A robust high-precision tracking control method and controller based on a dynamic sliding surface

Abstract

The application provides a kind of robust high-precision tracking control method based on dynamic sliding surface;The method is as follows: one, construct driver model.Two, introduce displacement error nonlinear dynamic function term, high-order error term and integral term in dynamic sliding surface.Three, based on the dynamic sliding surface and driver model obtained in step two, construct robust high-precision tracking sliding mode controller.Four, input the displacement error of the driver into the sliding mode controller;The sliding mode controller outputs a voltage signal to the piezoelectric driver.The application introduces displacement error nonlinear dynamic function term, high-order error term and integral term in dynamic sliding surface, the integral term improves the transient response speed of the sliding surface, reduces the steady-state error of the sliding surface;Nonlinear dynamic function improves the response speed and robustness of the sliding surface;High-order error term makes the sliding surface more sensitive to error changes.

Description

Technical Field

[0001] This invention belongs to the field of micro-nano drive technology, specifically relating to a robust high-precision tracking control method and controller based on dynamic sliding surfaces. Background Technology

[0002] In recent years, with the rapid development of microelectronic information device manufacturing, micro-nano manufacturing, and ultra-precision machining technologies, the precision requirements for manufacturing equipment have become increasingly stringent. Traditional motor actuators can no longer meet the demanding requirements of precision motion, necessitating the use of intelligent material actuators that can directly convert electrical or magnetic energy into mechanical energy. Piezoelectric ceramic actuators possess advantages such as high displacement resolution, high frequency response, high stiffness, small size, and high reliability, and are widely used in micro-nano processing equipment, precision instruments, atomic force microscopes, and ultra-precision machine tools.

[0003] However, the inherent nonlinear behavior of piezoelectric materials severely impacts the control performance of piezoelectric systems. Nonlinear behavior, particularly hysteresis, is a specific type of nonlinear phenomenon between input voltage and output displacement based on memory. This nonlinear behavior manifests not only as a static hysteresis loop but also as a dynamic frequency-dependent characteristic. This means that as the frequency of the input control signal to the piezoelectric actuator increases, the hysteresis loop becomes thicker and more rounded. This characteristic not only introduces system errors but also causes control difficulties due to the challenge of modeling it, and can even lead to instability in the closed-loop controller. This problem has become a major challenge for high-precision tracking control of piezoelectric drive platforms.

[0004] Currently, researchers have proposed many control methods to suppress the nonlinear behavior of piezoelectric actuators. Among them, sliding mode control is widely used to suppress the hysteresis characteristics of piezoelectric actuators due to its advantages such as high accuracy, strong robustness, good adaptability, and good disturbance rejection performance. However, current research on sliding mode controllers shows that the transient response speed of the controllers is relatively slow and the jitter is relatively large. Furthermore, current techniques are all designed under the condition that the upper bound of the disturbance is known, making it difficult to achieve accurate control of piezoelectric systems in application scenarios where the upper bound of the disturbance is unknown. Summary of the Invention

[0005] The purpose of this invention is to address the shortcomings of existing technologies by providing a robust high-precision tracking control method and controller based on dynamic sliding surfaces.

[0006] In a first aspect, the present invention provides a robust high-precision tracking control method based on a dynamic sliding surface, comprising the following steps:

[0007] Step 1: Construct a dynamic model of the piezoelectric micro-motion platform.

[0008] Step 2: Introduce a nonlinear dynamic function term, a higher-order error term, and an integral term of displacement error e(t) into the dynamic sliding surface s.

[0009] Step 3: Construct a robust, high-precision tracking sliding mode controller based on the dynamic sliding surface s and the actuator model obtained in Step 2.

[0010] Step 4: Input the displacement error e(t) of the driver with hysteresis effect into the sliding mode controller; the sliding mode controller outputs the control signal u(t) to the driver.

[0011] Preferably, the higher-order error terms in the dynamic sliding surface s are constructed based on the square of the displacement error e(t).

[0012] Preferably, the nonlinear dynamic function term in the dynamic sliding surface s is constructed using a hyperbolic tangent function.

[0013] Preferably, the expression for the dynamic sliding surface s is:

[0014]

[0015] Where α, β, and λ are the three gain parameters of the sliding surface.

[0016] Preferably, the hysteresis model of the piezoelectric actuator is any one of the PI model, Bouc-Wen model, and Preisach model.

[0017] Preferably, the hysteresis component of the piezoelectric micro-motion platform system adopts the PI model; the construction process of the dynamic model of the piezoelectric micro-motion platform is as follows:

[0018] Constructing the Play operator F r [u(t)] is as follows:

[0019] F r [u(t)]=max{u(t)-r,min[u(t)+r,F r [u(tT)]]}

[0020] Where r is the threshold of the play operator; T is the control period.

[0021] Based on the Play operator F r [u(t)] is the hysteresis component H[u(t)] of the piezoelectric micro-motion platform dynamic model.

[0022] A dynamic model of the piezoelectric stable platform is established based on the hysteresis component H[u(t)].

[0023] As a preferred option, the Play operator F r The initial value F of [u(t)] rThe value of [u(0)] is 0.

[0024] As a preferred embodiment, the sliding mode controller constructed in step three is as follows:

[0025]

[0026] Where, ω n δ and x represent the natural frequency and damping ratio of the piezoelectric micro-motion platform, respectively; d (t) represents the desired displacement; k smc k is the switching gain; d It is a fixed gain.

[0027] Preferably, the actuator described in step four is a piezoelectric actuator.

[0028] Secondly, the present invention provides a piezoelectric controller for executing the aforementioned robust high-precision tracking control method based on a dynamic sliding surface.

[0029] Thirdly, the present invention provides a piezoelectric drive control system, which includes a piezoelectric micro-motion platform and the aforementioned piezoelectric controller; the input voltage of the piezoelectric micro-motion platform is provided by the piezoelectric controller.

[0030] The beneficial effects of this invention are as follows:

[0031] 1. This invention introduces a nonlinear dynamic function term, a higher-order error term, and an integral term for displacement error into the dynamic sliding surface. The integral term improves the transient response speed of the sliding surface and reduces the steady-state error of the sliding surface; the nonlinear dynamic function improves the response speed and robustness of the sliding surface; and the higher-order error term makes the sliding surface more sensitive to changes in error.

[0032] 2. The robust high-precision tracking control method proposed in this invention can be applied to any piezoelectric micro-motion platform system model. The hysteresis model it targets is not limited to the PI model. Common models such as the Bouc-Wen model and the Preisach model can also be applied, which can effectively improve the control accuracy of the piezoelectric actuator.

[0033] 3. This invention is applicable not only to low-frequency trajectory tracking control but also to high-frequency trajectory tracking control. During operation, piezoelectric actuators inevitably exhibit dynamic hysteresis characteristics. The tracking control method provided by this invention can compensate for these dynamic hysteresis characteristics through feedback control, thereby improving the positioning accuracy of the piezoelectric micro-motion platform. Attached Figure Description

[0034] Figure 1 This is a flowchart of an embodiment of the present invention.

[0035] Figure 2 This is a block diagram of the control system according to an embodiment of the present invention.

[0036] Figure 3 This is a comparison chart of the experimental results of 10μm single-point tracking between the embodiments of the present invention and the traditional sliding mode controller.

[0037] Figure 4 This is a comparison chart of the tracking speed of the embodiment of the present invention and the traditional sliding mode controller under a sinusoidal signal.

[0038] Figure 5 This is a comparison chart of the expected displacement and actual displacement of a traditional sliding mode controller under a sinusoidal signal.

[0039] Figure 6 This is a comparison diagram of the expected displacement and actual displacement of the sliding mode controller provided in this embodiment of the invention under a sinusoidal signal. Detailed Implementation

[0040] To make the objectives, technical solutions, and beneficial effects of the present invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0041] like Figure 1 As shown, a robust high-precision tracking control method based on dynamic sliding surfaces includes the following steps:

[0042] 1) Considering the unknown disturbances such as unmodeled internal dynamics, parameter uncertainties, and external disturbance variables, a piezoelectric-driven micro-positioning platform system model with unknown disturbances is established based on the Prandtl-Ishlinskii model.

[0043] The established dynamic model of the piezoelectric micro-motion platform with unknown perturbation based on the Prandtl-Ishlinskii model is as follows:

[0044]

[0045]

[0046]

[0047] in, This is the second derivative of the actual displacement; ω is the first derivative of the actual displacement; n δ and δ represent the natural frequency and damping ratio of the piezoelectric micro-motion platform, respectively; H[u(t)] is the hysteresis component; F r [u(t)] is the Play operator; p0 and p(r) are the one-time coefficient and operator weight, respectively; u(t) is the control voltage of the piezoelectric actuator; F r[u(tT)] represents the Play operator value at the previous time step; P(t) represents the unknown disturbance components, including unmodeled internal dynamics, parameter uncertainties, and external disturbance variables. T is the control period. r is the threshold value of the Play operator.

[0048] 2) Design a robust high-precision tracking sliding mode controller based on dynamic sliding surfaces:

[0049] First, construct the dynamic sliding surface s of the controller:

[0050]

[0051] Where e(t) is the displacement error, expressed as e(t) = x(t) - x d (t); x d x(t) is the desired displacement of x(t), α, β and λ are the three gain parameters of the sliding surface, respectively; tanh(e) is the nonlinear dynamic function part; e 2 This is a higher-order error term; For the integral term of the sliding surface.

[0052] The expression for constructing a robust, high-precision tracking sliding mode controller based on a dynamic sliding surface s is as follows:

[0053]

[0054] Among them, H -1 ω is the inverse function of the hysteresis component of the driver model; n δ and δ are the natural frequency and damping ratio of the piezoelectric micro-motion platform, respectively; The second derivative of the desired displacement; The first derivative of the actual displacement. k is the first derivative of the error. smc The switching gain of the controller, sgn() is the sign function, and k is the value of the controller. d This represents the fixed gain of the sliding surface. The specific process of constructing the sliding controller is based on existing calculation methods and will not be elaborated here.

[0055] 3) such as Figure 2 As shown, the inputs to the sliding mode controller are the actual displacement x(t) and the desired displacement x. d The difference e(t) between the input voltage and the input voltage u(t) is calculated by inputting the sliding mode controller. The input voltage u(t) is then input into the piezoelectric actuator to track the desired displacement within a finite time, thus achieving high-precision and rapid positioning control of the piezoelectric actuator.

[0056] The proof that the above process can achieve finite-time trajectory tracking is as follows:

[0057] If the unknown disturbance P(t) is bounded by D, k2 > 0, k1 > D is the switching gain of the controller, and sgn(·) is the sign function. When the sliding surface is selected as a dynamic sliding surface, for the piezoelectric micro-motion platform, the tracking error e(t) approaches 0 under the condition that the control input law is:

[0058]

[0059] Choose the Lyapunov function Lyapunov's first derivative is:

[0060]

[0061] Substituting the first derivative of the sliding surface into the above equation and simplifying, we get:

[0062]

[0063] As can be seen from the above equation, the derivative of the Lyapunov function is negative definite. This means that, under the control law as shown in the above equation, when t→∞, s→0. According to the definition of a sliding surface, this implies that the system error e and its rate of change are... These values ​​will also approach zero. Therefore, the traditional sliding mode controller guarantees zero steady-state tracking error. Q.E.D.

[0064] This invention uses the Prandtl-Ishlinskii hysteresis model and the system transfer function obtained by the frequency sweep method as a cascaded model of the piezoelectric micro-motion platform for simulation experiments. The control flow diagram is as follows: Figure 2 As shown.

[0065] The specific parameters of the transfer function of the piezoelectric micro-motion platform system were obtained by the differential evolution algorithm and the frequency sweep method. The specific parameters of the system model of the piezoelectric micro-motion platform are shown in Table 1.

[0066] Table 1 Specific parameters of the piezoelectric micro-motion system

[0067] δ <![CDATA[ω n ]]> R r <![CDATA[p0]]> 0.2 2632.4 10 [0:1:9] 8.15 p(1) p(2) p(3) p(4) p(5) p(6) p(7) p(8) p(9) p(10) 0.15 3.01 0.86 0.73 0.20 0.01 0 0 0 0

[0068] The Prandtl-Ishlinskii hysteresis model was cascaded with the system transfer function to form a dynamic model of the piezoelectric micro-motion platform for simulation experiments. The following parameters were used in the simulation experiments: α = 18000, β = 600, λ = 60, k... smc =0.1 and k d =5000. Figure 3 This is a comparison chart of experimental results for single-point tracking of 10μm between the sliding mode controller of the present invention and the conventional sliding mode controller. Figure 4 This is a comparison chart of the tracking speed of the sliding mode controller of this invention and a conventional sliding mode controller under a sinusoidal signal. From Figure 3 and4 It can be seen that the PID-based adaptive finite-time trajectory tracking controller of this invention significantly improves the tracking speed accuracy compared to the traditional sliding mode controller. Simulation experiments were conducted using the following parameters: α = 18000, β = 600, λ = 60, k... smc =0.1 and k d =5000.

[0069] Figure 5 and Figure 6 The figures show a comparison of experimental results between the expected displacement and the actual displacement trajectory when tracking a sinusoidal signal using a traditional sliding mode controller and the controller of this invention. The results demonstrate that, with a fixed or sinusoidal expected displacement, the sliding mode controller of this invention reaches a stable state faster than the traditional sliding mode controller and exhibits better tracking performance.

Claims

1. A robust high-precision tracking control method based on a dynamic sliding surface, characterized in that: The method comprises the following steps: Step one, constructing a dynamic model of the piezoelectric micro-motion platform; Step two, introducing a nonlinear dynamic function term of displacement error e(t), a high-order error term and an integral term into the dynamic sliding mode surface s; Step three, constructing a robust high-precision tracking sliding mode controller based on the dynamic sliding mode surface s obtained in step two and a driver model; The expression of the dynamic sliding mode surface s is: ； where e(t) is the displacement error, expressed as ; is the desired displacement of , , and are three gain parameters of the sliding mode surface, respectively; is the nonlinear dynamic function part; is the high-order error term; is the integral term of the sliding mode surface; The construction process of the piezoelectric driver hysteresis model is: Constructing the play operator As follows: ； wherein, is the control voltage of the piezoelectric driver; is the value of the Play operator at the previous time instant; T is the control period; r is the threshold value of the play operator; Play operator based Constructing hysteresis model of piezoelectric actuator ; Hysteresis model based A dynamic model of the piezoelectric micro-motion platform is established; The sliding mode controller is as follows wherein, is the inverse function of the hysteresis part of the driver model; and are the natural frequency and damping ratio of the piezoelectric micro-motion stage, respectively; is the second derivative of the desired displacement; is the first derivative of the actual displacement, is the first derivative of the error, is the switching gain of the controller, is the sign function, is the fixed gain of the sliding surface; Step four, inputting the displacement error e(t) of the driver into the sliding mode controller; the sliding mode controller outputs a control signal u(t) to the driver. 2.The robust high-precision tracking control method based on dynamic sliding mode surface according to claim 1, characterized in that: The high-order error term in the dynamic sliding mode surface s is constructed based on the square of the displacement error e(t).

3. The robust high-precision tracking control method based on dynamic sliding mode surface according to claim 1, characterized in that: The nonlinear dynamic function term in the dynamic sliding mode surface s is constructed by a hyperbolic tangent function.

4. The robust high-precision tracking control method based on dynamic sliding mode surface according to claim 1, characterized in that: Play operator initial value of is 0.

5. The robust high-precision tracking control method based on dynamic sliding mode surface according to claim 1, characterized in that: The driver in step four is a piezoelectric driver.

6. A piezoelectric controller characterized by: A robust high-precision tracking control method based on a dynamic sliding mode surface for executing the method in claim 1.

7. A piezoelectric drive control system characterized by comprising: A piezoelectric controller as claimed in claim 6 and a piezoelectric micro-motion platform; the input voltage of the piezoelectric micro-motion platform is provided by the piezoelectric driver.

Images