Flexible macro-micro driving motion system preset output tracking control method based on active disturbance rejection

Abstract

The invention belongs to the technical field of flexible macro-micro drive motion system preset output tracking control, and particularly relates to a flexible macro-micro drive motion system preset output tracking control method based on active disturbance rejection, which specifically aims to estimate and compensate multi-modal vibration caused by a flexible hinge, an elastic structure and the like in real time through a linear extended state observer (LESO); the influence of high-frequency resonance on the system stability is reduced; an anti-saturation compensator is introduced to dynamically adjust the control quantity, the output saturation problem of the piezoelectric stack caused by input voltage amplitude limiting is avoided, and the physical realizability of a control instruction is ensured; and in combination with a preset performance function and a dynamic surface method, a tracking controller giving consideration to both transient response and steady-state precision is designed, and meanwhile, an LMI optimization control distribution strategy is utilized to coordinate the cooperative action of a macro-micro actuator under saturation constraint, so that the uncertainty of the system to the model and the robustness of the system to external disturbance are enhanced.

Description

Technical Field

[0001] The present invention belongs to the technical field of preset output tracking control of a flexible macro-micro drive motion system, and particularly relates to a method for preset output tracking control of a flexible macro-micro drive motion system based on active disturbance rejection. Background Art

[0002] In the prior art, for example, in existing literature,

[0003] "Zheng, J., A. Salton and M. Fu, Design and control of a rotary dual-stage actuator positioning system. Mechatronics, 2011. 21(6): p. 1003 - 1012." proposed a composite non-linear control method for a rotary dual-stage actuator (DSA), which suppresses the overshoot phenomenon of the main actuator when approaching the set point by designing a non-linear feedback law. Its technical solutions include: (1) designing feed-forward compensation based on the system dynamics model; (2) introducing a non-linear damping term to dynamically adjust the control gain; (3) verifying through experiments that the overshoot is significantly reduced in the step response. This method shows good transient performance in the rotary system, but has the following deficiencies: (1) It does not model and compensate for the structural vibrations caused by flexible components (such as hinges, elastic structures), resulting in limited stability in branch systems with flexible mechanisms such as FDLS; (2) It does not consider the actuator input voltage saturation constraint, and in practical applications, oscillations or a decrease in accuracy may be caused by the control quantity exceeding the physical limit. Another example is in existing literature,

[0004] "Dong, W., J. Tang and Y. El Deeb, Design of a linear-motion dual-stage actuation system for precision control. Smart Materials and Structures, 2009. 18(9): p. 095035." developed a VCM / piezoelectric stack composite linear drive system, which adopts a dual-loop control strategy: the inner loop is a precision position closed-loop based on a piezoelectric actuator, and the outer loop is a coarse-tuning tracking loop of the VCM. Its core measures include: (1) compensating for the low-frequency tracking error of the VCM by using the high bandwidth characteristics of the piezoelectric actuator; (2) allocating the control tasks of the macro-micro actuators according to the frequency domain separation principle. This scheme achieves sub-micron accuracy in long-stroke positioning, but has the following defects: (1) It relies on the frequency domain separation assumption, and when the system is subject to broadband disturbances (such as high-frequency resonance of flexible mechanisms), coupling interference is likely to occur between the control loops; (2) It does not integrate an anti-saturation mechanism, and the output saturation problem caused by the input voltage limitation of the piezoelectric actuator is not effectively handled, which may cause integrator saturation or tracking lag.

[0005] In summary, although the existing dual-drive system control methods have achieved certain results in specific scenarios, they generally have the following deficiencies: (1) Insufficient flexible vibration suppression: The dynamic characteristics of flexible components are not fully modeled, resulting in difficulty in actively compensating for high-frequency resonance; (2) Lack of actuator saturation handling: The control algorithm does not incorporate an anti-saturation strategy, making it prone to performance degradation due to input saturation; (3) Limited disturbance robustness: The online estimation and compensation capabilities for the lumped disturbances of the system (such as model uncertainty and external disturbances) are weak, affecting the reliability of high-precision tracking. In view of the above defects, the present invention proposes a preset output tracking control method for a flexible macro-micro drive motion system based on active disturbance rejection; a solution based on the active disturbance rejection method Summary of the Invention

[0006] The object of the present invention is to provide a preset output tracking control method for a flexible macro-micro drive motion system based on active disturbance rejection, aiming to solve the problems of vibration suppression, actuator saturation, and complex disturbance compensation in high-precision and large-stroke linear motion systems in precision engineering. The FDLS realizes millimeter-level stroke and nanometer-level resolution through a macro-micro dual-drive structure (such as a piezoelectric stack and a flexible displacement amplifier), and is suitable for fields with strict requirements for dynamic performance and steady-state accuracy such as semiconductor manufacturing and optical precision adjustment.

[0007] The technical solutions adopted by the present invention are specifically as follows:

[0008] A preset output tracking control method for a flexible macro-micro drive motion system based on active disturbance rejection, characterized in that it includes the following steps:

[0009] Step 1: Establish a tracking control model:

[0010] Similar to the spring-mass system, based on a compliant mechanism that transmits motion through the elastic deformation of a flexible hinge, it is classified as a second-order linear time-invariant system, and a force analysis based on the masses of the first-stage actuator m1 and the second-stage actuator m2; according to Newton's second law, the following dynamic equilibrium equation is derived:

[0011]

[0012] where F p is the internal force generated when the piezoelectric ceramic deforms, which is difficult to obtain; F p is eliminated by summing equations (1) and (2):

[0013]

[0014] where y1 is the actual displacement of m1, y2 is the actual displacement of m2, and y is the displacement of the system, expressed as: y = y2 = y1 + l2 (4)

[0016] And F c1 and F c2 are damping forces, whose directions are opposite to the direction of motion and are proportional to the velocity:

[0017]

[0018] where c1 and c2 are damping coefficients; F k2 is the elastic force generated by the equivalent spring compression:

[0019] F k2 = k2·y2 (6)

[0020] where k2 is the spring coefficient of the equivalent spring; F d1 is the thrust caused by the elongation of the piezoelectric stack and is calculated by the following formula:

[0021] F d1 = k1(A0l1 - y1) (7)

[0022] where A0 is the amplification ratio of the two-stage bridge amplifier without load; l i (i = 1, 2) represents the output displacements of the piezoelectric stack and the piezoelectric ceramic; according to the inverse piezoelectric effect, the external displacement of the piezoelectric material is approximately proportional to the voltage load at both ends, and the hysteresis effect is regarded as a system disturbance, that is:

[0023] l1 = p1v1 + d1

[0024] l2 = p2v2 + d2 (8) where p i and d i (i = 1, 2) are the piezoelectric coefficient and the interference respectively; the voltage applied to the piezoelectric ceramic v i is limited; therefore, the constraints on l1 and l2

[0025] 0 ≤ l1 ≤ L1, 0 ≤ l2 ≤ L2 (9)

[0026] Combining equations (3) - (8), the equation is obtained:

[0027]

[0028] where m = m1 + m2, c = c1 + c2, k = k1 + k2; the last two terms of equation (10) are the first and second derivatives of the output displacement of the piezoelectric ceramic, which are very small and can be regarded as interference terms; therefore, the dynamic equation can be written as follows:

[0029]

[0030] where

[0031]

[0032] Step 2: Complete the controller design, and the controller design includes the following steps:

[0033] Step 201: Transform the tracking control model in Step 1;

[0034] In Step 201, the state variables are defined as x1 = y, and v = [v1, v2], b = [k1A0P1, K1P2], and the dynamic formula (11) is rewritten as follows:

[0035]

[0036] where u = bv;

[0037] The goal is for x1 to track the desired output y of the system equation (13) using the control input u under unknown disturbances d ; for actuators l1 and l2 subject to the limitation of formula (9), there exist constants u max and u min such that the control u saturates

[0038]

[0039] where u c is the control command to be designed later;

[0040] According to the physical limitations, there is a finite difference between the required control input u and the nominal control input u c , which is defined as

[0041] Δu = u - u c (15)

[0042] where Δu is bounded, i.e., |Δu| ≤ u M with a constant u M , without loss of generality; Equation (13) is rewritten as follows,

[0043]

[0044] Step 202: Design a linear extended observer;

[0045] In Step 202, a linear extended observer is designed; by defining x3 = d and h(t) = d, a linear extended state observer (LESO) is introduced:

[0046]

[0047] where

[0048]

[0049] and ω0>0, α i (i = 1, 2, 3) needs to be selected to satisfy

[0050] s 3 +α1s 2 +α2s + α3 (19)

[0051] The polynomial (19) is Hurwitz, that is, the roots of the polynomial (19) are located in the left half of the complex plane or on the imaginary axis; for convenience, by solving

[0052] s 3 +α1s 2 +α2s + α3 = (s + 1) 3 (20)

[0053] obtain

[0054]

[0055] Step 203: Design a dynamic surface controller based on a linear extended state observer (LESO) for output tracking with error constraints;

[0056] In step 203, design a dynamic surface controller based on LESO for output tracking with error constraints; define the tracking error e = [e1, e2] T as e1 = x1 - y d , e2 = x2 - x 2d , where x 2d is the state variable, which will be given below;

[0057] To predefine the transient and steady-state performance of the tracking error e1, the specified performance is described by the following inequality

[0058] -η1ρ(t) < e1 < η2ρ(t) (22)

[0059] where ρ(t) = (ρ0 - ρ ∞ )e -γt +ρ ∞ , ρ0 > ρ ∞ > 0, γ > 0, η1, η2 > 0; and ρ0 satisfies -η1ρ0 < e1(0) < η2ρ0; to achieve the performance of (22), convert the constrained error into an equivalent unconstrained error; define

[0060] e1(t) = ρ(t)S[ε(t)] (23)

[0061] where ε(t) is the converted error, and the conversion function S[ε(t)] = (η1e ε(t) -η2e-ε(t) ) / (e ε(t) + e -ε(t) ) is smooth and strictly increasing, with the following properties: 1) -η1 < S[ε(t)] < η2; 2) lim ε→∞ S(ε) = η2 and lim ε→-∞ S(ε) = -η1. ε(t), written as:

[0062]

[0063] where λ = σ / ρ, and

[0064] Define the virtual controller as:

[0065]

[0066] where k1 is a positive constant; adopting the idea of dynamic surface, passing the virtual controller through a filter to obtain x 2d :

[0067]

[0068] Define The control input of the system is designed as

[0069]

[0070] where k i , i = 2, 3 are positive constants; and ξ is the output of the following input-limited compensator

[0071]

[0072] where k4 > 0.

[0073] Step 3: According to the control allocation method that distributes the control input u to the actuators v1 and v2, perform the control allocation for actuator saturation.

[0074] In the said Step 3, the control allocation method that distributes the control input u to the actuators v1, v2; define L = [L1, L2] T The control allocation problem for actuator saturation is expressed as:

[0075]

[0076] where h is a positive constant, denotes v T$M_v, M = \text{diag}\{M_1, M_2\}$ is a diagonal matrix with positive constants $M_1 > 0$ and $M_2 > 0$. To solve the above problems, the LMI method is adopted and the following optimal function is solved:

[0077]

[0078] s.t.

[0079] $\Upsilon_1 + \Upsilon_2 - \Upsilon < 0\ (31)$ where $\Upsilon > 0$, $\Upsilon_1 > 0$, $\Upsilon_2 > 0$, and

[0080]

[0081] The technical effects achieved by the present invention are as follows:

[0082] The present invention has high-precision tracking performance; in experimental verification, the present invention achieves nanometer-level steady-state accuracy and fast dynamic response: steady-state noise suppression: the steady-state output noise is only 5 nm, far lower than that of traditional dual-drive systems (usually in the order of dozens of nanometers), meeting the requirements of ultra-precision positioning; tracking error optimization: in the step response test, the system tracking error is stabilized within 2.5 nm.

[0083] The present invention has the effects of strong robustness and vibration suppression, disturbance estimation and compensation: the LESO estimates the lumped disturbance of the system in real time (including flexible mechanism vibration, model uncertainty and external interference), and the disturbance compensation efficiency is remarkable. Flexible vibration suppression: for the flexible hinge resonance problem of FDLS, the standard deviation of the tracking error remains at 2.5 nm, significantly better than that of PID control (the standard deviation of the error is 25 nm), proving its strong robustness. In the present invention, flexible vibration is suppressed: the multi-modal vibration caused by flexible hinges, elastic structures, etc. is estimated and compensated in real time through a linear extended state observer (LESO), reducing the influence of high-frequency resonance on the system stability; in the present invention, the disturbance robustness is improved: by combining a preset performance function and the dynamic surface method, a tracking controller that takes into account transient response and steady-state accuracy is designed, and at the same time, the LMI is used to optimize the control allocation strategy to coordinate the cooperative actions of macro-micro actuators under saturation constraints, enhancing the robustness of the system to model uncertainty and external disturbances.

[0084] The present invention has fast dynamic response and anti-saturation ability; step response speed: the system stabilization time is only 0.15 s, significantly improving the dynamic performance. Anti-saturation compensation: After introducing the anti-saturation compensator, the input voltage saturation rate of the piezoelectric actuator decreases, avoiding the tracking lag problem caused by integrator saturation. Control allocation optimization: By optimizing the cooperative control of the macro-micro actuator through the LMI method, under the condition of limited input voltage, the macro actuator undertakes 85% of the coarse adjustment task, and the micro actuator completes the remaining 15% of the precision compensation, achieving efficient resource allocation. In the present invention, to handle actuator saturation: introduce an anti-saturation compensator to dynamically adjust the control quantity, avoid the output saturation problem of the piezoelectric stack caused by input voltage limiting, and ensure the physical realizability of control instructions.

[0085] In summary, by integrating the active disturbance rejection control and the dynamic surface technology, the present invention realizes the high-precision, high-dynamic output tracking performance and strong robustness of the FDLS in a flexible dual-drive motion system, especially finally under the conditions of large stroke (millimeter level) and high resolution (nanometer level). Experimental data show that its performance is significantly better than existing methods, providing a reliable motion control solution for the field of precision manufacturing and measurement. BRIEF DESCRIPTION OF THE DRAWINGS

[0086] Figure 1 It is the force analysis diagram in the tracking control model in Step 1 of the present invention. In the figure, (a) represents the spring-damper-mass system; (b) represents the force analysis diagram of the first-stage actuator m1; (c) represents the force analysis diagram of the second-stage actuator m2.

[0087] Figure 2 It is the control mode diagram in the controller design in Step 2 of the present invention.

[0088] Figure 3 It is the experimental procedure diagram in the actual experimental process of the present invention.

[0089] Figure 4 It is the effect diagram under four different controllers in the actual experimental process of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0090] In order to make the objectives and advantages of the present invention clearer, the present invention will be specifically described below in conjunction with embodiments. It should be understood that the following text is only used to describe one or several specific implementation manners of the present invention, and does not strictly limit the specific protection scope claimed by the present invention.

[0091] As Figures 1-4 shown, the preset output tracking control method for a flexible macro-micro drive motion system based on active disturbance rejection includes the following steps:

[0092] Step 1: Establish a tracking control model:

[0093] Similar to the spring-mass system, based on compliant mechanisms, motion is transmitted through the elastic deformation of flexible hinges and is classified as a second-order linear time-invariant system, as shown in Figure 1 (a). Figure 1 (b) and (c) illustrate the force analysis based on the masses of the first-stage actuator m1 and the second-stage actuator m2; according to Newton's second law, the dynamic equilibrium equations are derived as follows:

[0094]

[0095] where F p is the internal force generated when the piezoelectric ceramic deforms, which is difficult to obtain; F p is eliminated by summing equations (1) and (2):

[0096]

[0097] where y1 is the actual displacement of m1, y2 is the actual displacement of m2, and the displacement of the system y is expressed as follows:

[0098] y = y2 = y1 + l2 (4) And F c1 and F c2 are damping forces, whose directions are opposite to the direction of motion and are proportional to the velocity:

[0099]

[0100] where c1, c2 are damping coefficients; F k2 is the elastic force generated by the equivalent spring compression:

[0101] F k2 = k2·y2 (6)

[0102] where k2 is the spring constant of the equivalent spring; F d1 is the thrust caused by the elongation of the piezoelectric stack and is calculated by the following formula:

[0103] F d1 = k1(A0l1 - y1) (7)

[0104] where A0 is the amplification ratio of the two-stage bridge amplifier when unloaded; l i (i = 1, 2) represents the output displacements of the piezoelectric stack and the piezoelectric ceramic; according to the inverse piezoelectric effect, the external displacement of the piezoelectric material is approximately proportional to the voltage load at both ends, and the hysteresis effect is regarded as a system perturbation, i.e.: l1 = p1v1 + d1

[0105] l2 = p2v2 + d2 (8)

[0106] where p i and d i(i = 1, 2) are the piezoelectric coefficient and the interference respectively; the voltage applied to the piezoelectric ceramic v i is limited; thus, the constraints on l1 and l2

[0107] 0 ≤ l1 ≤ L1, 0 ≤ l2 ≤ L2 (9)

[0108] Combining equations (3) - (8), the equation is obtained:

[0109]

[0110] where m = m1 + m2, c = c1 + c2, k = k1 + k2; the last two terms of equation (10) are the first - order and second - order derivatives of the output displacement of the piezoelectric ceramic, which are very small and can be regarded as interference terms; thus, the dynamic equation can be written as follows:

[0111]

[0112] where

[0113]

[0114] Step 2: Complete the controller design. This step introduces the controller design of FDLS. First, a linear extended - state observer is introduced to estimate the external disturbance, and an anti - saturation compensator is adopted to solve the input - saturation problem. Then, a dynamic surface controller is designed to achieve output tracking control with a prescribed error constraint. Finally, a control allocation method that maps the control input to the macro and micro actuators with actuator saturation is proposed and solved by the linear matrix inequality (LMI) method. Figure 2 Illustrates the scheme of the above - proposed control method, and the controller design includes the following steps:

[0115] Step 201: Transform the tracking - control model in Step 1;

[0116] In Step 201, the state variables are defined as x1 = y, and v = [v1, v2], b = [k1A0P1, K1P2], and the dynamic formula (11) is rewritten as follows:

[0117]

[0118] where u = bv;

[0119] The objective is for x1 to track the desired output y of the system equation (13) using the control input u under unknown disturbances d ; for the actuators l1 and l2 restricted by formula (9), there exist constants u max and u min , such that the control u saturates

[0120]

[0121] where u c is a control command designed later;

[0122] Due to physical limitations, there is a finite difference between the required control input u and the nominal control input u c which is defined as

[0123] Δu = u - u c (15)

[0124] where Δu is bounded, i.e., |Δu| ≤ u M with a constant u M , without loss of generality; Equation (13) is rewritten as follows,

[0125]

[0126] Step 202: Design a linear extended observer;

[0127] In Step 202, a linear extended observer is designed; by defining x3 = d and h(t) = d, a linear extended state observer (LESO) is introduced:

[0128]

[0129] where

[0130]

[0131] and ω0 > 0, the selection of α i (i = 1, 2, 3) needs to satisfy

[0132] s 3 + α1s 2 + α2s + α3 (19)

[0133] The polynomial (19) is Hurwitz, i.e., the roots of the polynomial (19) lie in the left half of the complex plane or on the imaginary axis; for convenience, by solving

[0134] s 3 + α1s 2 + α2s + α3 = (s + 1) 3 (20)

[0135] we get

[0136]

[0137] Step 203: Design a dynamic surface controller based on a linear extended state observer (LESO) for output tracking with error constraints;

[0138] In Step 203, design a dynamic surface controller based on LESO for output tracking with error constraints; define the tracking error e = [e1, e2] T where e1 = x1 - y d , e2 = x2 - x 2d , where x 2d is the state variable, which will be given below;

[0139] To predefine the transient and steady-state performance of the tracking error e1, the specified performance is described by the following inequality

[0140] -η1ρ(t) < e1 < η2ρ(t) (22)

[0141] where ρ(t) = (ρ0 - ρ ∞ )e -γt + ρ ∞ , ρ0 > ρ ∞ > 0, γ > 0, η1, η2 > 0; and ρ0 satisfies -η1ρ0 < e1(0) < η2ρ0; To achieve the performance of (22), convert the constrained error into an equivalent unconstrained error; define

[0142] e1(t) = ρ(t)S[ε(t)] (23)

[0143] where ε(t) is the converted error, and the conversion function S[ε(t)] = (η1e ε(t) -η2e -ε(t) ) / (e ε(t) + e -ε(t) ) is smooth and strictly increasing, with the following properties: 1) -η1 < S[ε(t)] < η2; 2) lim ε→∞ S(ε) = η2 and lim ε→-∞ S(ε) = -η1. ε(t), written as:

[0144]

[0145] where λ = σ / ρ, and

[0146] Define the virtual controller as:

[0147]

[0148] where k1 is a positive constant; Adopting the idea of dynamic surface, pass the virtual controller through a filter to obtain x2d :

[0149]

[0150] Define The control input of the system is designed as

[0151]

[0152] where k i , i = 2, 3 are positive constants; and ξ is the output of the following input-limited compensator

[0153]

[0154] where k4 > 0.

[0155] Step 3: According to the control allocation method of the control input u assigned to the actuator v1 and the actuator v2, perform the control allocation of actuator saturation.

[0156] In the said Step 3, the control allocation method of the control input u assigned to the actuators v1, v2; define L = [L1, L2] T The control allocation problem of actuator saturation is expressed as:

[0157]

[0158] where h is a positive constant, denotes v T Mv, M = diag{M1, M2} is a diagonal matrix with positive constants M1 > 0 and M2 > 0; to solve the above problem, the LMI method is adopted and solved by solving the following optimal function:

[0159]

[0160] s.t.

[0161] Υ1 + Υ2 - Υ < 0 (31)

[0162] where Υ > 0, Υ1 > 0, Υ2 > 0, and

[0163]

[0164] In the actual use of the present invention, an FDLS experimental prototype was built to verify the feasibility of the control method. Among them, the bridge displacement amplifier 1 is made of spring steel Mn65, and the rest of the components are made of aluminum alloy LY12. The macro displacement is provided by a piezoelectric stack actuator (model PST150 / 10 / 80, stroke 80μm, manufacturer: Piezomechanik), and the micro displacement is driven by a piezoelectric ceramic sheet (model PST150 / 7×7×2, stroke 3μm, manufacturer: Piezomechanik) as the secondary actuator. The driving power amplification factor of the piezoelectric actuator is 15 times, the input voltage limit is 150V, and the corresponding control signal saturation threshold is 10V. A linear encoder displacement sensor (model TONiC, resolution 1nm, manufacturer: Renishaw) is installed at the output end of the system to feedback the displacement signal to the control system.

[0165] Composition of the control system and experimental configuration: The control system is based on the PXI system of NI Corporation and includes the following components:

[0166] Real-time controller (model PXI-8862, main frequency 2.6GHz, 8 cores); multi-functional I / O card (model PXI-6363, 16-bit resolution, 4-channel analog output); eight-slot chassis.

[0167] The control algorithm is implemented and executed through LabVIEW programming. In the experiment, a capacitive micrometer (model capaNCDT_6500, manufacturer: Micro-Epsilon) is used as a third-party sensor to detect the output displacement. Figure 3 The schematic diagram of the experimental device is shown.

[0168] In the step response test, a step signal with an amplitude of 1mm is applied, and the displacement responses of the four control methods are as Figure 4 shown:

[0169] 1. Aggressive gain PID control (high proportional gain Kp and integral gain Ki): The response speed is the fastest, but the overshoot is the largest (up to 12%), and the settling time is the longest (0.35s);

[0170] 2. Gentle gain PID control: The overshoot is suppressed to 5%, but the response speed is significantly reduced, and the rise time is extended to 0.25s;

[0171] 3. IST control: The overshoot suppression effect is good (no overshoot), but the rise time is the longest (0.4s);

[0172] ADROTC method: with the optimal comprehensive performance, the response trajectory is strictly constrained within the preset output tracking boundary (black dashed line), without overshoot and the settling time is only 0.15 s. The steady-state noise is as low as 5 nm, and the tracking error is about 2.5 nm, significantly superior to the performance of traditional dual-drive systems.

[0173] The present invention discloses: the disturbance estimation and compensation of a linear extended state observer (LESO); the lumped disturbance of the system (including external disturbances, model uncertainties, and flexible mechanism vibrations) is estimated in real time by using a linear extended state observer and fed forward and compensated into the control law, significantly improving the robustness of the system to complex disturbances. This technology solves the defects of traditional methods that rely on accurate models and are difficult to compensate for broadband disturbances online.

[0174] The present invention discloses: the dynamic adjustment mechanism of an anti-saturation compensator; an anti-saturation compensator is introduced to dynamically correct the output of the control quantity. By predicting the saturation state of the actuator and adjusting the dynamics of the integrator, the problem of integral saturation caused by the input voltage limit of the piezoelectric stack is avoided. Compared with traditional PID control, the saturation rate of the piezoelectric actuator decreases, ensuring the physical realizability of the control command.

[0175] The present invention discloses: the co-design of dynamic surface control and a preset performance function; by combining the preset performance function (PPF) with the dynamic surface method (DSC), and by constraining the convergence boundary of the tracking error (such as the exponential decay rate and the steady-state error limit), the dual goals of transient overshoot suppression and steady-state accuracy optimization are achieved. Experiments show that the settling time of the step response is shortened to 0.15 s, and the steady-state error is controlled within 2.5 nm.

[0176] The preset output tracking method of a flexible macro-micro drive motion system (FDLS) based on active disturbance rejection control (ADRC) proposed by the present invention significantly improves the control performance and robustness of the system by integrating a linear extended state observer (LESO), an anti-saturation compensator, and dynamic surface control technology. The specific effects are as follows:

[0177] 1. High-precision tracking performance

[0178] In experimental verification, the present invention achieves nanoscale steady-state accuracy and fast dynamic response:

[0179] Steady-state noise suppression: The steady-state output noise is only 5 nm, much lower than that of traditional dual-drive systems (usually in the order of dozens of nanometers), meeting the requirements of ultra-precision positioning.

[0180] Tracking error optimization: In the step response test, the tracking error of the system is stabilized within 2.5 nm.

[0181] 2. Fast dynamic response and anti-saturation ability

[0182] Step response speed: The system stabilization time is only 0.15 s, significantly improving the dynamic performance.

[0183] Anti-saturation compensation: After introducing the anti-saturation compensator, the input voltage saturation rate of the piezoelectric actuator decreases, avoiding the tracking lag problem caused by integrator saturation.

[0184] Control allocation optimization: The cooperative control of the macro-micro actuator is optimized by the LMI method. Under the condition of limited input voltage, the macro actuator undertakes 85% of the coarse adjustment task, and the micro actuator completes the remaining 15% of the precise compensation, achieving efficient resource allocation.

[0185] 3. Strong robustness and vibration suppression

[0186] Disturbance estimation and compensation: The LESO estimates the lumped disturbance of the system in real time (including the vibration of the flexible mechanism, model uncertainty, and external interference), and the disturbance compensation efficiency is significant.

[0187] Flexible vibration suppression: Aiming at the resonance problem of the flexible hinge of the FDLS, the standard deviation of the tracking error remains at 2.5 nm, which is significantly better than the PID control (the standard deviation of the error is 25 nm), proving its strong robustness.

[0188] The present invention realizes the unity of high precision, high dynamics, and strong robustness in the flexible dual-drive motion system by integrating the active disturbance rejection control and the dynamic surface technology. Experimental data show that its performance is significantly better than the existing methods, providing a reliable motion control solution for the field of precision engineering.

[0189] The present invention discloses: Cooperative control allocation of macro-micro actuators based on LMI; Establish a linear matrix inequality (LMI) optimization model, and dynamically allocate the control amounts of the macro actuator (three-dimensional bridge amplifier) and the micro actuator (piezoelectric ceramic) under the condition of considering the actuator saturation constraint. The macro actuator undertakes 85% of the coarse adjustment task, and the micro actuator completes 15% of the precise compensation, maximizing the utilization rate of the system control resources.

[0190] The above are only the preferred embodiments of the present invention. It should be noted that for those of ordinary skill in the art, without departing from the principle of the present invention, several improvements and refinements can be made, and these improvements and refinements should also be regarded as the protection scope of the present invention. The structures, devices, and operation methods not specifically described and explained in the present invention are implemented according to the conventional means in the art without special description and limitation.

Claims

1. A preset output tracking control method for a flexible macro-micro drive motion system based on active disturbance rejection, characterized in that: It includes the following steps: Step 1: Establish a tracking control model: Based on the fact that the compliant mechanism transfers motion through the elastic deformation of the flexible hinge and is classified as a second-order linear time-invariant system, and the force analysis based on the masses of the first-stage actuator m1 and the second-stage actuator m2; according to Newton's second law, the dynamic equilibrium equation is derived as follows: Among them, F p is the internal force generated when the piezoelectric ceramic deforms; F p is eliminated by summing formulas (1) and (2): where y1 is the actual displacement of m1, y2 is the actual displacement of m2, and the displacement of the system y is expressed as follows: y = y2 = y1 + l2 (4) and F c1 and F c2 are damping forces, whose directions are opposite to the direction of motion and are proportional to the velocity: where c1 and c2 are damping coefficients; F k2 is the elastic force generated by the equivalent spring compression: F k2 = k2·y2 (6) where k2 is the spring constant of the equivalent spring; F d1 is the thrust caused by the elongation of the piezoelectric stack and is calculated by the following formula: F d1 = k1(A0l1 - y1) (7) where A0 is the amplification ratio of the two-stage bridge amplifier without load; l i (i = 1, 2) represents the output displacements of the piezoelectric stack and the piezoelectric ceramic; according to the inverse piezoelectric effect, the external displacement of the piezoelectric material is approximately proportional to the voltage load at both ends, and the hysteresis effect is regarded as a system disturbance, that is: l1 = p1v1 + d1 l2 = p2v2 + d2 (8) where p i and d i (i = 1, 2) are the piezoelectric coefficient and the interference respectively; The voltage applied to the piezoelectric ceramic v i is limited; thus, the constraints on l1 and l2 0 ≤ l1 ≤ L1, 0 ≤ l2 ≤ L2 (9) Combining equations (3) - (8), the equation is obtained: where m = m1 + m2, c = c1 + c2, k = k1 + k2; the last two terms of equation (10) are the first-order and second-order derivatives of the output displacement of the piezoelectric ceramic, regarded as interference terms; therefore, the dynamic equation can be written as follows: where Step 2: Complete the controller design, and the controller design includes the following steps: Step 201: Transform the tracking control model in Step 1; Step 202: Design a linear extended observer; Step 203: Design a dynamic surface controller based on the linear extended state observer (LESO) for output tracking with error constraints; Step 3: According to the control allocation method of the control input u to the actuators v1 and v2, perform the control allocation for actuator saturation.

2. The preset output tracking control method for the flexible macro-micro drive motion system based on active disturbance rejection according to claim 1, wherein: In the step 201, the state variables are defined as x1 = y, and v = [v1, v2], b = [k1A0P1, K1P2], and the dynamic formula (11) is rewritten as follows: Among them The goal is for x1 to track the desired output y of system equation (13) using the control input u under unknown disturbances d ; for actuators l1 and l2 subject to the limitation of formula (9), there exist constants u max and u min such that the control u saturates where u c is a control command designed later; Due to physical limitations, there is a finite difference between the required control input u and the nominal control input u c which is defined as Δu = u - u c (15) where Δu is bounded, i.e., |Δu| ≤ u M with constant u M , without loss of generality; Equation (13) is rewritten as follows, 3. The preset output tracking control method for the flexible macro-micro drive motion system based on active disturbance rejection according to claim 2, wherein: In Step 202, a linear extended observer is designed; by defining x3 = d and h(t) = d, a linear extended state observer (LESO) is introduced: where and ω0>0, α i (i = 1, 2, 3) needs to be selected to satisfy s 3 +α1s 2 +α2s+α3 (19) The polynomial (19) is Hurwitz, that is, the roots of the polynomial (19) are located in the left half of the complex plane or on the imaginary axis; by solving s 3 +α1s 2 +α2s+α3=(s + 1) 3 (20) obtain 4. The preset output tracking control method for the flexible macro-micro drive motion system based on active disturbance rejection according to claim 3, characterized in that: In step 203, design a dynamic surface controller based on LESO; define the tracking error e = [e1, e2] T where e1 = x1 - y d , e2 = x2 - x 2d , where x 2d is a state variable, which will be given below; The transient and steady-state performance of the predefined tracking error e1, and the specified performance is described by the following inequality -η1ρ(t) < e1 < η2ρ(t) (22) where ρ(t) = (ρ0 - ρ ∞ )e -γt + ρ ∞ , ρ0 > ρ ∞ > 0, γ > 0, η1, η2 > 0; and ρ0 satisfies -η1ρ0 < e1(0) < η2ρ0; To achieve the performance of (22), the restricted error is converted into an equivalent unrestricted error; Define e1(t) = ρ(t)S[ε(t)] (23) where ε(t) is the converted error, and the conversion function S[ε(t)] = (η1e ε(t) -η2e -ε(t) ) / (e ε(t) +e -ε(t) ) is smooth and strictly increasing, with the following properties: 1) -η1 < S[ε(t)] < η2; 2) lim ε→∞ S(ε) = η2 and lim ε→-∞ S(ε) = -η1. ε(t), written as: where λ = σ / ρ, and Define a virtual controller as follows: where k1 is a positive constant; adopting the idea of dynamic surface, the virtual controller passes through a filter to obtain x 2d : Definition The control input of the system is designed to where k i , i = 2, 3 are positive constants; and ξ is the output of the following input-limited compensator where k4 > 0.

5. The preset output tracking control method for a flexible macro-micro drive motion system based on active disturbance rejection according to claim 4, characterized in that: In the said step 3, a control allocation method for allocating the control input u to the actuators v1 and v2; define L = [L1, L2] T , the control allocation problem with actuator saturation is expressed as: where h is a positive constant, denotes v T Mv, M = diag{M1, M2} is a diagonal matrix with positive constants M1 > 0 and M2 > 0; Using the LMI method, solve by solving the following optimal function: where Υ > 0, Υ1 > 0, Υ2 > 0, and

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