any positive definite matrix Q2, there exists an equation that satisfies The positive definite matrix P2. Then equation (14) can be written as:

[0060]

[0061] The calculation of V0(k) is as follows:

[0062]

[0063] in ψ2=λ min (Q2)-(2+2 / γ). The convergence of the observer can be observed to be... Related to this item.

[0064] The specific process of step 4 is as follows:

[0065] Based on the estimated system state, disturbance, and approximation of the unknown function, a sliding surface capable of handling mismatched disturbances and uncertainties is designed, and the sliding surface is predicted. In the design process of enhanced sliding mode predictive control, a loss function is constructed by combining the defined sliding mode reaching law, the predicted sliding surface, and the sliding mode state error integrator. The optimal control sequence is then solved by minimizing the loss function through rolling optimization. The specific process is as follows:

[0066] Define the sliding surface as

[0067]

[0068] in H = [m2, m3, ..., m n [,0], parameter m n =1,m1,m2,…,m n-1 p(λ) = λ represents a polynomial with variable λ. n-1 +m n-1 λ n-1 The coefficients of +…m2λ+m1, and the eigenvalues ​​satisfy 2Re(λ) i )+|λ i | 2 T<0, where Re() represents the real part of the complex number.

[0069] The integrator for sliding mode error is defined as:

[0070]

[0071] in This is the sliding mode error. χ is a forgetting factor.

[0072] The sliding mode reaching law is defined as:

[0073]

[0074] Among them 0 <T s ω1<1,T s ω2>0, 0<ζ<1, δ>0, In addition, the function

[0075] Define extended state We can obtain:

[0076]

[0077] in

[0078] The expression for s(k) prediction is as follows:

[0079]

[0080] Where S(k)=[s(k+1),s(k+2),…,s(k+p)] T ,

[0081] U(k)=[u(k),u(k+1),…,u(k+p-1)] T , The prediction and control time domains are the same, denoted as p.

[0082] Combining the defined sliding mode convergence law, the predicted sliding surface, and the sliding mode state error integrator, a loss function is constructed, which is described as follows:

[0083] J(k)=S T (k)R1S(k)+U T (k)R2U(k) (23)

[0084] R1 and R2 represent the weight matrices.

[0085] The new loss function is derived as follows:

[0086]

[0087] in

[0088] By setting The optimal control sequence is derived as follows:

[0089]

[0090] Based on the principle of rolling time-domain optimization, the control signal applied to the system is derived as follows:

[0091]

[0092] Where G = [1, 0, ..., 0].

[0093] The adaptive law design for weights in a neural network is as follows:

[0094]

[0095] in and k i It is a normal number.

[0096] Theorem 1: For a system (1) affected by system uncertainty and disturbance, by designing a state observer (5), a generalized proportional-integral observer (8), a sliding surface (17), a control law (26), and an update law (27), the proposed predictive sliding mode control scheme can guarantee that: (1) the sliding state State estimation error Interference error and weighted error (2) The system output is bounded, which reduces the negative impact of mismatch uncertainty and disturbance on the system output.

[0097] Proof: For simplicity of analysis, the weight matrix of the control action can be assumed to be R² = 0. Based on the prediction of s(k) and the optimal control sequence (25), the sliding mode state... It is deduced as:

[0098]

[0099] Consider the Lyapunov function V c (k)=V0(k)+V3(k)+V4(k), where

[0100] Combining equation (28) with The calculation of △V3(k)=V3(k+1)-V3(k) is as follows:

[0101]

[0102] According to the update law (27), the estimation error of the weights in a neural network is derived as:

[0103]

[0104] in

[0105] V4(k) is calculated as follows:

[0106]

[0107] in According to Young's inequality, equation (31) can be written as:

[0108]

[0109] Combining equations (16), (29), and (32), V c (k) is derived as follows, V c (k) is calculated as follows:

[0110]

[0111] The definitions of ψ1 and ψ2 are the same as those in equation (16).

[0112] By selecting appropriate parameters such that ψ1>0, ψ2>0, ψ3>0, ψ4>0, and ψ5>0, the sliding mode state can be proven. estimation error and The boundedness of the variables. To simplify the description, the upper bound of the relevant variables is defined as...

[0113] Next, we analyze the boundedness of the system output and the impact of mismatch uncertainties and disturbances on the system output. The coordinate transformation condition is defined as:

[0114]

[0115] Combining equation (4) and The system is converted to:

[0116]

[0117] in

[0118] Then, we arrive at the conclusion. The characteristic polynomial is Based on the parameters defined in the equation, select the modulus η when applying conditional equation (17). i It is derived from the following formula:

[0119]

[0120] By combining the transformation matrix, we obtain Jordan matrix matrix The eigenvalues ​​are the same as those of J. Furthermore, assume the eigenvalue λ... i They are different, which means that the eigenvalue η i They are different. Therefore, matrix J can be written as J = diag[η1, η2, ..., η]. n-1 ].

[0121] Based on the Jordan matrix, the iterative equation (35) can be obtained. Then, its norm is derived as:

[0122]

[0123] It is worth noting that, Based on assumptions 1 and 2, and equation (33), The norm is derived as Then, for k→∞, The boundary is derived as follows:

[0124]

[0125] Observe the system output It is bounded, and its upper bound is independent of the bounds of disturbances and uncertainties. Although the compensation error still affects the system output, its magnitude may be much smaller than the magnitude of the disturbances and uncertainties themselves. Therefore, the negative impact of mismatch uncertainties and disturbances can be reduced in the system output.

[0126] The relevant parameters are selected according to the following principles:

[0127] The order of the piezoelectric active and passive vibration isolation system is taken as 3.

[0128] The system parameters are set to M = [8.8 × 10 7 7×10 4 ,1],H=[7×10 4 [,1,0],R1=I 10×10 R² = 8 × 10 11 I 10×10 χ=0.7, T s ω1=0.9, T s ω2=0.01, ζ=0.5, δ=1,

[0129] The parameters of the state observer based on the neural network are set as l1 = 2, l2 = 4 × 10. 3 l3 = 1 × 10 6 k1 = k2 = k3 = 0.5

[0130] The parameters of the generalized proportional-integral observer are set as follows:

[0131] The beneficial effects of this invention are as follows:

[0132] This invention designs an enhanced predictive sliding mode control scheme based on a generalized proportional-integral observer to achieve vibration suppression control of a piezoelectric-driven vibration isolation system. This method has four advantages: First, a neural network-based state observer is designed, enabling the approximation of unknown functions in the system and estimation of unknown states, facilitating the design of the output feedback controller and promoting the practical application of the algorithm. Second, a generalized proportional-integral observer is designed to estimate and predict matched and unmatched disturbances, providing sufficient disturbance information for controller design and eliminating the conservatism of predictive sliding mode control methods in handling time-varying disturbances. Third, a sliding surface incorporating estimated states, disturbances, and approximate uncertainties is designed, which can handle mismatched disturbances and uncertainties in the system, thereby improving the performance of the vibration isolation system. Fourth, the introduced sliding mode state error integrator allows sliding mode predictive control to correct the generated control signal based on past error information, reducing the negative impact of uncertainty on control performance and thus improving the performance of the piezoelectric-driven vibration isolation system under variable mass load conditions. Experimental results show that this scheme can effectively suppress various vibrations and maintain good performance under variable mass load conditions.

[0133] The method of this invention fully considers the problems of unmeasurable system state, mismatch disturbance caused by external vibration, and system uncertainty caused by variable mass load. It designs an enhanced sliding mode predictive control method based on a generalized proportional-integral observer for piezoelectric driven vibration isolation system, which improves the vibration suppression performance of piezoelectric driven vibration isolation system and promotes its practical engineering application. Attached Figure Description

[0134] Figure 1 Schematic diagram of the enhanced sliding mode predictive control method based on the generalized proportional-integral observer.

[0135] Figure 2 This is a schematic diagram of the experimental platform connection.

[0136] Figure 3 These are the amplitude-frequency and phase-frequency characteristics of a piezoelectric-driven active and passive vibration isolation system under variable mass load conditions.

[0137] Figure 4 This is a graph showing the experimental results of 5Hz suppression under a 150g load;

[0138] Figure 5 This is a graph showing the experimental results of 25Hz suppression under a 150g load;

[0139] Figure 6 This is a graph showing the results of a 50z suppression experiment under a 150g load;

[0140] Figure 7 This is a graph showing the experimental results of 75Hz suppression under a 150g load;

[0141] Figure 8 This is a diagram showing the results of a complex vibration suppression experiment under a 100g load;

[0142] Figure 9 This is a diagram showing the results of a complex vibration suppression experiment under a 150g load; Detailed Implementation

[0143] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0144] This embodiment describes an enhanced sliding mode predictive control method for a piezoelectric-driven vibration isolation system based on a generalized proportional-integral observer. The control block diagram is as follows: Figure 1 As shown. The specific design steps of the control method of the present invention are as follows:

[0145] Step 1: Based on the dynamic model of the piezoelectric-driven active-passive vibration isolation system, derive the system description of the piezoelectric-driven active-passive vibration isolator under variable mass load conditions. The variable mass load is considered as an uncertainty in the system and expressed as an unknown function. Vibration is considered as a disturbance in the system, and both mismatch disturbances and uncertainties exist in the system. The system description is as follows:

[0146]

[0147] Where x i For system state and n is the system order. u and y are the system input and output, b is the control gain, and h(u) is the hysteresis of the piezoelectric ceramic. The unknown function g... i (·) represents the system uncertainty caused by the variable mass load, d i ,i=1,…,n represents disturbances caused by external vibrations; where g1(·),…,g n-1 (·) and d1,…,d n-1 This refers to the mismatch, uncertainty, and disturbance in the system. Given a function with parameters a0, ..., a n-1 Related to and known physical properties of the system. Based on the Euler method, the discrete-time model of system (1) is derived as follows:

[0148]

[0149] Where T s The sampling period is The system state is unknown. The estimated value.

[0150] Using neural networks to approximate unknown functions is described as follows:

[0151]

[0152] Where ξ=[ξ1,ξ2,…,ξ q ] T ∈R q For the system input vector, Let the weight vector δ(ξ) be the Gaussian form of the active function vector, ε be the approximation error, and the optimal weights be...

[0153] By applying a neural network to approximate the unknown function, system (2) is rewritten in the following form:

[0154]

[0155] in This represents the composite disturbance in the system. u(k) + h(u(k)) + d n (k), b0 is the controller gain, ε i Let i = 1, 2, ..., n represent the approximation error. For simplicity, the function... Writing δ i (k), i = 1, ..., n.

[0156] Definition 1: D(k) = [D1(k), ..., D n (k)] T and ε(k)=[ε1(k),…,ε n (k)] T .

[0157] Assumption 1: The rate of change of the composite disturbance is uniformly bounded. Furthermore, the rate of change of ε(k) is bounded, defined as follows:

[0158] Assumption 2: For a piezoelectric active and passive vibration isolation system, the function The rate of change is uniformly bounded. Its norm is defined as...

[0159] The specific process of step 2 is as follows:

[0160] Based on the model derived in step 1, a neural network-based state observer is designed to estimate the unknown system state and disturbances, facilitating controller design under conditions where only the system output is measurable. The specific process is as follows:

[0161] For the discrete system (4), the state observer is designed as follows:

[0162]

[0163] in The estimates of the state and the disturbance are respectively, B = [0,…,0,T]. s b0], And l1,…,l n These are the design parameters in the state observer.

[0164] The state estimation error is:

[0165]

[0166] in and These represent the state, weight, and disturbance estimation errors, respectively.

[0167]

[0168] The specific process of step 3 is as follows:

[0169] A generalized proportional-integral observer is designed to estimate and predict time-varying disturbances of mismatch and matching in the system. The specific process is as follows:

[0170] First, define D i (k)=D i,0 (k), and its higher-order differences Where τ represents the order of the maximum difference. Assume D i,τ+1 (k) = 0, and further, define and The derivation is as follows:

[0171]

[0172] in

[0173] Subsequently, the generalized proportional-integral observer is designed as follows:

[0174]

[0175] in Indicates D i,0 (k),…,D i,τ The estimate of (k), These are the design parameters in the observer;

[0176] The expression for the estimated error is as follows:

[0177]

[0178] The matrix form of the estimation error is summarized as follows:

[0179]

[0180] in

[0181]

[0182] By combining equations (7) and (8), the p-step forward perturbation preview is derived as follows:

[0183]

[0184] in yes The estimated value, and G = [1,0,…,0].

[0185] By considering higher-order differences, the generalized proportional-integral (GSI) observer possesses the ability to accurately estimate time-varying perturbations. This study develops a GSI observer for systems with perturbations in multiple channels. It facilitates the application of GSI observers in complex systems. Furthermore, in the GSI observer, choosing a larger cost τ yields higher estimation accuracy, but also increases the computational burden. In this study, the order is set to τ = 2 to balance this trade-off.

[0186] Consider the Lyapunov function V o (k)=V1(k)+V2(k), where Both matrices P1 and P2 are positive definite.

[0187] definition And γ>0, the calculation of V1(k) is as follows:

[0188]

[0189] If the eigenvalues ​​of matrix A satisfy Then for any positive definite matrix Q1, there exists an equation satisfying Q1=-((1+γ)A T The positive definite matrix P1 is P1 of P1A-P1). Then formula (12) can be written as:

[0190]

[0191] Where λ min (Q1) represents the smallest eigenvalue of matrix Q1;

[0192] Define matrix The calculation for V2(k) is as follows:

[0193]

[0194] If the eigenvalues ​​of matrix A satisfy Then for any positive definite matrix Q2, there exists an equation that satisfies The positive definite matrix P2. Then equation (14) can be written as:

[0195]

[0196] The calculation of V0(k) is as follows:

[0197]

[0198] in ψ2=λ min (Q2)-(2+2 / γ). The convergence of the observer can be observed to be... This relates to the following topic. The stability of the closed-loop system is analyzed in Theorem 1 below.

[0199] The specific process of step 4 is as follows:

[0200] Based on the estimated system state, disturbance, and approximation of the unknown function, a sliding surface capable of handling mismatched disturbances and uncertainties is designed, and the sliding surface is predicted. In the design process of enhanced sliding mode predictive control, a loss function is constructed by combining the defined sliding mode reaching law, the predicted sliding surface, and the sliding mode state error integrator. The optimal control sequence is then solved by minimizing the loss function through rolling optimization. The specific process is as follows:

[0201] Based on the aforementioned observer, this section presents an enhanced predictive sliding mode method. The sliding surface is defined as...

[0202]

[0203] in H = [m2, m3, ..., m n [,0], parameter m n =1,m1,m2,…,m n-1 Let p(λ) = λ be a polynomial with independent variable λ. n-1 +m n-1 λ n-1 The coefficients of +…m2λ+m1, and the eigenvalues ​​satisfy 2Re(λ) i )+|λ i | 2 T<0, where Re() represents the real part of the complex number.

[0204] The integrator for sliding mode error is defined as:

[0205]

[0206] in This is the sliding mode error. χ is a forgetting factor.

[0207] The sliding mode reaching law is defined as:

[0208]

[0209] Among them 0 <T s ω1<1,T s ω2>0, 0<ζ<1, δ>0, In addition, the function

[0210] Define extended state We can obtain:

[0211]

[0212]

[0213] in

[0214] The expression for s(k) prediction is as follows:

[0215]

[0216] Where S(k)=[s(k+1),s(k+2),…,s(k+p)] T , U(k)=[u(k),u(k+1),…,u(k+p-1)] T , The prediction and control time domains are the same, denoted as p.

[0217] Compared to traditional sliding mode predictive control methods, the proposed method introduces an integrator for the sliding mode state error, enabling it to correct the control signal based on past error information. This process continues until the state matches its reference value, thereby improving the closed-loop system's performance in handling uncertainties.

[0218] The loss function is designed as follows:

[0219] J(k)=S T (k)R1S(k)+U T (k)R2U(k) (23)

[0220] R1 and R2 represent the weight matrices.

[0221] The new loss function is derived as follows:

[0222]

[0223] in

[0224] By setting The optimal control sequence is derived as follows:

[0225]

[0226] Based on the principle of rolling time-domain optimization, the control signal applied to the system is derived as follows:

[0227]

[0228] Where G = [1, 0, ..., 0].

[0229] The adaptive law design for weights in a neural network is as follows:

[0230]

[0231] in and k i It is a normal number.

[0232] The predicted disturbance information in the matrix Γ(k) of the control law can be obtained through equation (11). This eliminates the limitations of traditional predictive sliding mode control methods in dealing with time-varying disturbances.

[0233] Theorem 1: For a system (1) affected by system uncertainty and disturbance, by designing a state observer (5), a generalized proportional-integral observer (8), a sliding surface (17), a control law (26), and an update law (27), the proposed predictive sliding mode control scheme can guarantee that: (1) the sliding state State estimation error Interference error and weighted error (2) The system output is bounded, which reduces the negative impact of mismatch uncertainty and disturbance on the system output.

[0234] Proof: For simplicity of analysis, the weight matrix of the control action can be assumed to be R² = 0. Based on the prediction of s(k) and the optimal control sequence (25), the sliding mode state... It is deduced as:

[0235]

[0236] Consider the Lyapunov function V c (k)=V0(k)+V3(k)+V4(k), where

[0237] Combining equation (28) with The calculation of △V3(k)=V3(k+1)-V3(k) is as follows:

[0238]

[0239] According to the update law (27), the estimation error of the weights in a neural network is derived as:

[0240]

[0241] in

[0242] V4(k) is calculated as follows:

[0243]

[0244] in According to Young's inequality, equation (31) can be written as:

[0245]

[0246] Combining equations (16), (29), and (32), V c (k) is derived as follows, V c (k) is calculated as follows:

[0247]

[0248] The definitions of ψ1 and ψ2 are the same as those in equation (16).

[0249] By selecting appropriate parameters such that ψ1>0, ψ2>0, ψ3>0, ψ4>0, and ψ5>0, the sliding mode state can be proven. estimation error and The boundedness of the variables. To simplify the description, the upper bound of the relevant variables is defined as...

[0250] Next, we analyze the boundedness of the system output and the impact of mismatch uncertainties and disturbances on the system output. The coordinate transformation condition is defined as:

[0251]

[0252] Combining equation (4) and The system is converted to:

[0253]

[0254] in

[0255] Then, we arrive at the conclusion. The characteristic polynomial is Based on the parameters defined in the equation, select the modulus η when applying conditional equation (17). i It is derived from the following formula:

[0256]

[0257] By combining the transformation matrix, we obtain Jordan matrix matrix The eigenvalues ​​are the same as those of J. Furthermore, assume the eigenvalue λ... i They are different, which means that the eigenvalue η i They are different. Therefore, matrix J can be written as J = diag[η1, η2, ..., η]. n-1 ].

[0258] Based on the Jordan matrix, the iterative equation (35) can be obtained. Then, its norm is derived as:

[0259]

[0260] It is worth noting that, Based on assumptions 1 and 2, and equation (33), The norm is derived as Then, for k→∞, The boundary is derived as follows:

[0261]

[0262] Observe the system output It is bounded, and its upper bound is independent of the bounds of disturbances and uncertainties. Although the compensation error still affects the system output, its magnitude may be much smaller than the magnitude of the disturbances and uncertainties themselves. Therefore, the negative impact of mismatch uncertainties and disturbances can be reduced in the system output.

[0263] The relevant parameters are selected according to the following principles:

[0264] The order of the piezoelectric active and passive vibration isolation system is taken as 3.

[0265] The system parameters are set to M = [8.8 × 10 7 7×10 4 ,1],H=[7×10 4 [,1,0],R1=I 10×10 R² = 8 × 10 11 I 10×10 χ=0.7, T sω1=0.9, T s ω2=0.01, ζ=0.5, δ=1,

[0266] The parameters of the state observer based on the neural network are set as l1 = 2, l2 = 4 × 10. 3 l3 = 1 × 10 6 k1 = k2 = k3 = 0.5

[0267] The parameters of the generalized proportional-integral observer are set as follows:

[0268] Experimental platform: The experimental platform is as follows Figure 2 As shown, the system includes a computer, data acquisition card, piezoelectric drive power supply, displacement sensor, excitation source, piezoelectric actuator, passive vibration isolator, and sensitive load. The passive vibration isolator and piezoelectric actuator are connected in series to form an integrated active-passive vibration isolation system. The experimental platform is connected as follows... Figure 2 As shown. In this embodiment, the excitation source is also a piezoelectric device, driven by a piezoelectric power supply. Both the control signal and the vibration signal are generated by a computer, converted into analog signals by a data acquisition card, and amplified by the piezoelectric power supply. The difference is that the vibration signal is transmitted to the excitation source to generate vibration, while the control signal is used to control the output displacement of the piezoelectric actuator, thereby suppressing vibration. The system characteristics of the experimental system under different mass load conditions are shown in the figure. Figure 3 As shown, variable mass loads can cause changes in system characteristics, posing a challenge to vibration suppression.

[0269] Vibration Suppression Experiment: To verify the vibration suppression performance of the proposed control algorithm, periodic vibration suppression experiments at different frequencies (Experiment 1) and complex vibration suppression experiments under variable mass load conditions (Experiment 2) were conducted. In Experiment 1, periodic vibration signals at different frequencies (5Hz, 25Hz, 50Hz, 75Hz) with an effective load mass of 150g were used to verify the vibration suppression performance of the system. The experimental results are as follows: Figure 4-8 As shown, the proposed method can suppress vibration signals of different frequencies and loads. In Experiment 2, the control performance of the proposed control algorithm under variable mass load conditions in a piezoelectric-driven active-passive vibration isolation system was verified using complex vibration signals (40Hz, 60Hz, 80Hz) containing different frequencies under different mass load conditions (100g, 150g). The experimental results are shown in Figure 1. Figure 8-9 As shown in the figure, the control method proposed in this invention can suppress various vibration signals and exhibits good vibration isolation performance.

Claims

1. An enhanced sliding mode predictive control method for a piezoelectric-driven vibration isolation system based on a generalized proportional-integral observer, characterized in that, The specific steps of this method are as follows: Step 1: Describe the piezoelectric-driven active and passive vibration isolation system under variable mass load conditions based on the dynamic model of the piezoelectric-driven active and passive vibration isolation system. The system is described as follows: (1) in For system state and , Let be the system order. and These are the system's input and output, respectively. To control the gain, Hysteresis of piezoelectric ceramics; unknown function This represents the system uncertainty caused by variable mass loads. Interference caused by external vibration; among which and These are the mismatch uncertainty and disturbance in the system, respectively; Given a function, parameters Related to and known physical properties of the system; Based on the Euler method, the discrete-time model of system (1) is derived as follows: (2) in The sampling period is The system state is unknown. The estimated value; Step 2: Use a neural network to approximate the unknown function in the piezoelectric-driven active and passive vibration isolation system (2). The neural network is described as follows: (3) in For the system input vector, For weight vectors, Let the vector be an active function in Gaussian form. To approximate the error, the optimal weights are: ; By applying a neural network to approximate the unknown function, system (2) is rewritten in the following form: (4) in , This represents a complex disturbance in the system; , For controller gain, Represents approximation error; To simplify the explanation, the function writing System (4) is rewritten in the following form: Define the composite perturbation at time k Approximate error at time k The rate of change of the composite perturbation is uniformly bounded. ,also, The rate of change is bounded, defined as ; For piezoelectric active and passive vibration isolation systems, the function The rate of change is uniformly bounded, and the norm of the function is defined as follows: ; The state observer for the discrete system (4) is as follows: (5) in These are estimates of the state and the disturbance, respectively. ,and These are the design parameters in the state observer; The state estimation error is: (6) in and These are the state, weight, and disturbance estimation errors, respectively. , ； Step 3: First, define and its higher-order differences ,in Indicates the order of the maximum difference; Assumption Additionally, definition and The derivation is as follows: (7) in The generalized proportional-integral observer is used to estimate and predict the aforementioned time-varying disturbances of mismatch and matching in the system. The generalized proportional-integral observer is as follows: (8) in Indicates to The estimate, These are the design parameters in the observer; The expression for the estimated error is as follows: (9) The matrix form of the estimation error is summarized as follows: (10) in , , ; By combining equations (7) and (8), the p-step forward perturbation preview is derived as follows: (11) in yes The estimated value, and ; Consider Lyapunov functions ,in , ,matrix and All are positive definite; definition ,and ,for The calculation is as follows: (12) If the eigenvalues ​​of matrix A satisfy Then for any positive definite matrix There exists an equation that satisfies Positive definite matrix Then formula (12) can be written as: (13) in Representation matrix The smallest eigenvalue; Define matrix ,for The calculation is as follows: (14) If the eigenvalues ​​of matrix A satisfy Then for any positive definite matrix There exists an equation that satisfies Positive definite matrix Then equation (14) can be written as: (15) for The calculation is as follows: (16) in , , The convergence of the observer and Related to the item; Step 4: Define the sliding surface as (17) in , ,parameter Indicates that the independent variable is polynomial The coefficients of , whose eigenvalues ​​satisfy . Where Re() represents the real part of the complex number; The integrator for sliding mode error is defined as: (18) in It is the sliding mode state error; It is a forgetting factor; The sliding mode reaching law is defined as: (19) in , , It is a symbolic function; in addition, the function ; Define extended state , We can obtain: (20) (21) in , , , , , , , , , , ; The prediction is expressed as follows: (22) in , , , , , , , , , , The prediction and control time domains are the same, denoted as p; Combining the defined sliding mode convergence law, the predicted sliding surface, and the sliding mode state error integrator, a loss function is constructed, which is described as follows: (23) in and Represents the weight matrix; The new loss function is derived as follows: (24) in By setting The optimal control sequence is derived as follows: (25) Based on the principle of rolling time-domain optimization, the control signal applied to the system is derived as follows: (26) in ; The adaptive law design for weights in a neural network is as follows: (27) in and It is a normal number.

2. The enhanced sliding mode predictive control method for a piezoelectric-driven vibration isolation system based on a generalized proportional-integral observer according to claim 1, characterized in that, The order n of the piezoelectric active and passive vibration isolation system is taken as 3; The parameters in the system are set to , , , , , , , , , ; The parameters of the state observer based on the neural network are set as follows: , , , , , , ; The parameters of the generalized proportional-integral observer are set as follows: , , , , , , , , .

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