Displacement control method and device for single-degree-of-freedom parallel plate electrostatic driver

Abstract

The invention discloses a displacement control method and device for a single-degree-of-freedom parallel plate electrostatic actuator, and the method comprises the steps: building a physical model based on the single-degree-of-freedom parallel plate electrostatic actuator; based on the physical model, constructing a control system model to convert an output error constraint tracking problem of the single-degree-of-freedom parallel plate electrostatic actuator into a controller design problem based on an output feedback system; based on an output error needing to be tracked, filter expansion is introduced; designing a controller by using an internal model based on a filter expansion and control system model; in order to convert a security boundary into a mathematical constraint, a barrier Lyapunov function is introduced to be dynamically combined with an error, so that the stability of a closed-loop system is ensured, and an output error is always maintained in a preset security interval in the whole operation period. According to the invention, under the condition that large parameter changes exist, the movable plate can realize full-gap harmonic displacement under the condition that the movable plate is not in contact with the fixed plate.

Description

Technical Field

[0001] The present invention relates to the technical field of microelectromechanical system control, and particularly to a displacement control method, device, terminal device and computer-readable storage medium for a single-degree-of-freedom parallel-plate electrostatic actuator. Background Art

[0002] In recent years, single-degree-of-freedom (1-DOF) parallel-plate electrostatic actuators have been widely used in the field of microelectromechanical systems (MEMS) due to their significant advantages such as simple structure, mature manufacturing process, low power consumption, and flexible operation. Such actuators generate electrostatic force by applying voltage to drive the relative movement between the movable plate and the fixed plate, and are commonly used in precision devices such as micro sensors, optical switches, micro mirror arrays, and radio frequency switches. Their core working principle is based on the electrostatic effect of parallel-plate capacitors, and the position of the movable plate is controlled by adjusting the voltage, thereby achieving precise micro-displacement operation.

[0003] However, the inherent nonlinear characteristics of electrostatic actuators lead to the well-known "pull-in" phenomenon, which severely limits the stable operating range of the movable plate, usually only reaching one-third of the full gap. After exceeding the pull-in position, a sudden collision will occur between the movable plate and the fixed plate, resulting in system failure. Therefore, analyzing and eliminating pull-in instability has become the main goal of studying such devices.

[0004] Existing control methods include open-loop control and closed-loop control. Open-loop control methods such as input shaping control, although simple, are not robust enough to parameter uncertainties and have a steady-state error under external disturbances. Closed-loop control methods, especially nonlinear control techniques such as design based on control Lyapunov functions and flatness-based control, can theoretically expand the stable operating range of the movable plate to the entire gap. However, in addition to the pull-in phenomenon, 1-DOF parallel-plate electrostatic actuators also face two challenges: one is parameter variations due to inconsistent manufacturing processes, and the other is that the movable plate may come into contact with the fixed plate multiple times during the transient process, resulting in surface damage and affecting the device life. Summary of the Invention

[0005] To solve the above deficiencies of the prior art, the present invention provides a displacement control method, device, terminal device and computer-readable storage medium for a single-degree-of-freedom parallel-plate electrostatic actuator, which realizes tracking control with output error constraint by constructing an output feedback system, ensuring that the movable plate can achieve harmonic displacement of the full gap without contacting the fixed plate in the presence of large parameter variations.

[0006] The first object of the present invention is to provide a displacement control method for a single-degree-of-freedom parallel-plate electrostatic actuator.

[0007] The second object of the present invention is to provide a displacement control device for a single-degree-of-freedom parallel-plate electrostatic actuator.

[0008] The third object of the present invention is to provide a terminal device.

[0009] The fourth object of the present invention is to provide a computer-readable storage medium.

[0010] The first object of the present invention can be achieved by adopting the following technical solutions:

[0011] Based on a single-degree-of-freedom parallel-plate electrostatic actuator, a physical model is constructed;

[0012] Based on the physical model, a control system model is constructed to transform the output error constraint tracking problem of the single-degree-of-freedom parallel-plate electrostatic actuator into a controller design problem based on an output feedback system;

[0013] Based on the required output error to be tracked, filter expansion is introduced; based on the filter expansion and the control system model, an output feedback system-based controller is designed using the internal model;

[0014] In order to transform the safety boundary into a mathematical constraint, an obstacle Lyapunov function is introduced and combined with the error dynamics to ensure the stability of the closed-loop system and that the output error remains within a preset safety interval throughout the entire operating cycle.

[0015] Furthermore, the physical model is:

[0016]

[0017] In the formula, ω, ξ, q, and r respectively represent velocity, damping ratio, electric charge quantity, and resistance;

[0018] The target trajectory under the physical model is:

[0019] F(t) = A f + A m sin(σt + φ)

[0020] In the formula, A f , A m , σ, and φ are all given parameters;

[0021] At the same time, it is ensured that the tracking error |x(t) - F(t)| < L, where L is a set constant greater than 0; to achieve global stability, it is required that the trajectory starting from any initial state of the closed-loop system is globally bounded;

[0022] To achieve full-gap operation, it is allowed that the amplitude of the target trajectory exceeds the stable range under traditional open-loop control, and the maximum dynamic displacement can reach the full gap.

[0023] Furthermore, constructing a control system model based on the physical model includes:

[0024] The sinusoidal reference trajectory to be tracked is generated by the following system:

[0025]

[0026] Wherein, is the external system trajectory, i.e., the state space generating the sine wave;

[0027] The target trajectory is expressed as:

[0028] F(t) = v1(t) + v3(t)

[0029] Then the corresponding relationship between the initial state v(0) and the target trajectory parameters is:

[0030] A f = v3(0)

[0031]

[0032] φ = arctan(v1(0) / v2(0))

[0033] Define the system parameter vector Ω = (ξ, r) and decompose it into:

[0034]

[0035] Wherein, is the nominal parameter; is the change of the parameter relative to the nominal value, R 2 is the two-dimensional Euclidean space, and W is a compact subset of R 2 ;

[0036] Redefine the state variables of the parallel plate electrostatic actuator model:

[0037]

[0038] And define the control input transformation:

[0039]

[0040] Then the output error constrained tracking problem is regarded as the controller design problem of the following control system model:

[0041]

[0042] e = y - v1 - v3

[0043] Design a controller for the control system model to achieve asymptotic tracking of the target trajectory, i.e., lim t→∞e(t) = 0, and the error constraint |e(t)| ≤ L is satisfied for all times t > 0.

[0044] Furthermore, to eliminate the need for velocity measurement, the filter is designed as:

[0045]

[0046] where λ1 = λ2 = 1, and ξ1, ξ2 are both generated auxiliary states.

[0047] Furthermore, based on the filter-extended and control system model, an output feedback controller is designed using internal model, including:

[0048] Based on the filter-extended and control system model, a new control system model is obtained;

[0049] Based on the new control system model and internal model, a controller is designed.

[0050] Furthermore, the obtaining of the new control system model based on the filter-extended and control system model includes:

[0051] The control system model is combined with the filter and the following transformation is adopted:

[0052]

[0053] where

[0054] In the new coordinates (z, y, ξ1, ξ2), the new control system model is obtained as:

[0055]

[0056] e = y - q(v)

[0057] where H(w) = [1 0],

[0058]

[0059] q(v) = v1 + v3.

[0060] Furthermore, the designing of the controller based on the new control system model and internal model includes:

[0061] The established internal model is:

[0062]

[0063] The internal model is combined with the new control system model to obtain an augmented system, and the following coordinate transformation is performed on the augmented system:

[0064]

[0065] e = y - v1 - v3

[0066]

[0067] The output feedback system obtained is as follows:

[0068]

[0069] where (x1, x2, x3) = (e, ξ1, ξ2), Ψ σ = ΨT -1 ,

[0070] Construct the initial Lyapunov function as follows:

[0071]

[0072] In the formula, and are both positive constants, and P are both positive definite solutions of the Lyapunov equation;

[0073] Since F(w) and M are both Hurwitz matrices, there are:

[0074]

[0075] PM + M T P = -I3

[0076] Since and are both real-valued continuous functions and w ∈ W, there exist positive constants q1, q2 such that:

[0077]

[0078] Then the time derivative of the output feedback system along the trajectory of the subsystem is:

[0079]

[0080] where,

[0081]

[0082] Define x4 = α3 = u, then the controller design of the output feedback system is:

[0083]

[0084] Among them,

[0085]

[0086] Furthermore, by combining the barrier Lyapunov function with the error dynamics, the constructed composite Lyapunov function is:

[0087]

[0088] In the formula, is a positive constant to be specified;

[0089] Define

[0090] At the end of the backstepping design, define

[0091] Since w ∈ W, there exist positive constants and satisfying l - 5||H(w)|| 2 ≥ 1 and h - 5||b(ω)Ψ σ || 2 ≥ 1; Select such that when t ≥ 0, there is Then there is Therefore, when t ≥ 0, it satisfies:

[0092]

[0093] According to Barbalat’s Lemma, when t → ∞, and approach zero; According to and the definitions of, and because v and ω are bounded, so z and η are bounded, x2 and u are bounded; Therefore, all states and derivatives of the closed-loop system are bounded, and further it can be known that the trajectory error

[0094] According to the characteristics of the barrier Lyapunov function, if the initial condition of the tracking error is within the pre-required region, that is then the output tracking error constraint can be satisfied

[0095] The second object of the present invention can be achieved by adopting the following technical solutions:

[0096] A displacement control device for a single-degree-of-freedom parallel-plate electrostatic actuator, the device includes:

[0097] A first construction module, configured to construct a physical model based on a single-degree-of-freedom parallel-plate electrostatic actuator;

[0098] A second construction module, configured to construct a control system model based on a physical model, so as to transform the output error constraint tracking problem of a single-degree-of-freedom parallel-plate electrostatic actuator into a controller design problem based on an output feedback system;

[0099] A design module, configured to introduce a filter expansion based on the output error to be tracked; and based on the filter expansion and the control system model, design a controller for the output feedback system by using an internal model;

[0100] A control module, configured to introduce a barrier Lyapunov function in combination with the error dynamics in order to transform the safety boundary into a mathematical constraint, so as to ensure the stability of the closed-loop system and that the output error always remains within a preset safety interval during the entire operation period.

[0101] The third objective of the present invention can be achieved by adopting the following technical solution:

[0102] A terminal device includes a processor and a memory for storing executable programs of the processor. When the processor executes the programs stored in the memory, the displacement control method of the single-degree-of-freedom parallel-plate electrostatic actuator described above is implemented.

[0103] The fourth objective of the present invention can be achieved by adopting the following technical solution:

[0104] A computer-readable storage medium stores a program, and when the program is executed by a processor, the displacement control method of the single-degree-of-freedom parallel-plate electrostatic actuator described above is implemented.

[0105] The present invention has the following beneficial effects compared with the prior art:

[0106] The present invention constructs a hierarchical robust control system to comprehensively improve the system performance through multi-dimensional technology integration. First, a normalized dynamic model including mechanical-electrical coupling characteristics is constructed to quantify the dynamic relationship between the displacement of the moving plate, the damping ratio, and the charge quantity, providing an accurate physical basis for subsequent control design and overcoming the description deviation of traditional linearized models under non-steady-state working conditions. Based on the physical model, the displacement tracking problem is transformed into a robust output regulation problem of an output feedback system, and a system model is constructed based on the robust adaptive control technology, effectively overcoming the dependence of traditional control methods on accurate mathematical models. Then, through filter expansion, the unmeasurable speed information is reconstructed to avoid direct measurement of the moving plate speed, reducing the hardware complexity to optimize the engineering implementation. Furthermore, through internal model design, dynamic compensation and system state information reconstruction are realized, and an output feedback architecture is designed in combination with filter expansion to achieve closed-loop control without installing a speed sensor. On this basis, the barrier Lyapunov function technology is creatively introduced to expand the traditional error convergence control into a high-performance robust control with constraint guarantee. By constructing a time-varying constraint boundary and a potential energy function, it is ensured that the output error of the system always remains within the preset safety interval throughout the entire operation cycle (including transient processes such as startup and load mutation), avoiding safety hazards such as mechanical contact failure to the greatest extent, and forming a complete technical closed-loop from theoretical methods to engineering implementation. Brief Description of the Drawings

[0107] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on the structures shown in these drawings.

[0108] Figure 1 It is a flowchart of the displacement control method for a single-degree-of-freedom parallel-plate electrostatic actuator according to Embodiment 1 of the present invention;

[0109] Figure 2 It is a control framework for output error control of a single-degree-of-freedom parallel-plate electrostatic actuator according to Embodiment 1 of the present invention;

[0110] Figure 3 It is a schematic structural diagram of a single-degree-of-freedom parallel-plate electrostatic actuator according to Embodiment 1 of the present invention;

[0111] Figure 4 It is a curve graph of the target trajectory and the actual trajectory of a single-degree-of-freedom parallel-plate electrostatic actuator according to Embodiment 1 of the present invention;

[0112] Figure 5 It is a curve graph of the tracking error of a single-degree-of-freedom parallel-plate electrostatic actuator according to Embodiment 1 of the present invention;

[0113] Figure 6 This is the structural block diagram of the displacement control device for the single-degree-of-freedom parallel-plate electrostatic actuator according to Embodiment 2 of the present invention;

[0114] Figure 7 This is the structural block diagram of the terminal device according to Embodiment 3 of the present invention. Detailed implementation manners

[0115] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Apparently, the described embodiments are some, but not all, of the embodiments of the present invention. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention. It should be understood that the specific embodiments described are only used to explain the present application and not to limit the present application.

[0116] Embodiment 1:

[0117] As Figure 1 、 2 shown, this embodiment provides a displacement control method for a single-degree-of-freedom parallel-plate electrostatic actuator, including the following steps:

[0118] S101. Based on the single-degree-of-freedom parallel-plate electrostatic actuator, construct a physical model.

[0119] As Figure 3 shown, let G(t) be the air gap, G0 be the zero-voltage gap, and A be the plate area, then the output displacement is expressed as x = 1 - G / G0. Based on the mechanical-electrical coupling characteristics of the parallel-plate electrostatic actuator, establish a normalized dynamic model including the output displacement, damping ratio, and electric charge amount as follows:

[0120]

[0121] where ω, ξ, q, and r respectively represent velocity, damping ratio, electric charge amount, and resistance.

[0122] Under this model, clarify the control objective, that is, a feedback controller needs to be designed so that the normalized displacement x(t) of the actuator can asymptotically track the target trajectory. The target trajectory is as follows:

[0123] F(t) = A f + A m sin(σt + φ) (2)

[0124] where A f 、A m 、σ, and φ are all given parameters.

[0125] Meanwhile, ensure that the tracking error |x(t)-F(t)| < L (L>0) to avoid contact between the moving plate and the fixed plate during the transient process. In addition, to achieve global stability, it is required that the trajectory starting from any initial state of the closed-loop system is globally bounded.

[0126] In terms of performance, to achieve full-gap operation, the amplitude A of the target trajectory is allowed f +A m to exceed the stable range under traditional open-loop control (i.e., the pull-in position x < 1 / 3), and the maximum dynamic displacement can reach the full gap (x→1).

[0127] S102. Based on the physical model, construct a control system model to transform the output error constrained tracking problem of the single-degree-of-freedom parallel-plate electrostatic actuator into a robust output regulation problem based on the output feedback system.

[0128] To reformulate the output error constrained tracking problem of the parallel-plate electrostatic actuator as a robust output regulation problem based on the output feedback system, the sinusoidal reference trajectory to be tracked is generated by the following system:

[0129]

[0130] where is the external system trajectory, i.e., the state space that generates the sine wave;

[0131] The target trajectory can be expressed as:

[0132] F(t) = v1(t)+v3(t) (4)

[0133] Then the correspondence between the initial state v(0) and the target trajectory parameters is:

[0134] A f = v3(0)

[0135]

[0136] φ = arctan(v1(0) / v2(0))

[0137] To make the controller robust, it is necessary to model the parameter uncertainty for effective processing. Define the system parameter vector Ω = (ξ, r), and decompose it into:

[0138]

[0139] where is the nominal parameter, represents the change of the parameter relative to the nominal value (R 2 represents the 2D Euclidean space, and W is R2 compact subsets).

[0140] Next, redefine the state variables of the parallel plate electrostatic actuator model as follows:

[0141]

[0142] And define the control input transformation:

[0143]

[0144] Then the above output error constrained tracking problem can be regarded as a controller design problem for the following control system model:

[0145]

[0146] Next, design a controller u for this system to achieve asymptotic tracking of the target trajectory, i.e., lim t→∞ e(t)=0, and the error constraint |e(t)|≤L is satisfied for any time t>0.

[0147] S103. Based on the required tracking output error, introduce filter expansion; based on the filter expansion and the control system model, use internal model to design the controller.

[0148] To eliminate the measurement requirement of velocity v, design the filter as follows:

[0149]

[0150] where λ1 = λ2 = 1 (fixed parameters), generating auxiliary states ξ1, ξ2.

[0151] Combine the systems of (6) and (7) and adopt the following transformation:

[0152]

[0153] where,

[0154] In the new coordinates (z, y, ξ1, ξ2), the system can be obtained as follows:

[0155]

[0156] where, H(w)=[1 0],

[0157]

[0158] q(v)=v1 + v3.

[0159] Assume that there exists a sufficiently smooth function \(z(v, w)\) that satisfies the regulation equation:

[0160]

[0161] Then the solution of the regulator equation for system (9) can be calculated as:

[0162]

[0163] To verify this assumption, note that \(G(q(v), w)\) in (10) can be expressed as:

[0164] \(G(v_1, w)=G_1(w)v\) [1] (12)

[0165] where The result is:

[0166]

[0167] Define \(z(v, w)=Z_1(w)v\) [1] , and we can further obtain:

[0168] \(Z_1(w)A\) [1] \(=F(w)Z_1(w)+G_1(w)\) (14)

[0169] where Define We get the following equation:

[0170]

[0171]

[0172] Furthermore, we have:

[0173]

[0174] where:

[0175]

[0176] Define Then we have:

[0177]

[0178] We obtain \(\Psi = [1\ 0\ 0]\), select any three-dimensional Hurwitz matrix \(M\) and three-dimensional column matrix \(N\) such that \((M, N)\) is controllable, then there exists a unique non-singular matrix \(T\) that satisfies the Sylvester equation:

[0179] \(T\Phi - MT = N\Psi\) (17)

[0180] The internal model is established as follows:

[0181]

[0182] Combining (9) and (18) can obtain the augmented system. The following coordinate transformation is performed on the augmented system:

[0183]

[0184] The output feedback system can be obtained as follows:

[0185]

[0186] where, (x1, x2, x3) = (e, ξ1, ξ2),

[0187] Construct the initial Lyapunov function as follows:

[0188]

[0189] where and are positive constants, and P is the positive definite solution of the Lyapunov equation.

[0190] Since F(w) and M are Hurwitz matrices, there are:

[0191]

[0192] Due to and are real-valued continuous functions and w ∈ W, there exist positive constants q1, q2 such that:

[0193]

[0194] Then the time derivative of the system (20) along the trajectory of the subsystem has:

[0195]

[0196] where,

[0197]

[0198] Define x4 = α3 = u, then the controller for the system (20) can be designed as follows:

[0199]

[0200] Among them,

[0201]

[0202] S104. By introducing the barrier Lyapunov function technique, prove the stability of the constructed closed-loop system to ensure that the output error of the system remains within the preset safety interval throughout the entire operation cycle.

[0203] To transform the safety boundary into a mathematical constraint, introduce the barrier Lyapunov function and combine it with the error dynamics. Construct the composite Lyapunov function as follows:

[0204]

[0205] Among them, is a positive constant to be specified, and the time derivative of V1 is calculated as follows:

[0206]

[0207] Define Furthermore, obtain:

[0208]

[0209] At the end of the backstepping design, define Then there is:

[0210]

[0211] Among them,

[0212] Because w ∈ W within a set W range, there exist positive constants and satisfying l - 5||H(w)|| 2 ≥ 1 and h - 5||b(ω)Ψ σ || 2 ≥ 1. Select such that when t ≥ 0, there is Then there is Therefore, according to the above relationship, it can be obtained that when t ≥ 0, it satisfies:

[0213]

[0214] According to Barbalat’s Lemma, when t → ∞, and approach zero. According to and By the definition, since \(v\) and \(\omega\) are bounded, \(z\) and \(\eta\) are bounded, and \(x_2\) and \(u\) are bounded. Therefore, all states and derivatives of the closed-loop systems (1), (7), (18), and (25) are bounded. Furthermore, it can be known that the trajectory error

[0215] According to the characteristics of the barrier Lyapunov function, if the initial conditions of the tracking error are within a pre-specified region, i.e., the output tracking error constraint can be satisfied

[0216] In this embodiment, by introducing the barrier Lyapunov function, it is ensured that the output error always remains within the preset range, avoiding the contact between the movable plate and the fixed plate, thereby improving the reliability and long-term service life of the MEMS device under complex working conditions.

[0217] S105. Prove the effectiveness of the output feedback controller through MATLAB simulation.

[0218] In this step, the effectiveness of the feedback controller for output error control of the single-degree-of-freedom parallel-plate electrostatic actuator designed in this embodiment will be verified through MATLAB simulation.

[0219] The nominal damping ratio and resistance parameters are selected as The parameter perturbation is selected as \(w=(0.1, 0.1)\). The harmonic displacement is selected as \(F(t)=0.6 + 0.1\sin(t+(\pi / 2))\), and this trajectory represents a harmonic oscillation with an amplitude of 0.1 superimposed near the 60% position of the full-gap stroke. The initial conditions of the external system are set as \(v_1(0)=0.1\), \(v_2(0)=0\), \(v_3(0)=0.6\). To avoid the contact between the movable plate and the fixed plate, the maximum allowable tracking error \(L = 0.3\) is set. The remaining initial conditions are set as \(x_1(0)=0\), \(x_2(0)=0.01\), \(y(0)=0.7\), \(b(0)=0\), \(k(0)=0\), \(\eta(0)=\text{col}(0,0)\).

[0220] According to the selected parameters, the simulation results are as Figure 4 、 5 shown. Figure 4 represents the curves of the actual displacement of the actuator and the target trajectory changing with time, Figure 5 represents the output tracking error curve. According to Figure 5 it can be observed that the output tracking error of the single-degree-of-freedom parallel-plate electrostatic actuator can converge to the origin. This simulation result shows that the feedback controller designed in this embodiment can effectively control and regulate the output of the single-degree-of-freedom parallel-plate electrostatic actuator.

[0221] Those skilled in the art can understand that all or part of the steps in the methods of the above embodiments can be completed by instructing relevant hardware through a program, and the corresponding program can be stored in a computer-readable storage medium.

[0222] It should be noted that although the method operations of the above embodiments are described in a specific order in the drawings, this does not require or imply that these operations must be performed in that specific order, or that all the operations shown must be performed to achieve the desired result. On the contrary, the described steps can be changed in the execution order. Additionally or alternatively, some steps can be omitted, multiple steps can be combined into one step for execution, and / or one step can be decomposed into multiple steps for execution.

[0223] Embodiment 2:

[0224] As Figure 6 shown, this embodiment provides a displacement control device for a single-degree-of-freedom parallel-plate electrostatic actuator. The device includes a first construction module 601, a second construction module 602, a design module 603, and a control module 604, where:

[0225] The first construction module 601 is used to construct a physical model based on a single-degree-of-freedom parallel-plate electrostatic actuator;

[0226] The second construction module 602 is used to construct a control system model based on the physical model to transform the output error constraint tracking problem of the single-degree-of-freedom parallel-plate electrostatic actuator into a controller design problem based on an output feedback system;

[0227] The design module 603 is used to introduce a filter expansion based on the desired output error to be tracked; based on the filter expansion and the control system model, an output feedback system-based controller is designed using an internal model;

[0228] The control module 604 is used to introduce a barrier Lyapunov function in combination with the error dynamics in order to transform the safety boundary into a mathematical constraint, so as to ensure the stability of the closed-loop system and that the output error always remains within a preset safety interval throughout the entire operation cycle.

[0229] For the specific implementation of each module in this embodiment, reference can be made to Embodiment 1 above, and details will not be repeated here; it should be noted that the device provided in this embodiment is only illustrated by the above division of each functional module. In practical applications, the above functions can be allocated to different functional modules as needed, that is, the internal structure can be divided into different functional modules to complete all or part of the functions described above.

[0230] Embodiment 3:

[0231] This embodiment provides a terminal device, which can be a computer, such asFigure 7 As shown in the figure, it includes a processor 702, a memory, an input device 703, a display 704, and a network interface 705 connected through a system bus 701. The processor is used to provide computing and control capabilities. The memory includes a non-volatile storage medium 706 and an internal memory 707. The non-volatile storage medium 706 stores an operating system, a computer program, and a database. The internal memory 707 provides an environment for the operation of the operating system and the computer program in the non-volatile storage medium. When the processor 702 executes the computer program stored in the memory, it implements the displacement control method of the single-degree-of-freedom parallel-plate electrostatic actuator in the above-mentioned Embodiment 1.

[0232] Embodiment 4:

[0233] This embodiment provides a computer-readable storage medium that stores a computer program. When the computer program is executed by a processor, it implements the displacement control method of the single-degree-of-freedom parallel-plate electrostatic actuator in the above-mentioned Embodiment 1.

[0234] It should be noted that the computer-readable storage medium in this embodiment can be a computer-readable signal medium, a computer-readable storage medium, or any combination of the two. A computer-readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination of the above. More specific examples of a computer-readable storage medium can include, but are not limited to: an electrical connection with one or more wires, a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disk read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the above.

[0235] As described above, only the preferred embodiments of this invention patent are presented, but the protection scope of this invention patent is not limited thereto. Any person skilled in the art within the scope disclosed by this invention patent, according to the technical solution and inventive concept of this invention patent, makes equivalent substitutions or changes, and all belong to the protection scope of this invention patent.

Claims

1. A displacement control method for a single-degree-of-freedom parallel-plate electrostatic actuator, characterized in that The method includes: Constructing a physical model based on a single-degree-of-freedom parallel-plate electrostatic actuator; Constructing a control system model based on the physical model to transform the output error constraint tracking problem of the single-degree-of-freedom parallel-plate electrostatic actuator into a controller design problem based on an output feedback system; Introducing a filter extension based on the required tracking output error; designing a controller for the output feedback system using internal model design based on the filter extension and the control system model; To transform the safety boundary into a mathematical constraint, an obstacle Lyapunov function is introduced and combined with the error dynamics to ensure the stability of the closed-loop system and that the output error remains within a preset safety interval throughout the entire operating cycle.

2. The displacement control method according to claim 1, characterized in that The physical model is: Where ω, ξ, q, and r represent velocity, damping ratio, charge quantity, and resistance respectively; The target trajectory under the physical model is: F(t) = A f + A m sin(σt + φ) where A f , A m , σ, and φ are all given parameters; At the same time, ensure that the tracking error |x(t) - F(t)| < L, where L is a set constant greater than 0; To achieve global stability, it is required that the trajectory starting from any initial state of the closed-loop system is globally bounded; To achieve full-gap operation, the amplitude of the target trajectory is allowed to exceed the stable range under traditional open-loop control, and the maximum dynamic displacement can reach the full gap.

3. The displacement control method according to claim 2, characterized in that The constructing of the control system model based on the physical model includes: The sinusoidal reference trajectory to be tracked is generated by the following system: Among them, is the external system trajectory, that is, the state space that generates the sine wave; The target trajectory is expressed as: F(t) = v1(t) + v3(t) Then the corresponding relationship between the initial state v(0) and the target trajectory parameters is: A f = v3(0) φ = arctan(v1(0) / v2(0)) Define the system parameter vector Ω = (ξ, r) and decompose it as: Among them, is the nominal parameter; is the change of the parameter relative to the nominal value, R 2 is the two-dimensional Euclidean space, and W is a 2 compact subset of R; Redefine the state variables of the parallel-plate electrostatic actuator model: And define the control input transformation: Then the output error constraint tracking problem is regarded as a controller design problem for the following control system model: e = y - v1 - v3 Design a controller for the control system model to achieve asymptotic tracking of the target trajectory, i.e., lim t→∞ e(t) = 0, and the error constraint |e(t)| ≤ L is satisfied for all times t > 0.

4. The displacement control method according to claim 3, wherein To eliminate the need for velocity measurement, design a filter as: Where λ1 = λ2 = 1, and ξ1, ξ2 are both generated auxiliary states.

5. The displacement control method according to claim 4, wherein The designing of the controller for the output feedback system using internal model design based on the filter extension and the control system model includes: Based on the filter extension and the control system model, obtain a new control system model; Design a controller based on the new control system model and the established internal model.

6. The displacement control method according to claim 5, characterized in that, The obtaining of the new control system model based on the filter extension and the control system model includes: The control system model is combined with the filter and the following transformation is adopted: Among them, In the new coordinates (z, y, ξ1, ξ2), the new control system model obtained is: e = y - q(v) Among them, H(w) = [10]; q(v) = v1 + v3.

7. The displacement control method according to claim 6, characterized in that, The designing of the controller based on the new control system model and the internal model includes: The established internal model is: Combine the internal model with the new control system model to obtain an augmented system, and perform the following coordinate transformation on the augmented system: e = y - v1 - v3 Obtain the output feedback system as: where (x1, x2, x3) = (e, ξ1, ξ2), Ψ σ = ΨT -1 , Construct an initial Lyapunov function as follows: wherein, and are both positive constants, and P are both positive definite solutions of the Lyapunov equation; Since F(w) and M are both Hurwitz matrices, there is: PM+M T P = -I3 Since and are both real-valued continuous functions and \(w\in W\), there exist positive constants \(q_1\) and \(q_2\) such that: Then the time derivative of the output feedback system along the subsystem trajectory is: Among them, Definition If x4 = α3 = u, then the controller of the output feedback system is designed as follows: Among them, 8. The displacement control method according to claim 7, wherein, Combine the obstacle Lyapunov function with the error dynamics, and the constructed composite Lyapunov function is: wherein, is a positive constant to be specified; Definition At the end of the backstepping design, define Since w ∈ W, there exist positive constants and such that l-5||H(w)|| 2 ≥ 1 and h-5||b(ω)Ψ σ || 2 ≥ 1; select such that for t ≥ 0 there is Then there is Therefore, for t ≥ 0 there is: According to Barbalat’s Lemma, when t→∞, and approach zero; according to the definitions of and , and since v and ω are bounded, z and η are bounded, and x2 and u are bounded; thus all states and derivatives of the closed-loop system are bounded, and it can be further known that the trajectory error According to the characteristics of the barrier Lyapunov function, if the initial condition of the tracking error is within the pre-specified region, i.e., then the output tracking error constraint can be satisfied 9. A displacement control device for a single-degree-of-freedom parallel-plate electrostatic actuator, characterized in that, The device includes: A first construction module for constructing a physical model based on a single-degree-of-freedom parallel-plate electrostatic actuator; A second construction module, configured to construct a control system model based on a physical model to transform the output error constraint tracking problem of a single-degree-of-freedom parallel plate electrostatic actuator into a controller design problem for an output feedback system; A design module, configured to introduce a filter extension based on the desired output error to be tracked; and based on the filter extension and the control system model, design a controller for the output feedback system using internal model control; A control module, configured to introduce a barrier Lyapunov function in combination with the error dynamics to transform the safety boundary into a mathematical constraint, so as to ensure the stability of the closed-loop system and that the output error always remains within a preset safety interval throughout the entire operating cycle.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, the control method according to any one of claims 1 to 8 is implemented.

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