Output feedback sliding mode prediction control method of piezoelectric driving active and passive integrated vibration isolation system

Abstract

The invention discloses an output feedback sliding mode prediction control method of a piezoelectric driving active and passive integrated vibration isolation system. The invention aims to solve the problems that the input and output characteristics of an actuator are neglected in the control of a vibration isolation system, the system state is not measurable, and the vibration suppression performance is influenced by mismatching disturbance, and optimization and input constraints are considered in a control algorithm. According to the method, an output feedback sliding mode predictive control algorithm based on a control guide model is designed by considering the neglected actuator input and output characteristics in an active and passive integrated vibration isolation system, the unmeasurable state of the system and the vibration suppression effect of the sliding mode predictive control algorithm which is limited by unconsidered mismatching disturbance; therefore, the suppression of the vibration signal is realized. According to the invention, the vibration suppression performance is improved, and the practical engineering application is promoted.

Description

Technical Field

[0001] The invention relates to the technical field of control engineering, and in particular to an output feedback sliding mode prediction control method for a piezoelectric driven active and passive integrated vibration isolation system. Background Art

[0002] Sensitive instruments on precision manufacturing equipment require a stable operating environment to ensure equipment performance. However, precision equipment is often affected by micro-vibrations with wide frequency distribution, small vibration amplitude, and complex vibration sources, which seriously reduces the performance of precision equipment. It is of great significance to conduct a detailed analysis of the micro-vibration environment and vibration source characteristics of sensitive precision equipment and design a high-performance vibration suppression system. At present, the commonly used vibration suppression methods mainly include passive vibration isolation, active vibration isolation, and active and passive integrated vibration isolation. Passive vibration isolation does not require external energy input, but its adaptability is limited. Active vibration isolation includes sensors and actuators. The vibration of sensitive loads is measured by sensors, and then the actuators are controlled to suppress vibration based on the corresponding control algorithm. However, its robustness and stability are completely dependent on the control algorithm, and its reliability needs to be strengthened. Active and passive integrated vibration isolation combines the advantages of passive vibration isolation and active vibration isolation, which can make the system adaptable to various vibrations, suppress low-frequency vibrations that passive vibration isolation cannot handle, and ensure the reliability of the system. Therefore, the present invention conducts vibration suppression research based on the active and passive integrated vibration isolation system.

[0003] For active and passive integrated vibration isolation systems, accurate models are of great significance to the design of active vibration suppression control algorithms. Conventional active and passive integrated models mainly include finite element models, dynamic models that describe the relationship between actuator output force and system output displacement, or dynamic decoupling models. However, finite element models are generally used to analyze the modal vibration shapes of sensitive loads, and the input and output characteristics of the actuator are generally not considered in the above models, which will hinder the design of active vibration suppression control algorithms. In the active and passive integrated vibration isolation system, the input and output characteristics of the actuator are coupled with the characteristics of the passive vibration isolation system, making the establishment of a control-oriented model for the active and passive integrated vibration isolation system still an open problem.

[0004] At present, scholars have carried out research on vibration suppression using robust control, adaptive control, and sliding mode control. Among them, sliding mode control is widely used due to its fast response and strong robustness. However, in the existing sliding mode control algorithms for vibration suppression, optimization and constraints are less considered. The sliding mode predictive control algorithm combines the advantages of sliding mode control and model predictive control algorithms, and can explicitly handle constraints while ensuring robustness. In addition, thanks to the rolling optimization idea of ​​model predictive control, the inherent vibration phenomenon in the algorithm is reduced. However, the existing sliding mode predictive control algorithms generally assume that the system state is known and only consider matching disturbances. In the vibration suppression system, the system state is unmeasurable, and external vibrations will bring unmatched disturbances to the system.

[0005] Therefore, in the present invention, the problems of difficulty in establishing a control-oriented model, unmeasurable system state and mismatched disturbances in the active and passive integrated vibration isolation system are considered, a control-oriented model of the system is derived, and based on this, an output feedback sliding mode predictive control algorithm considering mismatched disturbances is designed to improve the performance of the active and passive integrated vibration isolation system and promote the practical application of the control algorithm. Summary of the invention

[0006] The present invention mainly solves the control difficulties in the piezoelectric driven active and passive integrated vibration isolation system, derives a control-oriented model of the active and passive integrated vibration isolation system, and designs a high-performance output feedback sliding mode predictive control method using only the output information of the system to achieve the suppression of various vibration signals under actual working conditions.

[0007] The technical solution adopted by the present invention is as follows:

[0008] Step 1: Combining the electromechanical model of the piezoelectric actuator and the input-output characteristics of the vibration isolation system, a control-oriented model of the active-passive integrated vibration isolation system is derived, and the disturbances are fully considered as matched and unmatched disturbances in the system;

[0009] Combining the circuit equivalent model and the dynamic model, the system equation is as follows:

[0010]

[0011] Among them, V in is the input voltage, k amp is the fixed voltage amplifier gain, R 0 is the equivalent resistance, V h The hysteresis effect generates voltage, H represents the hysteresis effect of piezoelectric, V a is the voltage for the piezoelectric effect, C a is the system equivalent capacitance, q 1 ,q 2 ,q 3 is the charge in the system, T emIndicates the conversion ratio of mechanical energy to electrical energy during the piezoelectric effect, x a is the output displacement of the piezoelectric actuator; m a is the mass of the piezoelectric actuator, m p is the mass of the sensitive load, x p is the displacement of the sensitive load, c 1 , c 2 , k 1 , k 2 are the equivalent damping and stiffness in the system respectively;

[0012] V in -(H(q 1 ) / k amp ) is regarded as the equivalent input signal u(t), and the Laplace transform of the above formula (1) can be obtained:

[0013]

[0014] Based on the above formula (2), from the equivalent input voltage signal u(s) to the sensitive load displacement x p The transfer function of (s) is derived as:

[0015]

[0016] To facilitate controller design, the model can be reduced to:

[0017]

[0018] The model parameter b 0 ,a 0 ,...,a n-1 , parameter identification needs to be performed according to the characteristics of the vibration isolation system;

[0019] Taking the vibration signal into full consideration as the matching and mismatching disturbances in the system, the state equation is written as:

[0020]

[0021] x 1 ~x n is the system state, u is the controller input voltage, and y is the sensitive load displacement output by the controller;

[0022] The discrete expression of the system is:

[0023]

[0024] k represents the current time, T is the time interval adopted by the system;

[0025] Formula (10) can be written in matrix form as follows:

[0026]

[0027] in B=[0,…,b 0 T] T , C=[1,…,0] T , d = [Td 1 ,Td 2 ,…,Td n ];

[0028] Definition 1: The variable μ satisfies and When, it can be defined as O(T l ), l is the corresponding order;

[0029] Assumption 1: The error is bounded ||d(k)||≤d * ,||d(k)-d(k-1)|≤θ * ;

[0030] Step 2: Based on the derived control-oriented model, design state and disturbance observers to estimate the unknown system states and disturbances in the system;

[0031] The state and disturbance observers are designed for the discrete expression (10) of the system as follows:

[0032]

[0033] The function z i (k) = d i (k)-γ i x i (k) is the auxiliary variable introduced in the observer, β i and γ i is the parameter to be designed in the observer, x i , d i , z i Estimates of , i = 1, 2, …, n;

[0034] Formula (12) can be written in matrix form as follows:

[0035]

[0036] The definitions of matrices A and B are the same as in formula (11), γ=diag[γ 1 ,γ 2 ,…,γ n ], D=diag[1-Tγ 1 ,1-Tγ2 ,…1-Tγ n ],N=[0,…,-Tγ n b 0 ] T ,

[0037] The estimated error of the observer is defined as: And the bound of the estimation error is defined as:

[0038] Step 3: Design a sliding surface that contains system state and disturbance estimation information to eliminate the impact of mismatched disturbances on system performance, and predict the sliding surface at future times based on the derived control-oriented model;

[0039] The discrete time sliding surface is designed as:

[0040]

[0041] Where M = [m 1 ,…,m n-1 ,1],H=[m 2 ,…,m n-1 ,1,0]; parameter m 1 ,…,m n-1 The polynomial p(λ)=λ n-1 +m n-1 λ n-1 +…m 2 λ+m 1 The coefficients of the polynomial, and the characteristic roots of the polynomial satisfy 2Re(λ i )+|λ i 2 T<0, where Re(·) represents the real part of the complex number; combined with the state and disturbance observer (12), a one-step prediction of the sliding surface is obtained:

[0042]

[0043] Its p-step prediction formula is derived as follows:

[0044]

[0045] Where s(k)=[s(k+1),s(k+1),…s(k+p)] T ,u(k)=[u(k),u(k+1),…u(k+p-1)] T , Φ=[MA;MA 2 ;…;MA p ],

[0046] Step 4: According to the rolling optimization principle of model predictive control, a loss function is constructed to make the predicted sliding surface track the designed enhanced exponential reaching law, and the optimal control sequence is obtained by minimizing the constructed loss function. The first signal of the optimal control sequence obtained at each moment is applied to the piezoelectric actuator to suppress vibration;

[0047] The enhanced discrete-time sliding mode reaching law is designed as:

[0048]

[0049] Where 0 <Tω 1 <1,ω 2 >0,0<χ<1,δ>0, is the parameter in the sliding mode reaching law; when When the sliding mode state s r (k) is bounded, and |s r (k)|≤O(T 2 );

[0050] According to the design principle of sliding mode predictive control, the loss function is designed so that the predicted sliding mode state tracks the designed sliding mode reaching law, and the optimal control sequence is obtained by minimizing the loss function; combining the predicted sliding surface (16) and the reaching law (17), the following loss function is defined as follows:

[0051] J(k)=[s r (k)-s(k)] T Q[s r (k)-s(k)]+u T (k)Ru(k) (18)

[0052] where s r (k) = [s r (k+1),…s r (k+p)] T , s(k)=[s(k+1),…s(k+p)] T u(k)=[u(k),u(k+1),…u(k+p-1)] T , Q, R are weight matrices; the prediction time domain and the control time domain are the same, both are p; Substituting the predicted sliding surface formula into the loss function (18) yields:

[0053]

[0054] Convert to: J(k) = 2u T (k)Θ T Qξ+u T (k)(Θ T QΘ+R)u(k)+ξT Q

[0055] in Considering that T There is no relevant term of the control signal u(k) in ξ, which is ignored in the optimization process, and the loss function is simplified to:

[0056]

[0057] Minimize the loss function so that its derivative The optimal control sequence is:

[0058]

[0059] in The disturbance estimate at the future time contained in F(k) is difficult to obtain, so in actual control, Substitute F(k); according to the rolling optimization principle of predictive control, only the first control signal of the optimal control sequence is applied to the system, and the control law is derived as follows:

[0060]

[0061] Step 5: Consider the problem that the system is subject to output constraints, transform the constructed loss function into a quadratic programming problem, and use the Hildreth QP algorithm to solve the optimal control sequence under input constraints;

[0062] Consider the input constraint u in the controller design process min ≤u(k+j-1)≤u max (i=1,…,p), where u min ,u max are the upper and lower bounds of the control signal, respectively, and u min ≤0≤u max ;

[0063] Convert the optimization problem into a quadratic programming problem:

[0064]

[0065] where K = Θ T QΘ+R, The matrices Z and Ω are derived according to the actual constraints; the above quadratic programming problem is solved using Hildreth's QP algorithm, and the first element of the optimal solution is used as the optimal control signal to apply to the system.

[0066] The relevant parameters are selected according to the following principles:

[0067] The control-oriented model is obtained by using the least squares method: The parameter in the observer is set to β 1 =100,β 2 =500,β 3 =1000,γ 1 =1×10 3 , γ 2 =1×10 6 , γ 3 =6×10 8 The parameter design in the sliding mode reaching law is Tω 1 =0.9,ω 2 =0.1,χ=0.2,δ=1, The prediction domain and the control domain are equal, and p = 5. The parameter design in the controller is Q = 10. - 8 I 5×5 , R=I 5×5 , M = [1 × 10 6 ,40,1], H=[40,1,0].

[0068] The beneficial effects of the present invention are:

[0069] The method of the present invention takes into account that the input-output characteristics of the actuator ignored in the active-passive integrated vibration isolation system, the unmeasurable state of the system, and the unconsidered mismatched disturbance will limit the vibration suppression effect of the sliding mode predictive control algorithm, and designs an output feedback sliding mode predictive control algorithm based on the control-oriented model to achieve the suppression of vibration signals. The method has the following three main advantages: First, the control-oriented model of the active-passive integrated vibration isolation system is derived in the present invention. Different from the conventional vibration isolation system dynamics equation, the input-output characteristics of the actuator and the dynamic relationship of the vibration isolation system are fully considered in the present invention, and the obtained control-oriented model is more convenient for controller design. Second, the present invention designs an output feedback sliding mode predictive control algorithm to suppress vibration. The sliding mode predictive control algorithm combines the advantages of sliding mode control and predictive control, can ensure the robustness of the system and can explicitly handle constraints. In addition, the observer is used to estimate the unmeasurable state of the system, and the output feedback sliding mode predictive control algorithm is designed, and the convergence is proved; compared with the control algorithm that requires a known state, the algorithm proposed in the present invention is more suitable for the practical application of the vibration suppression system. Third, the present invention considers vibration as matching and mismatching disturbances in the system. Compared with the method of directly treating disturbances as matching disturbances in the system, the present invention fully considers the vibration signal. By designing a sliding surface containing state and disturbance estimation, the mismatching disturbance is eliminated in the output channel of the system, which improves the performance of the control algorithm in suppressing vibration.

[0070] The method of the present invention fully considers the coupling problem between the input-output characteristics of the active actuator and the system dynamic characteristics in the active-passive integrated vibration isolation system, the unmeasurable problem of the system state, and the mismatch disturbance problem caused by vibration, derives the control-oriented model, and designs an output feedback sliding mode predictive control algorithm considering the mismatch disturbance, thereby improving the performance of vibration suppression and promoting its practical engineering application. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 It is the principle block diagram of the output feedback sliding mode predictive control algorithm of the piezoelectric driven active and passive integrated vibration isolation system;

[0072] Figure 2 It is the equivalent circuit and force analysis principle diagram of the piezoelectric driven active and passive integrated vibration isolation system;

[0073] Figure 3 is a schematic diagram of the experimental platform;

[0074] Figure 4 It is the connection schematic diagram of the experimental platform;

[0075] Figure 5 is the zero-pole distribution diagram of the model;

[0076] Figure 6 It is the experimental result diagram of system frequency response;

[0077] Figure 7 is the result diagram of free vibration suppression experiment;

[0078] Figure 8 This is the result of the 5Hz periodic vibration suppression experiment;

[0079] Fig. 9 This is the result of the 25Hz periodic vibration suppression experiment;

[0080] Fig.10 This is the result of the 50Hz periodic vibration suppression experiment;

[0081] Fig.11 This is the result of the complex vibration suppression experiment;

[0082] Fig.12 This is a graph of vibration suppression test results with input constraints. DETAILED DESCRIPTION

[0083] The present invention will be further specifically described below in conjunction with the accompanying drawings and embodiments.

[0084] The output feedback sliding mode predictive control method of the piezoelectric driven active and passive integrated vibration isolation system described in this embodiment has a control block diagram as shown in FIG. Figure 1 The specific steps are:

[0085] Step 1:

[0086] The equivalent circuit and force analysis principle diagram of the piezoelectric driven active and passive integrated vibration isolation system are as follows: Figure 2 As shown in the figure, based on the electromechanical model of the piezoelectric actuator and the dynamic model of the vibration isolation system, the control-oriented model of the active and passive integrated vibration isolation system is derived, and the specific process of fully considering the disturbance as the matching and mismatching disturbance in the system is as follows:

[0087] Consider the equivalent circuit model of the piezoelectric actuator as follows:

[0088]

[0089] Where V in is the input voltage, k amp is the fixed voltage amplifier gain, R 0 is the equivalent resistance, V h The hysteresis effect generates voltage, H represents the hysteresis effect of piezoelectric, V a is the voltage for the piezoelectric effect, C a is the system equivalent capacitance, q 1 ,q 2 ,q 3 is the charge in the system, T em Indicates the conversion ratio of mechanical energy to electrical energy during the piezoelectric effect, x a is the output displacement of the piezoelectric actuator. The circuit model can be simplified as:

[0090]

[0091] According to the dynamic model of the system and the piezoelectric effect, the following equations are listed:

[0092]

[0093] Based on formula (1), the voltage V a Convert to

[0094] V a (t) = q 1 (t) / C a -T em x a (t) / C a (4)

[0095] Combining the circuit equivalent model (2) and the dynamic model (3), the system equation is written as follows:

[0096]

[0097] V in -(H(q1 ) / k amp ) is regarded as the equivalent input signal, written as u(t). Performing Laplace transform on the above equation yields:

[0098] From the equivalent input voltage signal u(s) to the sensitive load displacement x p The transfer function of (s) is derived as:

[0099]

[0100] To facilitate controller design, the model can be reduced to:

[0101]

[0102] The model parameter b 0 ,a 0 ,...,a n-1 , parameter identification needs to be performed based on the characteristics of the actual object.

[0103] Taking the vibration signal into full consideration as the matching and mismatching disturbances in the system, the state equation is written as:

[0104]

[0105] x 1 ~x n is the system state, u is the controller input voltage, and y is the sensitive load displacement output by the controller; the discrete expression of the system is:

[0106]

[0107] k represents the current time, T is the time interval adopted by the system;

[0108] Written in matrix form:

[0109]

[0110] in B=[0,…,b 0 T] T , C=[1,…,0] T , d=[d 1 ,d 2 ,…,d n ].

[0111] Definition 1: The variable μ satisfies and When, it can be defined as O(T l ), l is the corresponding order.

[0112] Assumption 1: The error is bounded ||d(k)||≤d * ,||d(k)-d(k-1)||≤θ * .

[0113] Step 2:

[0114] The specific process of designing state and disturbance observers based on the control-oriented model derived in step 1 to estimate the unknown system states and disturbances in the system is:

[0115] The state and disturbance observers are designed for the discrete system (10) as follows:

[0116]

[0117] The function z i (k) = d i (k)-γ i x i (k) is the auxiliary variable introduced in the observer, β i and γ i is the parameter to be designed in the observer, x i , d i , z i Estimates.

[0118] Written in matrix form:

[0119]

[0120] The definitions of matrices A and B are the same as in formula (11), γ=diag[γ 1 ,γ 2 ,…,γ n ], D=diag[1-Tγ 1 ,1-Tγ 2 ,…1-Tγ n ],N=[0,…,-Tγ n b 0 ] T ,

[0121] The estimated error of the observer is defined as: And the bound of the estimation error is defined as:

[0122] Step 3:

[0123] The specific process of designing a sliding surface that contains system state and disturbance estimation information and predicting the sliding surface at future moments based on the derived control-oriented model is as follows:

[0124] The discrete time sliding surface is designed as:

[0125]

[0126] Where M = [m 1 ,…,m n-1 ,1],H=[m 2 ,…,m n-1 ,1,0]. Parameter m 1 ,…,m n-1 The polynomial p(λ)=λ n-1 +m n-1 λ n-1 +…m 2 λ+m 1 The coefficient of , and its characteristic root satisfies 2Re(λ i )+|λ i 2 T<0, where Re(·) represents the real part of the complex number. Combined with observer (12), a one-step prediction of the sliding surface yields:

[0127]

[0128] Its p-step prediction formula is derived as follows:

[0129]

[0130] Where s(k)=[s(k+1),s(k+1),…s(k+p)] T ,u(k)=[u(k),u(k+1),…u(k+p-1)] T , Φ=[MA;MA 2 ;…;MA p ],

[0131] Step 4:

[0132] The loss function is constructed by combining the enhanced exponential reaching law and the predicted sliding surface, and the optimal control sequence is obtained by minimizing the constructed loss function. The specific process of applying the first signal of the optimal control sequence obtained at each moment to the piezoelectric driver to suppress vibration is as follows:

[0133] The discrete-time sliding mode reaching law is designed as:

[0134]

[0135] Where 0 <Tω1 <1,ω 2 >0,0<χ<1,δ>0, when When , the sliding mode state is bounded, and |s r (k)|≤Ω=O(T 2 ).

[0136] According to the design principle of sliding mode predictive control, the loss function is designed so that the predicted sliding mode state tracks the designed sliding mode reaching law, and the optimal control sequence is obtained by minimizing the loss function. Combining the sliding surface and the reaching law, the following loss function is defined as follows:

[0137] J(k)=[s r (k)-s(k)] T Q[s r (k)-s(k)]+u T (k)Ru(k) (18)

[0138] where s r (k) = [s r (k+1),…s r (k+p)] T , s(k)=[s(k+1),…s(k+p)] T ,u(k)=[u(k),u(k+1),…u(k+p-1)] T , Q, R are weight matrices. The prediction time domain and the control time domain are the same, both are p. Substituting the predicted sliding surface formula into the loss function, we get:

[0139]

[0140] Convert to: J(k) = 2u T (k)Θ T Qξ+u T (k)(Θ T QΘ+R)u(k)+ξ T Q

[0141] in Considering that T There is no relevant term of the control signal u(k) in ξ, which is ignored in the optimization process, and the loss function is simplified to:

[0142]

[0143] Minimize the loss function so that its derivative The optimal control sequence is:

[0144]

[0145] in The disturbance estimate at the future time contained in F(k) is difficult to obtain, so in actual control, Substitute F(k). According to the rolling optimization principle of predictive control, only the first control signal of the optimal control sequence is applied to the system, and the control law is derived as follows:

[0146]

[0147] Stability proof: Substituting the optimal control sequence into the predicted sliding surface yields:

[0148]

[0149] Taking the first element of the sliding mode state, we get

[0150]

[0151] in Combined with the boundedness of the sliding mode reaching law, we can get:

[0152]

[0153] According to the parameter design principle in the sliding surface (14), the system state can be written as Substituting it into the system equation, we get

[0154]

[0155] Written in matrix form:

[0156]

[0157] in matrix The characteristic polynomial of We can get:

[0158] [(η-1) / T] n-1 +m n-1 [(η-1) / T] n-2 +…+m 2 [(η-1) / T]+m 1 =0 (28)

[0159] Use λ i ,i=1,…,n represents the polynomial p(λ)=λ n-1 +m n-1 λ n-1 +…m 2 λ+m 1 The corresponding eigenvalue of the above characteristic polynomial is η i =1+Tλ i,i=1,…,n-1. According to m 1 ,…,m n-1 The parameter selection conditions are as follows:

[0160]

[0161] definition Then the system equation can be written as follows:

[0162]

[0163] According to matrix theory, the matrix The Jordan matrix is ​​derived as Where P is the state transition matrix, and J = diag[η 1 ,η 2 ,…,η n-1 ]and have the same characteristic eigenvalues.

[0164] Iterating the above formula from k = 0, we can get:

[0165]

[0166] Taking the norm on both sides of the equation gives:

[0167]

[0168] It should be noted that ||J||=η max =max{η i =1+Tλ i ,i=1,…,n-1}<1, and Taking the norm of g(k) yields:

[0169]

[0170] Combining the formula, when k→∞, The bound of is derived as:

[0171]

[0172] where λ max =max{λ i ,i=1,*,n-1}, and ε * , …, δ * , d * are all bounded. Therefore, the boundedness of the state of the closed-loop system can be proved.

[0173] Next, it is demonstrated that the mismatched disturbance can be cancelled in the output channel of the system.

[0174] The coordinate transformation of the system state is defined as follows:

[0175]

[0176] The converted system writes:

[0177]

[0178] Worth noting:

[0179]

[0180] The system equation (36) can be rewritten as follows:

[0181]

[0182] in Ξ=[0,d 1 (k+1)-d 1 (k),…,d n-1 (k+1)-d n-1 (k)] T .definition Then its norm can be calculated as:

[0183] ||f(k)||≤O(T 3 )+2||H||ε * T+||H||θ * T+||M||σ * T+||H||ε * +θ * T (39)

[0184] By calculations similar to formulas (31)-(34), the boundedness of the system state after coordinate transformation can be calculated as:

[0185]

[0186] It can be seen that the bounds of the system state and the perturbations are unrelated, and Therefore, it can be shown that the disturbance is cancelled on the output channel of the system.

[0187] Step 5:

[0188] Considering the problem that the system is subject to output constraints, the constructed loss function is transformed into a quadratic programming problem, the Hildreth QP algorithm is used to solve the optimal control sequence under the input constraints, and the specific process of applying the first signal to the control system is:

[0189] Consider the input constraint u in the controller design process min≤u(k+j-1)≤u max (i=1,…,p), where u min ,u max are the upper and lower bounds of the control signal, respectively, and u min ≤0≤u max . Convert the optimization problem into a quadratic programming problem:

[0190]

[0191] where K = Θ T QΘ+R, The matrices Z and Ω are based on actual constraints. Hildreth's QP algorithm is used to solve the above quadratic programming problem, and the first element of the optimal solution is used as the optimal control signal to apply to the system.

[0192] The stability of the system is proved by introducing terminal constraints. The loss function with input constraints and terminal constraints can be written as follows:

[0193]

[0194] where s r (k+p+j)-s(k+p+j)=0 is the terminal constraint. At time k, the optimal solution of the above optimization problem can be written as u(k)=[u(k),u(k+1),…,u(k+p-1)] T , the corresponding loss function is expressed as J * In addition, assume that the feasible solution at time k+1 is u(k+1)=[u(k+1),…,u(k+p-1),0] T , and its corresponding loss function is defined as And it is derived as:

[0195]

[0196] Combined terminal constraints r (k+p+j)-s(k+p+j)=0 and Available In addition, the loss function of the optimal control sequence at time k+1 is optimized to satisfy Therefore, we can get That is, the loss function is monotonically decreasing, which proves the stability of the closed-loop system.

[0197] The relevant parameters are selected according to the following principles:

[0198] The control-oriented model is obtained by using the least squares method: The parameter in the observer is set to β 1 =100,β2 =500,β 3 =1000,γ 1 =1×10 3 , γ 2 =1×10 6 , γ 3 =6×10 8 The parameter design in the sliding mode reaching law is Tω 1 =0.9,ω 2 =0.1,χ=0.2,δ=1, The prediction domain and the control domain are equal, and p = 5. The parameter design in the controller is Q = 10. - 8 I 5×5 , R=I 5×5 , M = [1 × 10 6 ,40,1], H=[40,1,0].

[0199] The beneficial effects of the present invention are verified by the following specific examples.

[0200] Example: Figure 3 Taking the piezoelectric driven active and passive integrated vibration isolation system shown as an example, the output feedback sliding mode predictive control algorithm designed by the present invention is used to control the vibration isolation system to suppress various vibrations, thereby verifying the performance of the constructed vibration isolation system and the proposed control algorithm.

[0201] Experimental platform: The experimental platform is as follows Figure 3 As shown in the figure, it 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. The passive vibration isolator and the piezoelectric actuator are connected in series to form an active and passive integrated vibration isolation system. The connection method of the experimental platform is as follows Figure 4 As shown. In this embodiment, the excitation source is also a piezoelectric device, and is also driven by a piezoelectric drive 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 a 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, thereby suppressing vibration.

[0202] Verification of control-oriented model: The control-oriented model of the active-passive integrated vibration suppression system is derived as shown in formula (7). To reduce the model order, the equivalent resistance can be assumed to be 0. The 1-400 Hz sweep signal is used as the input of the piezoelectric actuator, the output displacement of the system is collected, and the least squares method is used to identify the model parameters. The system model is obtained as follows: The zero and pole distribution of the model is as follows Figure 5(As shown in (a)), it can be seen that there are overlapping zeros and poles. By canceling the zeros and poles, the order of the model can be further reduced. The reduced-order system model obtained is The distribution of its zeros and poles is as Figure 5 (shown in (b)). In addition, the amplitude-frequency and phase-frequency response results of the model and the experimental system are as Figure 6 shown, which proves that the obtained model can describe the system characteristics with high accuracy.

[0203] Vibration suppression experiment: To verify the vibration suppression performance of the proposed control algorithm, vibration suppression experiments were carried out under free vibration (Experiment 1), periodic vibration with different frequencies (Experiment 2), complex vibration (Experiment 3), and input constraint (Experiment 4), respectively. When the system deviates from the equilibrium point due to an external impact force, the system will vibrate under its internal elastic force or gravity, that is, free vibration. Under the condition of no control, the free vibration will last for a period of time. In Experiment 1, a pulse voltage of 120 V was applied to the excitation source to generate an impact force, and the experimental results are as Figure 7 shown. The experimental results prove that the proposed method can respond quickly and suppress the free vibration of the system in a short time. In Experiment 2, periodic vibration signals with different frequencies (5 Hz, 25 Hz, 50 Hz) were used to verify the vibration suppression performance of the system, and the experimental results are as Figure 8-Figure 10 shown, which proves that the proposed method can suppress vibration signals with different frequencies. In Experiment 3, a noise signal was used for the vibration suppression performance of the system under complex working conditions, and the experimental results are as Fig.11 shown. The root mean square value of the vibration signal was suppressed to 0.1025 μm, which proves the effectiveness of the proposed method for complex vibration signals. In Experiment 4, the input constraint was set as -6V < u < 6V, and the experimental results are as Fig.12 shown. The experimental results prove that the input voltage is effectively controlled within the constraint range, and can effectively suppress vibration and ensure the stability of the system. 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 output feedback sliding mode predictive control method for a piezoelectric driven active and passive integrated vibration isolation system, characterized in that: The steps of this method are as follows: Step 1: Combining the electromechanical model of the piezoelectric actuator and the input-output characteristics of the vibration isolation system, a control-oriented model of the active-passive integrated vibration isolation system is derived, and the disturbances are fully considered as matched and unmatched disturbances in the system; Combining the circuit equivalent model and the dynamic model, the system equation is as follows: Among them, V in is the input voltage, k amp is a fixed voltage amplifier gain, R0 is an equivalent resistor, V h The hysteresis effect generates voltage, H represents the hysteresis effect of piezoelectric, V a is the voltage for the piezoelectric effect, C a is the system equivalent capacitance, q1, q2, q3 are the charges in the system, T em Indicates the conversion ratio of mechanical energy to electrical energy during the piezoelectric effect, x a is the output displacement of the piezoelectric actuator; m a is the mass of the piezoelectric actuator, m p is the mass of the sensitive load, x p is the displacement of the sensitive load, c1, c2, k1, k2 are the equivalent damping and stiffness in the system respectively; V in -(H(q1) / k amp ) is regarded as the equivalent input signal u(t), and the Laplace transform of the above formula (1) can be obtained: Based on the above formula (2), from the equivalent input voltage signal u(s) to the sensitive load displacement x p The transfer function of (s) is derived as: To facilitate controller design, the model can be reduced to: The model parameters are b0, a0, ..., a n-1 , parameter identification needs to be performed according to the characteristics of the vibration isolation system; Taking the vibration signal into full consideration as the matching and mismatching disturbances in the system, the state equation is written as: x1~x n is the system state, u is the controller input voltage, and y is the sensitive load displacement output by the controller; The discrete expression of the system is: k represents the current time, T is the time interval adopted by the system; Formula (10) can be written in matrix form as follows: in B=[0,...,b0T] T , C=[1,...,0] T , d=[Td1,Td2,…,Td n ]; Definition 1: The variable μ satisfies and When, it can be defined as O(T l ), l is the corresponding order; Assumption 1: The error is bounded ||d(k)||≤d * ,||d(k)-d(k-1)||≤θ * ; Step 2: Based on the derived control-oriented model, design state and disturbance observers to estimate the unknown system states and disturbances in the system; The state and disturbance observers are designed for the discrete expression (10) of the system as follows: The function z i (k) = d i (k)-γ i x i (k) is the auxiliary variable introduced in the observer, β i and γ i is the parameter to be designed in the observer, x i , d i , z i Estimates of , i = 1, 2, …, n; Formula (12) can be written in matrix form as follows: The definitions of matrices A and B are the same as in formula (11), γ=diag[γ1,γ2,…,γ n ], D=diag[1-Tγ1,1-Tγ2,…1-Tγ n ],N=[0,…,-Tγ n b0] T , The estimated error of the observer is defined as: And the bound of the estimation error is defined as: Step 3: Design a sliding surface that contains system state and disturbance estimation information to eliminate the impact of mismatched disturbances on system performance, and predict the sliding surface at future times based on the derived control-oriented model; The discrete time sliding surface is designed as: Where M=[m1,…,m n-1 ,1],H=[m2,…,m n-1 ,1,0]; parameters m1,…,m n-1 The polynomial p(λ)=λ n-1 +m n-1 λ n-1 +…m2λ+m1 coefficients, and the characteristic roots of the polynomial satisfy 2Re(λ i )+|λ i 2 T<0, where Re(·) represents the real part of the complex number; combined with the state and disturbance observer (12), a one-step prediction of the sliding surface is obtained: Its p-step prediction formula is derived as follows: Where s(k)=[s(k+1),s(k+1),…s(k+p)] T ,u(k)=[u(k),u(k+1),…u(k+p-1)] T , Φ=[MA;MA 2 ;…;MA p ], Step 4: According to the rolling optimization principle of model predictive control, a loss function is constructed to make the predicted sliding surface track the designed enhanced exponential reaching law, and the optimal control sequence is obtained by minimizing the constructed loss function. The first signal of the optimal control sequence obtained at each moment is applied to the piezoelectric actuator to suppress vibration; The enhanced discrete-time sliding mode reaching law is designed as: Where 0 <Tω1<1,ω2> 0, 0<χ<1, δ>0, is the parameter in the sliding mode reaching law; when When the sliding mode state s r (k) is bounded, and |s r (k)|≤O(T 2 ); According to the design principle of sliding mode predictive control, the loss function is designed so that the predicted sliding mode state tracks the designed sliding mode reaching law, and the optimal control sequence is obtained by minimizing the loss function; combining the predicted sliding surface (16) and the reaching law (17), the following loss function is defined as follows: J(k)=[s r (k)-s(k)] T Q[s r (k)-s(k)]+u T (k)Ru(k) (18) where s r (k) = [s r (k+1),…s r (k+p)] T , s(k)=[s(k+1),…s(k+p)] T u(k)=[u(k),u(k+1),…u(k+p-1)] T , Q, R are weight matrices; the prediction time domain and the control time domain are the same, both are p; Substituting the predicted sliding surface formula into the loss function (18) yields: converted to:J(k)=2u T (k)I T Qξ+u T (k)(Θ) T QΘ+R)u(k)+ξ T Qx in Considering that T There is no relevant term of the control signal u(k) in ξ, which is ignored in the optimization process, and the loss function is simplified to: Minimize the loss function so that its derivative The optimal control sequence is: in The disturbance estimate at the future time contained in F(k) is difficult to obtain, so in actual control, Substitute F(k); according to the rolling optimization principle of predictive control, only the first control signal of the optimal control sequence is applied to the system, and the control law is derived as follows: Step 5: Consider the problem that the system is subject to output constraints, transform the constructed loss function into a quadratic programming problem, and use the Hildreth QP algorithm to solve the optimal control sequence under input constraints; Consider the input constraint u in the controller design process min ≤u(k+j-1)≤u max (i=1,…,p), where u min ,u max are the upper and lower bounds of the control signal, respectively, and u min ≤0≤u max ; Convert the optimization problem into a quadratic programming problem: where K = Θ T QΘ+R, The matrices Z and Ω are derived according to the actual constraints; the above quadratic programming problem is solved using Hildreth's QP algorithm, and the first element of the optimal solution is used as the optimal control signal to apply to the system.

2. The output feedback sliding mode predictive control method of a piezoelectric driven active and passive integrated vibration isolation system according to claim 1 is characterized in that: The relevant parameters are selected according to the following principles: 1) Using the least squares method, the control-oriented model is obtained as 2) To ensure the accuracy and convergence of the observer, the parameters in the observer are set to β1 = 100, β2 = 500, β3 = 1000, γ1 = 1×10 3 ,γ2=1×10 6 ,γ3=6×10 8 ; 3) To ensure the response speed of the algorithm and reduce chattering, the parameters in the sliding mode reaching law are designed as Tω1=0.9, ω2=0.1, χ=0.2, δ=1, 4) Considering both computational complexity and control accuracy, the prediction domain and control domain in the control algorithm are equal, and p = 5, and the parameter design in the controller is Q = 10 -8 I 5×5 , R=I 5×5 , M = [1 × 10 6 ,40,1], H=[40,1,0].

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