Enhanced sliding mode predictive control method based on generalized proportional integral observer for piezoelectric driving vibration isolation system

Abstract

The invention provides an enhanced sliding mode predictive control method based on a generalized proportional integral observer for a piezoelectric driving vibration isolation system, and aims to solve the problem of vibration suppression of the piezoelectric driving vibration isolation system under the condition of variable mass load. The method comprises the following steps: firstly, considering a variable mass load in a piezoelectric driving active and passive vibration isolator system under a variable mass load condition as uncertainty in the system and representing the uncertainty as an unknown function, and considering vibration as disturbance in the system; then, estimating an unknown system state by using a state observer of the neural network, and approaching an unknown function in the system by the neural network; estimating and predicting mismatching and matching time-varying disturbance in the system by using a generalized proportional-integral observer; an enhanced sliding mode prediction method is adopted to predict a sliding mode surface, an optimal control sequence is solved through a rolling optimization minimization loss function, and a control signal applied to the system is exported.

Description

Technical Field

[0001] The present invention relates to the technical field of control engineering, and particularly to an enhanced sliding mode predictive control method based on a generalized proportional integral observer for a piezoelectric drive vibration isolation system. Background Art

[0002] Currently, high-tech fields such as aerospace and optical instruments are booming, and the accuracy and reliability issues of precision instrument equipment during operation have received extensive attention from domestic and foreign research scholars. From the current research situation, vibration isolation technologies mainly include: passive vibration isolation, active vibration isolation, semi-active vibration isolation, and hybrid main-passive vibration isolation. Among them, the piezoelectric drive main-passive vibration isolation system has high resolution, reliability, and controllability, and is widely used for micro-vibration suppression. However, for the piezoelectric drive main-passive vibration isolator system 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, the present invention conducts research on an active vibration control method for a piezoelectric drive vibration isolation system under variable mass load conditions.

[0003] To ensure the performance of the vibration isolation system, the design of an active control method is of great significance. Currently, scholars have carried out research on vibration suppression using robust control, adaptive control, and sliding mode control. Among them, the sliding mode control method based on an observer has been widely used to suppress external vibrations due to its robustness to uncertainties and disturbances. However, the existing observers have room for improvement in estimating time-varying disturbances. In addition, the sliding mode control method has an inevitable chattering phenomenon. Therefore, improving the sliding mode control performance of complex uncertain systems remains a research topic worthy of study.

[0004] In the sliding mode predictive control method, by fully utilizing system information to predict the sliding mode state and making it track the reference trajectory, and deriving the control law through rolling horizon optimization, the control accuracy can be improved and the chattering phenomenon can be reduced. For a piezoelectric drive vibration isolation system, its system state information is difficult to obtain, and time-varying mismatched system uncertainties and disturbances continuously exist in the system. However, in the conventional sliding mode predictive control method, the system state is generally directly used for controller design, and only matched disturbances are considered. In addition, in the prediction horizon of the traditional sliding mode predictive control method, the disturbance is generally assumed to be constant, which is also conservative for dealing with time-varying disturbances.

[0005] Therefore, in the present invention, an enhanced sliding mode predictive control based on a neural network state observer and a generalized proportional integral observer is designed under the condition of only using the measured output of the system, which promotes the practical engineering application of the algorithm. In addition, mismatched uncertainties and disturbances are considered in the controller design, the time-varying disturbance is predicted through a generalized proportional integral observer, and a sliding mode state error integrator is introduced to improve the performance of the sliding mode predictive control on the piezoelectric drive vibration isolation system. Summary of the invention

[0006] The present invention mainly solves the control difficulties in the piezoelectric driven active and passive vibration isolation system under variable load conditions, fully considers the mismatch disturbance and uncertainty in the system, and designs an enhanced sliding mode predictive control method based on the 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 by the present invention is as follows:

[0008] An enhanced sliding mode predictive control method for a piezoelectric driven vibration isolation system based on a generalized proportional integral observer, the specific steps are as follows:

[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 the uncertainty in the system and expressed as an unknown function, the vibration is considered as the disturbance in the system, and 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, and the unknown function in the system is approximated by the neural network;

[0011] Step 3: Use the generalized proportional-integral observer to estimate and predict the mismatched and matched time-varying disturbances in the system;

[0012] Step 4: Based on the estimated system state, disturbance and approximated unknown function, a sliding surface that can handle mismatched disturbances and uncertainties is designed, and the sliding surface is predicted; the loss function is constructed by combining the defined sliding convergence law, the predicted sliding surface and the sliding state error integrator, and the optimal control sequence is solved by minimizing the loss function through rolling optimization, and the control signal applied to the system is derived.

[0013] The specific process of step 1 is:

[0014] The piezoelectric driven active and passive vibration isolation system is described according to the dynamic model of the piezoelectric driven active and passive vibration isolation system under variable mass load conditions. The system is described as follows:

[0015]

[0016] where x i is the system status and n is the system order. u and y are the input and output of the system, b is the control gain, and h(u) is the hysteresis of the piezoelectric ceramic. Unknown function g i (·) represents the system uncertainty caused by the variable mass load, d i, i = 1, …, n are disturbances caused by external vibrations; where g1(·), …, g n-1 (·) and d1, …, d n-1 are mismatched uncertainties and disturbances in the system. are known functions, and the parameters a0, …, a n-1 are related to the physical characteristics of the system and are known..

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

[0018]

[0019] where T s is the sampling period, is the estimated value of the unknown system state .

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

[0021] Use a neural network to approximate the unknown function in the piezoelectric-driven main and passive vibration isolation system (2), and the neural network is described as:

[0022]

[0023] where ξ = [ξ1, ξ2, …, ξ q T ∈ R q is the system input vector, is the weight vector, δ(ξ) is the active function vector in the Gaussian form, ε is the approximation error, and the optimal weight value is

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

[0025]

[0026] where 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 errors.

[0027] For the sake of simplicity in description, the function is written as δ i (k), i = 1, …, n, and system (4) is rewritten in the following form:

[0028]

[0029] Define the composite disturbance at time k as D(k) = [D1(k), …, D​n (k)] T , the approximation error ε(k) at time k = [ε1(k), …, ε n (k)] T ; where the change rate of the composite disturbance is uniformly bounded, In addition, the change rate of ε(k) is bounded and is defined as For the piezoelectric active and passive vibration isolation system, the function has a uniformly bounded change rate, and the norm of this function is defined as

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

[0031]

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

[0033] The state estimation error is:

[0034]

[0035] where and are 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 difference where τ represents the maximum difference order; assume D i,τ+1 (k) = 0, in addition, define and The derivation is as follows:

[0039]

[0040] where

[0041] Use the generalized proportional-integral observer to estimate and predict the above-mentioned mismatched and matched time-varying disturbances in the system. The generalized proportional-integral observer is as follows:

[0042]

[0043] Among them represents the estimation of D i,0 (k), …, D i,τ (k); is the design parameter in the observer.

[0044] The estimation error expression is as follows:

[0045]

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

[0047]

[0048] Among them

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

[0050]

[0051] Among them is the estimated value of, and G = [1, 0, …, 0].

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

[0053] Define 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 a positive definite matrix P1 that satisfies the equation Q1 = -((1 + γ)A T P1A - P1). Then formula (12) can be written as:

[0055]

[0056] where λ min (Q1) represents the minimum eigenvalue of matrix Q1;

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

[0058]

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

[0060]

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

[0062]

[0063] where ψ2 = λ min (Q2)-(2 + 2 / γ). It can be observed that the convergence of the observer is related to the term.

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

[0065] Design a sliding mode surface that can handle mismatched disturbances and uncertainties based on the estimated system state, disturbance, and approximated unknown function, and predict the sliding mode surface. In the process of enhanced sliding mode predictive control design, combine the defined sliding mode reaching law, predicted sliding mode surface, and sliding mode state error integrator to construct a loss function, and solve the optimal control sequence by minimizing the loss function through rolling optimization. The specific process is as follows:

[0066] Define the sliding mode surface as

[0067]

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

[0069] Define the integrator of the sliding mode state error as:

[0070]

[0071] where is the sliding mode state error. χ is a forgetting factor.

[0072] Define the sliding mode reaching law as:

[0073]

[0074] where \(0 < T\) s \(\omega_1 < 1\), \(T\) s \(\omega_2>0\), \(0 < \zeta < 1\), \(\delta>0\), In addition, the function

[0075] defines the extended state It can be obtained that:

[0076]

[0077] where

[0078] The predictive expression of \(s(k)\) is:

[0079]

[0080] where \(S(k)=[s(k + 1), s(k + 2), \cdots, s(k + p)]\) T ,

[0081] \(U(k)=[u(k), u(k + 1), \cdots, u(k + p - 1)]\) T ,

[0082] The prediction and control horizons are the same, denoted as \(p\).

[0083] Combining the defined sliding mode reaching law, the predicted sliding mode surface, and the sliding mode state error integrator to construct a loss function, which is described as:

[0084] \(J(k)=S\) T (k) \(R_1 S(k)+U\) T (k) \(R_2 U(k)\) (23)

[0085] where \(R_1\) and \(R_2\) represent weight matrices.

[0086] The new loss function is obtained as:

[0087]

[0088] where

[0089] By setting The optimal control sequence is derived as:

[0090]

[0091] According to the principle of rolling horizon optimization, the control signal applied to the system is derived as:

[0092]

[0093] where \(G = [1, 0, \ldots, 0]\).

[0094] The adaptation law of the weights in the neural network is designed as:

[0095]

[0096] where and \(k\) i are positive constants.

[0097] Theorem 1: For the system (1) affected by system uncertainties and disturbances, by designing the state observer (5), the generalized proportional-integral observer (8), the sliding mode surface (17), the control law (26) and the update law (27), the proposed predictive sliding mode control scheme can ensure that: (1) the sliding mode state the state estimation error the disturbance error and the weighted error are bounded; (2) the system output is bounded, reducing the negative impact of mismatched uncertainties and disturbances on the system output.

[0098] Proof: For simplicity of analysis, the weight matrix of the control action can be assumed to be \(R_2 = 0\). According to the prediction of \(s(k)\) and the optimal control sequence (25), the sliding mode state is derived as:

[0099]

[0100] Consider the Lyapunov function \(V\) c (k)=V0(k)+V3(k)+V4(k), where

[0101] Combining equation (28) with \(\Delta V_3(k)=V_3(k + 1)-V_3(k)\) is calculated as follows:

[0102]

[0103] According to the update law (27), the estimation error of the weights in the neural network is derived as:

[0104]

[0105] where

[0106] V4(k) is calculated as follows:

[0107]

[0108] Where According to Young's inequality, equation (31) can be written as:

[0109]

[0110] Combining equations (16), (29) and (32), V c (k) is derived as follows, V c (k) is calculated as follows:

[0111]

[0112] where the definitions of ψ1 and ψ2 are the same as those in equation (16).

[0113] By choosing appropriate parameters such that ψ1>0, ψ2>0, ψ3>0, ψ4>0, ψ5>0, the boundedness of the sliding mode state estimation error and can be proven. For simplicity of description, the upper bounds of the relevant variables are defined as

[0114] Next, the boundedness of the system output and the effects of mismatched uncertainties and disturbances on the system output are analyzed. The coordinate transformation conditions are defined as:

[0115]

[0116] Combining equation (4) and the system is transformed into:

[0117]

[0118] where

[0119] Then, it is obtained that the characteristic polynomial is According to the parameter selection conditions defined in the equation, when choosing equation (17), the modulus η i is derived from the following equation:

[0120]

[0121] Combining the transformation matrix, the Jordan matrix of is obtained matrix has the same eigenvalues as J. Additionally, assume that the eigenvalues λ i are distinct, which means that the eigenvalues η i are distinct. Therefore, the matrix J can be written as J = diag[η1, η2, …, η n-1 .

[0122] Based on the Jordan matrix, the iterative equation (35) can be obtained Then, its norm is derived as follows:

[0123]

[0124] It should be noted that based on Assumptions 1 and 2, and Equation (33), the norm of is derived as Then, for k → ∞, the bound of

[0125]

[0126] It is observed that the system output is bounded, and its upper bound is independent of the bounds of the perturbation and uncertainty. Although the compensation error still affects the system output, its magnitude may be much smaller than the magnitudes of the disturbance and uncertainty itself. Therefore, the negative impacts of the mismatched uncertainty and disturbance can be reduced in the system output.

[0127] The relevant parameters are selected according to the following principles:

[0128] The order of the piezoelectric main - passive vibration isolation system is taken as 3.

[0129] The parameters in the system are set as 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,

[0130] The parameters of the neural - network - based state observer are set as l1 = 2, l2 = 4×10 3 , l3 = 1×10 6 , k1 = k2 = k3 = 0.5,

[0131] The parameters of the generalized proportional - integral observer are set as

[0132] The beneficial effects of the present invention are as follows:

[0133] The present 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 state observer based on a neural network is designed, which can achieve the approximation of unknown functions in the system and estimate the unknown states of the system, facilitating the design of an 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 conservativeness of the predictive sliding mode control method in dealing with time-varying disturbances. Third, a sliding mode surface containing the estimated states, disturbances, and approximate uncertainties is designed, which can handle unmatched disturbances and uncertainties in the system, thereby improving the performance of the vibration isolation system. Fourth, the introduced sliding mode state error integrator enables the sliding mode predictive control to correct the generated control signal based on past error information, which can weaken the negative impact of uncertainties on control performance, thereby 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.

[0134] The method of the present invention fully considers the problems of unmeasurable system states, unmatched disturbances caused by external vibrations, and system uncertainties brought about by variable mass loads, designs an enhanced sliding mode predictive control method based on a generalized proportional integral observer for a piezoelectric-driven vibration isolation system, improves the vibration suppression performance of the piezoelectric-driven vibration isolation system, and promotes its practical engineering application. Brief Description of the Drawings

[0135] Figure 1 is the schematic diagram of the enhanced sliding mode predictive control method based on a generalized proportional integral observer

[0136] Figure 2 is the schematic diagram of the connection of the experimental platform;

[0137] Figure 3 are the amplitude-frequency and phase-frequency characteristics of the piezoelectric-driven main passive vibration isolation system under variable mass load conditions;

[0138] Figure 4 is the experimental result diagram of the 5Hz suppression experiment under a 150g load;

[0139] Figure 5 is the experimental result diagram of the 25Hz suppression experiment under a 150g load;

[0140] Figure 6 is the experimental result diagram of the 50Hz suppression experiment under a 150g load;

[0141] Figure 7 It is the result diagram of the 75Hz suppression experiment under a 150g load;

[0142] Figure 8 It is the result diagram of the complex vibration suppression experiment under a 100g load;

[0143] Figure 9 It is the result diagram of the complex vibration suppression experiment under a 150g load; Detailed implementation manners

[0144] The content of the present invention will be further specifically described below in conjunction with the accompanying drawings and embodiments.

[0145] The enhanced sliding mode predictive control method based on a generalized proportional integral observer for a piezoelectric drive vibration isolation system described in this embodiment has a control block diagram as Figure 1 shown. The design steps of the control method of the present invention are specifically as follows:

[0146] Step 1: Combining the dynamic model of the piezoelectric drive active and passive vibration isolation system, deduce the system description of the piezoelectric drive active and passive vibration isolator under variable mass load conditions, consider the variable mass load as the uncertainty in the system and represent it as an unknown function, and consider the vibration as the disturbance in the system. Both mismatched disturbances and uncertainties exist in the system. The system description is as follows:

[0147]

[0148] where x i is the system state and n is the system order. u and y are the input and output of the system, 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, and d i , i = 1,..., n are the disturbances caused by external vibrations; where g1(·),..., g n-1 (·) and d1,..., d n-1 are the mismatched uncertainties and disturbances in the system. is a known function, and the parameters a0,..., a n-1 are related to the physical characteristics of the system and are known.. Based on the Euler method, the discrete-time model of system (1) is deduced as

[0149]

[0150] where T s is the sampling period, is the estimated value of the unknown system state .

[0151] The use of a neural network to approximate an unknown function is described as follows:

[0152]

[0153] where ξ = [ξ1, ξ2, …, ξ q T ∈R q is the system input vector, is the weight vector, δ(ξ) is the activation function vector in Gaussian form, ε is the approximation error, and the optimal weight value is

[0154] By applying the neural network to approximate the unknown function, system (2) is rewritten in the following form

[0155]

[0156] where represents the composite disturbance in the system. u(k) + h(u(k)) + d n (k), b0 is the controller gain, ε i , i = 1, 2, …, n represent the approximation errors. For simplicity of explanation, the function is written as δ i (k), i = 1, …, n.

[0157] Definition 1: D(k) = [D1(k), …, D n (k)] T and ε(k) = [ε1(k), …, ε n (k)] T .

[0158] Assumption 1: The rate of change of the composite disturbance is uniformly bounded, In addition, the rate of change of ε(k) is bounded and is defined as

[0159] Assumption 2: For the piezoelectric primary and passive vibration isolation system, the rate of change of the function is uniformly bounded. Its norm is defined as

[0160] The specific process of Step 2 is as follows:

[0161] Design a neural network-based state observer according to the model derived in Step 1 to estimate the unknown system state and disturbance in the system, facilitating the design of the controller under the condition that only the system output is measurable. The specific process is as follows:

[0162] Design a state observer for the discrete system (4) as follows:

[0163] ​

[0164] where are the estimates of the state and the disturbance respectively, and B = [0, …, 0, T s b0], and l1, …, l n are the design parameters in the state observer.

[0165] The state estimation error is as follows:

[0166]

[0167] where and are the state, weight, and disturbance estimation errors respectively.

[0168]

[0169] The specific process of step 3 is as follows:

[0170] Design a generalized proportional-integral observer to estimate and predict the mismatched and matched time-varying disturbances in the system. The specific process is as follows:

[0171] First, define D i (k) = D i,0 (k), and its higher-order differences where τ represents the maximum order of differences. Assume D i,τ+1 (k) = 0. Additionally, define and The derivation is as follows:

[0172]

[0173] where

[0174] Subsequently, the generalized proportional-integral observer is designed as follows:

[0175]

[0176] where represents the estimate of D i,0 (k), …, D i,τ (k), are the design parameters in the observer;

[0177] The expression of the estimation error is as follows:

[0178]

[0179] The matrix form of the estimation error is summarized as follows:

[0180]

[0181] Among them

[0182]

[0183] By combining equations (7) and (8), the p-step forward perturbation preview is derived as:

[0184]

[0185] Among them is the estimated value of, and G = [1, 0, …, 0].

[0186] By considering the high-order difference, the generalized proportional-integral observer has the ability to accurately estimate the time-varying perturbation. In this study, a generalized proportional-integral observer was developed for a system with perturbations in multiple channels. It promotes the application of the generalized proportional-integral observer in complex systems. In addition, in the generalized proportional-integral observer, choosing a larger cost τ results in higher estimation accuracy, but also increases the computational burden. In this study, the order was set to τ = 2 to balance this trade-off.

[0187] Considering the Lyapunov function V o (k) = V1(k) + V2(k), where the matrices P1 and P2 are both positive definite.

[0188] Define and γ > 0, the calculation of V1(k) is as follows:

[0189]

[0190] If the eigenvalues of matrix A satisfy then for any positive definite matrix Q1, there exists a positive definite matrix P1 that satisfies the equation Q1 = -((1 + γ)A T P1A - P1). Then formula (12) can be written as:

[0191]

[0192] where λ min (Q1) represents the minimum eigenvalue of matrix Q1;

[0193] Define the matrix The calculation of V2(k) is as follows:

[0194]

[0195] If the eigenvalues of matrix A satisfy Then for any positive definite matrix Q2, there exists a positive definite matrix P2 that satisfies the equation . Then equation (14) can be written as:

[0196]

[0197] The calculation of V0(k) is as follows:

[0198]

[0199] where ψ2 = λ min (Q2)-(2 + 2 / γ). It can be observed that the convergence of the observer is related to the term. In Theorem 1 below, the stability of the closed-loop system is analyzed.

[0200] The specific process of Step 4 is as follows:

[0201] Design a sliding mode surface that can handle mismatched disturbances and uncertainties based on the estimated system state, disturbance, and approximated unknown function, and predict the sliding mode surface. In the design process of enhanced sliding mode predictive control, combine the defined sliding mode reaching law, predicted sliding mode surface, and sliding mode state error integrator to construct a loss function, and solve the optimal control sequence by minimizing the loss function through rolling optimization. The specific process is as follows:

[0202] Based on the above observer, an enhanced predictive sliding mode method is designed in this section. Define the sliding mode surface as

[0203]

[0204] where H = [m2, m3, …, m n , 0], and the parameter m n = 1, m1, m2, …, m n-1 represents the coefficients of the polynomial p(λ) = λ n-1 + m n-1 λ n-1 + … m2λ + m1 with the independent variable λ, and the eigenvalues satisfy 2Re(λ i ) + |λ i | 2 T < 0, where Re() represents the real part of a complex number.

[0205] The integrator of the sliding mode state error is defined as:

[0206]

[0207] where is the sliding mode state error. χ is a forgetting factor.

[0208] The sliding mode reaching law is defined as:

[0209]

[0210] where \(0 < T\) s \(\omega_1 < 1\), \(T\) s \(\omega_2>0\), \(0 < \zeta < 1\), \(\delta>0\), In addition, the function

[0211] defines the extended state It can be obtained that:

[0212]

[0213]

[0214] where

[0215] The predicted expression form of \(s(k)\) is:

[0216]

[0217] where \(S(k)=[s(k + 1),s(k + 2),\cdots,s(k + p)] T , \(U(k)=[u(k),u(k + 1),\cdots,u(k + p - 1)] T , The prediction and control time domains are the same, denoted as \(p\).

[0218] Compared with the traditional sliding mode predictive control method, the integrator of the introduced sliding mode state error enables the proposed method to correct the control signal based on past error information. This process continues until the state is consistent with its reference value, thereby improving the performance of the closed-loop system in dealing with uncertainties.

[0219] The loss function is designed as:

[0220] \(J(k)=S T (k)R_1S(k)+U T (k)R_2U(k)\ (23)

[0221] where \(R_1\) and \(R_2\) represent weight matrices.

[0222] The new loss function is obtained as:

[0223]

[0224] where

[0225] By setting The optimal control sequence is derived as:

[0226]

[0227] According to the principle of receding horizon optimization, the control signal applied to the system is derived as:

[0228]

[0229] where \(G = [1, 0, \ldots, 0]\).

[0230] The adaptation law of the weights in the neural network is designed as:

[0231]

[0232] where and \(k\) i are positive constants.

[0233] The preview disturbance information in the control law matrix \(\Gamma(k)\) can be obtained through Equation (11). This eliminates the limitations of the traditional predictive sliding mode control method in dealing with time-varying disturbances.

[0234] Theorem 1: For the system (1) affected by system uncertainties and disturbances, by designing the state observer (5), the generalized proportional-integral observer (8), the sliding mode surface (17), the control law (26) and the update law (27), the proposed predictive sliding mode control scheme can ensure that: (1) the sliding mode state the state estimation error the disturbance error and the weighted error are bounded; (2) the system output is bounded, reducing the negative impact of unmatched uncertainties and disturbances on the system output.

[0235] Proof: For the simplicity of analysis, the weight matrix of the control action can be assumed to be \(R_2 = 0\). According to the prediction of \(s(k)\) and the optimal control sequence (25), the sliding mode state is derived as:

[0236]

[0237] Consider the Lyapunov function \(V\) c (k)=V0(k)+V3(k)+V4(k), where

[0238] Combining Equation (28) with △V3(k) = V3(k + 1) - V3(k) is calculated as follows:

[0239]

[0240] According to the update law (27), the estimation error of the weights in the neural network is derived as:

[0241]

[0242] where

[0243] V4(k) is calculated as follows:

[0244]

[0245] where According to Young's inequality, equation (31) can be written as:

[0246]

[0247] Combining equations (16), (29) and (32), V c (k) is derived as follows, V c (k) is calculated as follows:

[0248]

[0249] where the definitions of ψ1 and ψ2 are the same as those in equation (16).

[0250] By choosing appropriate parameters such that ψ1 > 0, ψ2 > 0, ψ3 > 0, ψ4 > 0, ψ5 > 0, the boundedness of the sliding mode state estimation error and can be proved. For simplicity of description, the upper bounds of the relevant variables are defined as

[0251] Next, the boundedness of the system output and the effects of the mismatched uncertainties and disturbances on the system output are analyzed. The coordinate transformation conditions are defined as:

[0252]

[0253] Combining equation (4) and the system is transformed into:

[0254]

[0255] where

[0256] Then, obtain The characteristic polynomial is Select the modulus η when choosing the conditional equation (17) according to the parameters defined in the equation i It is derived from the following formula as:

[0257]

[0258] Combined with the transformation matrix, obtain The Jordan matrix of Matrix Has the same eigenvalues as J. Additionally, assume the eigenvalue λ i Is different, which means the eigenvalue η i Is different. Therefore, the matrix J can be written as J = diag[η1, η2, …, η n-1 .

[0259] Based on the Jordan matrix, the iterative equation (35) can be obtained Then, its norm is derived as:

[0260]

[0261] It should be noted that Based on Assumptions 1 and 2, and Equation (33), The norm of is derived as Then, for k → ∞, The bound of is derived as:

[0262]

[0263] It is observed that the system output Is bounded, and its upper bound is independent of the bounds of the perturbation and uncertainty. Although the compensation error still affects the system output, its magnitude may be much smaller than the magnitudes of the disturbance and uncertainty itself. Therefore, the negative impacts of the mismatched uncertainty and disturbance can be reduced in the system output.

[0264] The relevant parameters are selected according to the following principles:

[0265] The order of the piezoelectric main - passive vibration isolation system is taken as 3.

[0266] The parameters in the system are set as 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,

[0267] The parameters of the state observer based on neural network are set as l1 = 2, l2 = 4×10 3 , l3 = 1×10 6 , k1 = k2 = k3 = 0.5,

[0268] The parameters of the generalized proportional-integral observer are set as

[0269] Experimental platform: The experimental platform is as Figure 2 shown, and includes a computer, a data acquisition card, a piezoelectric drive power supply, a displacement sensor, an excitation source, a piezoelectric actuator, a passive vibration isolator and a sensitive load. Among them, the passive vibration isolator and the piezoelectric actuator form a main-passive integrated vibration isolation system in series. The connection mode of the experimental platform is as Figure 2 shown. In this embodiment, the excitation source is also a piezoelectric device and is also driven by a piezoelectric drive power supply. The control signal and the vibration signal are both generated by the computer, converted into analog signals by the data acquisition card, and amplified by the piezoelectric drive 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 to suppress vibration. The system characteristics under different mass load conditions of the experimental system are as Figure 3 shown. The variable mass load will bring changes in system characteristics, posing challenges to vibration suppression.

[0270] Vibration suppression experiment: To verify the vibration suppression performance of the proposed control algorithm, different frequency periodic vibration suppression experiments (Experiment 1) and vibration suppression experiments for complex vibrations under variable mass load conditions (Experiment 2) were respectively carried out. In Experiment 1, periodic vibration signals with different frequencies (5 Hz, 25 Hz, 50 Hz, 75 Hz) of a payload with a mass of 150 g were used to verify the vibration suppression performance of the system. The experimental results are as Figures 4 - 8 shown, proving that the proposed method can suppress vibration signals with different frequencies and different loads. In Experiment 2, under different mass load conditions (100 g, 150 g), complex vibration signals containing different frequencies (40 Hz, 60 Hz, 80 Hz) were used to verify the control performance of the proposed control algorithm under variable mass load conditions of the piezoelectric drive main-passive vibration isolation system. The experimental results are as Figures 8 - 9 shown. It can be seen from the figure that the control method proposed by the present invention can suppress various vibration signals and exhibits good vibration isolation performance.

Claims

1. An enhanced sliding mode predictive control method based on a generalized proportional integral observer for a piezoelectric-driven vibration isolation system, characterized in that The specific steps of this method are as follows: Step 1: Describe the piezoelectric-driven main and passive vibration isolation system under variable mass load conditions according to the dynamic model of the piezoelectric-driven main 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 mismatched disturbances and uncertainties exist in the system; Step 2: Use the state observer of the neural network to estimate the unknown system state, and the unknown function in the system is approximated by the neural network; Step 3: Use the generalized proportional integral observer to estimate and predict the mismatched and matched time-varying disturbances in the system; Step 4: Design a sliding mode surface that can handle mismatched disturbances and uncertainties based on the estimated system state, disturbances, and approximated unknown function, and predict the sliding mode surface; Combine the defined sliding mode reaching law, the predicted sliding mode surface, and the sliding mode state error integrator to construct a loss function, and solve the optimal control sequence by minimizing the loss function through rolling optimization, and derive the control signal applied to the system.

2. The enhanced sliding mode predictive control method based on a generalized proportional integral observer for the piezoelectric drive vibration isolation system according to claim 1, characterized in that, The specific process of Step 1 is: Describe the piezoelectric-driven main and passive vibration isolation system under variable mass load conditions according to the dynamic model of the piezoelectric-driven main and passive vibration isolation system. This system is described as follows: where x i is the system state and n is the system order, u and y are the input and output of the system respectively, 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 are the disturbances caused by external vibrations; where g1(·), …, g n-1 (·) and d1, …, d n-1 are the mismatched uncertainty and disturbance in the system respectively. is a known function, and the parameters a0, …, a n-1 are related to the physical characteristics of the system and are known; Based on the Euler method, the discrete-time model of system (1) is derived as: where T s is the sampling period, is the estimated value of the unknown system state .

3. The enhanced sliding mode predictive control method based on a generalized proportional integral observer for the piezoelectric drive vibration isolation system according to claim 2, wherein, The specific process of Step 2 is: Use the neural network to approximate the unknown function in the piezoelectric-driven main and passive vibration isolation system (2). The neural network is described as: where ξ = [ξ1, ξ2, …, ξ q T ∈ R q is the system input vector, is the weight vector, δ(ξ) is the active function vector in Gaussian form, ε is the approximation error, and the optimal weight is ​ By applying the neural network to approximate the unknown function, system (2) is rewritten in the following form where 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 represents the approximation error; For simplicity of explanation, the function is written as δ i (k), i = 1, …, n, and the system (4) is rewritten in the following form: Define the composite disturbance D(k) at time k as D(k)=[D1(k),…,D n (k)] T , and the approximation error ε(k) at time k as ε(k)=[ε1(k),…,ε n (k)] T ; where the change rate of the composite disturbance is uniformly bounded, In addition, the change rate of ε(k) is bounded, defined as For the piezoelectric active and passive vibration isolation system, the change rate of the function is uniformly bounded, and the norm of this function is defined as The state observer for the discrete system (4) is as follows: where are the estimates of the state and the disturbance respectively, and B = [0, …, 0, T s b0], and l1, …, l n are design parameters in the state observer; The state estimation error is: wherein and are the state, weight, and disturbance estimation error, respectively; 4. The enhanced sliding mode predictive control method based on a generalized proportional integral observer for the piezoelectric drive vibration isolation system according to claim 3, characterized in that The specific process of Step 3 is: First, define D i (k) = D i,0 (k), and its higher-order differences where τ represents the maximum order of difference; Assume D i,τ+1 (k) = 0. Additionally, define and The derivation is as follows: Among them Use the generalized proportional integral observer to estimate and predict the above-mentioned mismatched and matched time-varying disturbances in the system. The generalized proportional integral observer is as follows: Among them denotes the estimation of D i,0 (k),…,D i,τ (k); is the design parameter in the observer; The estimation error expression is as follows: The matrix form of the estimation error is summarized as follows: Among them By combining equations (7) and (8), the p-step forward disturbance preview is derived as: Among them is 's estimated value, and G = [1, 0, …, 0]; Consider the Lyapunov function V o (k)=V1(k)+V2(k), where the matrices P1 and P2 are both positive definite; Definition and γ > 0, the calculation of V1(k) is as follows: If the eigenvalues of matrix A satisfy then for any positive definite matrix Q1, there exists a positive definite matrix P1 that satisfies the equation Q1 = -((1 + γ)A T P1A - P1); then formula (12) can be written as: where λ min (Q1) represents the minimum eigenvalue of matrix Q1; Define matrix The calculation of V2(k) is as follows: If the eigenvalues of matrix A satisfy then for any positive definite matrix Q2, there exists a positive definite matrix P2 that satisfies the equation ; then equation (14) can be written as: The calculation for V0(k) is as follows: where Ξ1 = (2 + 2 / γ)||P1||T s 2 +(2 + 2 / γ)||P2||T s 2 , ψ1 = λ min (Q1)-(2 + 2 / γ)||P2||T s 2 , ψ2 = λ min (Q2)-(2 + 2 / γ); The convergence of the observer is related to terms.

5. The enhanced sliding mode predictive control method based on a generalized proportional integral observer for the piezoelectric drive vibration isolation system according to claim 4, characterized in that The specific process of Step 4 is: Define the sliding mode surface as Among them H = [m2, m3, …, m n , 0], the parameter m n = 1, m1, m2, …, m n-1 represents the coefficients of the polynomial p(λ) = λ n-1 + m n-1 λ n-1 + … m2λ + m1 with the independent variable λ, and its eigenvalues satisfy 2Re(λ i ) + |λ i | 2 T < 0, where Re() represents the real part of a complex number; Define the integrator of the sliding mode state error as: wherein is the sliding mode state error; χ is a forgetting factor; Define the sliding mode reaching law as: where 0 < T s ω1 < 1, T s ω2 > 0, 0 < ζ < 1, δ > 0, sgn is the sign function; in addition, the function Define the extended state It can be obtained that: Among them The predicted expression form of s(k) is: where S y(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 horizons are the same and denoted as p; Combine the defined sliding mode reaching law, the predicted sliding mode surface, and the sliding mode state error integrator to construct a loss function, which is described as: J(k) = S T (k)R1S(k) + U T (k)R2U(k) (23) Where R1 and R2 represent weight matrices; The new loss function is obtained as: Among them By setting The optimal control sequence is derived as: According to the principle of rolling horizon optimization, the control signal applied to the system is derived as: Where G = [1, 0, …, 0]; The adaptive law for the weights in the neural network is designed as: wherein and k i are positive constants.

6. According to the enhanced sliding mode predictive control method based on the generalized proportional integral observer for the piezoelectric-driven vibration isolation system described in claim 1, it is characterized in that The order n of the piezoelectric main and passive vibration isolation system is taken as 3; The parameters in the system are set as M = 8 .8 × 10 7 , 7 × 104 , 1 , H = 7 × 104 , 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, The parameters of the neural network-based state observer are set as l1 = 2, l2 = 4×10 3 , l3 = 1×10 6 , k1 = k2 = k3 = 0.5, The parameter settings of the generalized proportional-integral observer are as follows

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