Robust high-precision tracking control method and controller based on dynamic sliding mode surface

Abstract

The invention provides a robust high-precision tracking control method based on a dynamic sliding mode surface. The method comprises the following steps: 1, constructing a driver model; 2, introducing a nonlinear dynamic function term, a high-order error term and an integral term of a displacement error into the dynamic sliding mode surface; and 3, constructing a robust high-precision tracking sliding mode controller based on the dynamic sliding mode surface and the driver model obtained in the step 2. 4, inputting the displacement error of the driver into the sliding mode controller; and the sliding mode controller outputs a voltage signal to the piezoelectric actuator. The nonlinear dynamic function term, the high-order error term and the integral term of the displacement error are introduced into the dynamic sliding mode surface, the transient response speed of the sliding mode surface is improved through the integral term, and the steady-state error of the sliding mode surface is reduced; the nonlinear dynamic function improves the response speed and robustness of the sliding mode surface; the high-order error term enables the sliding mode surface to be more sensitive to error changes.

Description

Technical Field

[0001] The present invention belongs to the field of micro-nano drive technology, and in particular relates to a robust high-precision tracking control method and a controller based on a dynamic sliding surface. Background Art

[0002] In recent years, with the rapid development of microelectronic information device manufacturing, micro-nano manufacturing, and ultra-precision processing technology, the requirements for precision in manufacturing equipment have become increasingly higher. Traditional motor drivers can no longer meet the stringent requirements of precision motion, and it is necessary to use smart material drivers that can directly convert electrical or magnetic energy into mechanical energy. Piezoelectric ceramic drivers have the advantages of high displacement resolution, high frequency response, high stiffness, small size, and high reliability. They are widely used in micro-nano processing equipment, precision instruments, atomic force microscopes, ultra-precision machine tools, and other fields.

[0003] However, the control performance of the piezoelectric system is seriously affected by the undesirable nonlinear behavior of the piezoelectric material itself. Nonlinear behavior, especially hysteresis, is a special memory-based nonlinear phenomenon between the input voltage and the output displacement. The nonlinear behavior not only manifests itself as a static hysteresis loop, but also as a dynamic frequency-dependent characteristic. This means that when the frequency of the input control signal of the piezoelectric actuator increases, the hysteresis loop becomes thicker and rounder. This characteristic not only causes system errors, but also causes control difficulties due to the difficulty in modeling, and even leads to instability of 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 precision, strong robustness, good adaptability, and good anti-disturbance performance. However, in the current research on sliding mode controllers, the transient response speed of the controller is slow and the jitter is large. In addition, the current technology designs sliding mode controllers under the condition of known upper bounds of disturbances, which makes it difficult to achieve precise control of piezoelectric systems in scenarios where the upper bounds of disturbances are unknown. Summary of the invention

[0005] The object of the present invention is to provide a robust high-precision tracking control method and a controller based on a dynamic sliding surface in view of the deficiencies in the prior art.

[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 the dynamic model of the piezoelectric micro-motion platform.

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

[0009] Step 3: Based on the dynamic sliding surface s and the driver model obtained in step 2, a robust and high-precision tracking sliding mode controller is constructed.

[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 a 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 quadratic displacement error e(t).

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

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

[0014]

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

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

[0017] Preferably, the hysteresis part of the piezoelectric micro-motion platform system adopts a PI model; the construction process of the piezoelectric micro-motion platform dynamics model is:

[0018] Constructing 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] Among them, r is the threshold of the play operator; T is the control period.

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

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

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

[0024] As a preference, the sliding mode controller constructed in step 3 is as follows

[0025]

[0026] Among them, ω n and δ are the natural frequency and damping ratio of the piezoelectric micro-motion platform respectively; x d (t) is the expected displacement; k smc is the switching gain; k d is a fixed gain.

[0027] Preferably, the driver in step 4 is a piezoelectric driver.

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

[0029] In a third aspect, 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 present invention has the beneficial effects:

[0031] 1. The present invention introduces nonlinear dynamic function terms, high-order error terms and integral terms of displacement errors 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; the high-order error term makes the sliding surface more sensitive to changes in errors.

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

[0033] 3. The present invention is not only applicable to low-frequency trajectory tracking control, but also to high-frequency trajectory tracking control. During use, the piezoelectric driver will definitely have dynamic hysteresis characteristics, and the tracking control method provided by the present invention can perform feedback control compensation for the dynamic hysteresis characteristics, thereby improving the positioning accuracy of the piezoelectric micro-motion platform. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1 is a flow chart of an embodiment of the present invention.

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

[0036] Figure 3 It is a comparison chart of the experimental results of 10 μm single-point tracking performed by the embodiment of the present invention and the traditional sliding mode controller.

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

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

[0039] Figure 6 It is a comparison diagram of the expected displacement and the actual displacement of the sliding mode controller provided by an embodiment of the present invention under a sinusoidal signal. DETAILED DESCRIPTION

[0040] In order to make the purpose, technical solution and beneficial effects of the present invention more clear, the present invention is 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 a dynamic sliding surface comprises the following steps:

[0042] 1) Considering the unknown disturbance parts such as internal unmodeled dynamics, parameter uncertainty 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 dynamic model of the piezoelectric micro-motion platform with unknown disturbance based on the Prandtl-Ishlinskii model is:

[0044]

[0045]

[0046]

[0047] in, is the second-order derivative of the actual displacement; is the first-order derivative of the actual displacement; ω n and δ are the natural frequency and damping ratio of the piezoelectric micro-motion platform respectively; H[u(t)] is the hysteresis part; 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)] is the Play operator value at the previous moment; P(t) is the unknown disturbance part such as internal unmodeled dynamics, parameter uncertainty and external disturbance variables. T is the control period. r is the threshold of the Play operator.

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

[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 (t) is the expected 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 is the higher-order error term; is the integral term of the sliding surface.

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

[0053]

[0054] Among them, H -1 is the inverse function of the hysteresis part of the driver model; ω n and δ are the natural frequency and damping ratio of the piezoelectric micro-motion platform, respectively; is the second-order derivative of the desired displacement; is the first-order derivative of the actual displacement, is the first-order derivative of the error, k smc is the switching gain of the controller, sgn() is the sign function, k d is the fixed gain of the sliding surface. The specific process of constructing the sliding mode controller is an existing extrapolation method and will not be described in detail here.

[0055] 3) If Figure 2 As shown, the input of the sliding mode controller is the actual displacement x(t) and the desired displacement x d The difference e(t) of (t) is input into the sliding mode controller for calculation to obtain the input voltage u(t). The input voltage u(t) is input into the piezoelectric actuator to track the desired displacement within a limited time and realize high-precision and fast positioning control of the piezoelectric actuator.

[0056] The proof process 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 switch gain of the controller, and sgn(·) is the sign function. When the sliding surface is selected as the dynamic sliding surface, for the piezoelectric micro-motion platform, the condition for the tracking error e(t) to approach 0 is that the control input law is:

[0058]

[0059] Select the Lyapunov function Then Lyapunov's first-order derivative is:

[0060]

[0061] Substituting the first-order derivative of the sliding surface into the above formula, we can get:

[0062]

[0063] It can be seen from the above formula that the derivative of the Lyapunov function is negative definite. This means that when the control law is as shown in the above formula, s→0 when t→∞, according to the definition of the sliding surface, this means that the system error e and its rate of change are will also approach zero respectively. Therefore, the traditional sliding mode controller guarantees zero steady-state tracking error.

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

[0065] The specific parameters of the transfer function of the piezoelectric micro-motion platform system are obtained by differential evolution algorithm and 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 and the system transfer function cascade are used as the piezoelectric micro-motion platform dynamic model for simulation experiments. The following parameters are used for simulation experiments: α = 18000, β = 600, λ = 60, k smc = 0.1 and k d =5000. Figure 3 This is a comparison chart of the experimental results of the single-point tracking of 10 μm between the sliding mode controller of the present invention and the traditional sliding mode controller. Figure 4 This is a comparison chart of the tracking speed of the sliding mode controller of the present invention and the traditional sliding mode controller under a sinusoidal signal. Figure 3 and4 It can be seen that the PID-based adaptive finite time trajectory tracking controller of the present invention has significantly improved the tracking speed accuracy compared with the traditional sliding mode controller. The following parameters are used for simulation experiments: α = 18000, β = 600, λ = 60, k smc = 0.1 and k d =5000.

[0069] Figure 5 and Figure 6 The experimental results of the comparison of the expected displacement and the actual displacement trajectory of the traditional sliding mode controller and the controller of the present invention when tracking the sinusoidal signal are shown in Figure 2. After testing, it is found that when the expected displacement is fixed or sinusoidally changing, the sliding mode controller of the present invention can reach a stable state faster than the traditional sliding mode controller and has better tracking performance.

Claims

1. A robust high-precision tracking control method based on dynamic sliding surface, characterized in that: The following steps are involved: Step 1: Construct a dynamic model of the piezoelectric micro-motion platform; Step 2: Introduce nonlinear dynamic function terms, high-order error terms and integral terms of the displacement error e(t) into the dynamic sliding surface s; Step 3: construct a robust high-precision tracking sliding mode controller based on the dynamic sliding mode surface s and the driver model obtained in step 2; Step 4: Input 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 surface according to claim 1, characterized in that: The higher-order error terms in the dynamic sliding surface s are constructed based on the square of the displacement error e(t).

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

4. The robust high-precision tracking control method based on dynamic sliding surface according to claim 1, characterized in that: The expression of dynamic sliding surface s is: Among them, α, β and λ are three gain parameters of the sliding surface.

5. The robust high-precision tracking control method based on dynamic sliding surface according to claim 4, characterized in that: The construction process of the piezoelectric actuator hysteresis model is: Constructing Play Operator F r [u(t)] is as follows: F r [u(t)]=max{u(t)-r,min[u(t)+r,F r [u(t-T)]]} Among them, r is the threshold of the play operator; T is the control period; Based on Play operator F r [u(t)] constructs the hysteresis model H[u(t)] of the piezoelectric actuator; The dynamic model of the piezoelectric micro-motion platform is established based on the hysteresis model H[u(t)].

6. The robust high-precision tracking control method based on dynamic sliding surface according to claim 5, characterized in that: Play Operator F r The initial value F of [u(t)] r The value of [u(0)] is 0.

7. The robust high-precision tracking control method based on dynamic sliding surface according to claim 5, characterized in that: The sliding mode controller constructed in step 3 is as follows Among them, ω n and δ are the natural frequency and damping ratio of the piezoelectric micro-motion platform respectively; x d (t) is the expected displacement; k smc is the switching gain; k d is a fixed gain.

8. The robust high-precision tracking control method based on dynamic sliding surface according to claim 1, characterized in that: The driver described in step 4 is a piezoelectric driver.

9. A piezoelectric controller, characterized in that: Used to execute a robust high-precision tracking control method based on dynamic sliding surface as described in claim 1.

10. A piezoelectric drive control system, characterized in that: It comprises a piezoelectric micro-motion platform and a piezoelectric controller as claimed in claim 8; the input voltage of the piezoelectric micro-motion platform is provided by a piezoelectric driver.

Images