A design method for active disturbance rejection feedback controller of ultra-low frequency air-floating vibration isolation system

Abstract

The present invention proposes a method for designing a self-disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system. The method comprises the following steps: establishing the dynamic equations of the ultra-low frequency air-floating vibration isolation system of the present invention; deriving the state-space equations of the vibration isolation system; designing a state observer based on the state space to estimate the total disturbance; expressing the state-space observer as a linearly extended state observer; simplifying the observer-compensated model to a unit-gain dual integrator; and designing a proportional-differential controller for the simplified model. This method does not rely on accurate system parameters because it has a linearly extended state observer, resulting in stronger anti-disturbance capabilities. It is also simple and easy to use, effectively improving the vibration isolation effect of the ultra-low frequency air-floating vibration isolation system.

Description

Technical Field

[0001] The invention relates to a design method of an auto-disturbance rejection feedback controller, and belongs to the technical field of air-floating micro-vibration isolation systems. Background Art

[0002] Traditional feedback controllers are widely used in feedback control of vibration isolation systems because of their simple design, simple implementation, and independence from system models. However, traditional feedback controllers lack flexibility, good robustness, and anti-interference capabilities in design, and require certain engineering experience and a large number of debugging times to achieve the desired effect. LADRC can estimate the total disturbance through an extended state observer, thereby compensating for the total disturbance, and therefore has good anti-interference capabilities and greater flexibility. As a model-independent control algorithm, LADRC inherits the advantages of PID control, has a simple controller structure, and is easy to implement. Existing research on the LADRC method has not been directly used in ultra-low frequency vibration isolation systems, and has not designed observers for external and internal disturbances of the system to compensate for the disturbance, thereby improving the vibration suppression effect and anti-interference capability of the feedback control at the system's natural frequency.

[0003] Therefore, it is urgent to propose a design method for an auto-disturbance rejection feedback controller of an ultra-low frequency air-floating vibration isolation system to solve the above technical problems. Summary of the Invention

[0004] To address the aforementioned issues, a design method for an active disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system is provided. A brief overview of the invention is provided below to provide a basic understanding of certain aspects of the invention. It should be understood that this overview is not an exhaustive overview of the invention. It is not intended to identify key or important aspects of the invention, nor is it intended to limit the scope of the invention.

[0005] The technical solution of the present invention:

[0006] A method for designing an auto-disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system includes the following steps:

[0007] Step 1: Establish the dynamic equation of the ultra-low frequency air flotation vibration isolation system;

[0008] Step 2: Based on the dynamic equation, derive the state space equation of the vibration isolation system;

[0009] Step 3: Design a state space observer for the ultra-low frequency vibration isolation system;

[0010] Step 4: Express the state space observer as a linear extended state observer;

[0011] Step 5: Design a proportional-derivative controller for the model after linear extended state observer compensation.

[0012] Preferred: Based on the knowledge of rigid body dynamics, the dynamic equation of the vibration isolation system is established as follows:

[0013]

[0014] In the expression, M1, C1, and K1 are the mass, damping, and stiffness of the vibration isolation system, respectively; q0 and q1 are the displacements of the base frame and the vibration isolation platform, respectively; and F is the actuator output of the vibration isolation system.

[0015] The dynamic equation of the vibration isolation system is rewritten as follows:

[0016]

[0017] In the expression, a=C1 / M1, b=K1 / M1, u represents the control variable, where f is called generalized disturbance or disturbance because it includes the unknown internal dynamics of the active vibration isolation system. and external disturbance w;

[0018] c is an unknown quantity, and c0 is a known constant set by oneself. Its function is to express all the unknown quantities in the formula with f and hand them over to the state observer for prediction. In this way, only the known constant c0 remains before the control quantity u.

[0019] Preferably, in step 2, the state space equation of the vibration isolation system is written as follows:

[0020]

[0021] In the expression x1=q1, x3=f is added as an augmented state, As an unknown disturbance.

[0022] Preferably, in step 3, a state observer based on state space is used to estimate the total disturbance f, and the state observer is written as follows:

[0023]

[0024] y=Cz

[0025] in

[0026]

[0027] x is the state variable, which contains x1, x2, and x3; A is the system matrix, B is the control matrix, C is the observation matrix, and E is the disturbance matrix; z is the observation vector, specifically x predicted by the observer, and z is the prediction of x.

[0028] Preferably, in step 4, the state space observer in step 3 is expressed as a linear extended state observer, which is constructed as follows:

[0029]

[0030] In the expression, L is the observer gain vector

[0031] T is the symbol of the matrix transpose, w0 is the bandwidth of the observer;

[0032] With a properly designed observer, the controller can be given by:

[0033]

[0034] Ignoring the estimation error in z3, the model can be simplified to a unity-gain double integrator:

[0035]

[0036] The z3 observer is an estimate of x3, which is also an estimate of f. Since the model needs to be simplified into a double integrator with unity gain, u needs to be rewritten in a form without f, leaving only the control quantity. Therefore, u0 is also a control quantity, but it has a conversion relationship with u.

[0037] Preferably: in step 5, the proportional differential controller can be given by the following formula:

[0038] u0=k p (r-z1)-k d z2

[0039] In the expression, r is the reference signal, k d =2ξω c ,ω c and ξ are the closed-loop natural frequency and damping ratio of the desired system. ξ is chosen to avoid any oscillations. The z1 observer is an estimate of x1, and the z2 observer is an estimate of x2.

[0040] Preferred: Design method of linear active disturbance rejection feedback controller for ultra-low frequency air-floating vibration isolation system.

[0041] The present invention has the following beneficial effects:

[0042] 1. The method described in this invention systematically and comprehensively introduces the design method of a linear active disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system. A linear extended state observer for the ultra-low frequency vibration isolation system is designed, which can compensate for disturbances and uncertainties in the system in real time, greatly enhancing the flexibility, robustness, and anti-interference ability of the feedback control algorithm.

[0043] 2. The control method of the present invention is independent of the model, simple in design and easy to implement. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 This is a flow chart of the design method of the automatic disturbance rejection feedback controller of the ultra-low frequency air-floating vibration isolation system;

[0045] Figure 2 is a schematic diagram of the vibration isolation system;

[0046] Figure 3 This is a diagram of an ultra-low frequency air-floating vibration isolation unit;

[0047] Figure 4 It is a low-frequency vibration isolation system test bench.

[0048] In the figure, k is the system stiffness, c is the system damping, 1 is the load of the vibration isolation system, 2 is the ultra-low frequency air-floating vibration isolation unit, 3 is the expansion cylinder of the vibration isolation unit, 4 is the base of the vibration isolation unit, 5 is the basic frame of the vibration isolation system, and 6 is the air-floating guide device of the vibration isolation system. DETAILED DESCRIPTION

[0049] To make the objectives, technical solutions, and advantages of the present invention more clearly apparent, the present invention is described below using specific embodiments shown in the accompanying drawings. However, it should be understood that these descriptions are merely illustrative and are not intended to limit the scope of the present invention. In addition, in the following description, descriptions of well-known structures and technologies are omitted to avoid unnecessary confusion of the concepts of the present invention.

[0050] Specific implementation method 1: Combination Figure 1-4 This embodiment describes a method for designing an active disturbance rejection feedback controller for an ultra-low frequency air floating vibration isolation system, which is applied to a method for designing a linear active disturbance rejection feedback controller for an ultra-low frequency air floating vibration isolation system, and includes the following steps:

[0051] Step 1: Establish the dynamic equation of the ultra-low frequency air flotation vibration isolation system;

[0052] In step 1, the ultra-low frequency air-floating vibration isolation system (abbreviated as the vibration isolation system) is modeled to obtain a mathematical description of its dynamic behavior. This involves using Newton's second law and mass, spring stiffness, and damping coefficients to construct a state-space model of the system, providing a basis for subsequent modeling and analysis.

[0053] According to the knowledge of rigid body dynamics, the dynamic equation of the vibration isolation system is established as follows:

[0054]

[0055] In the expression, M1, C1, and K1 are the mass, damping, and stiffness of the vibration isolation system, respectively; q0 and q1 are the displacements of the base frame (vibration isolation system base frame 5) and the vibration isolation platform (vibration isolation system load 1), respectively; and F is the output of the vibration isolation system actuator (the motor of the ultra-low frequency air floating vibration isolation unit 2).

[0056] The dynamic equation of the vibration isolation system is rewritten as follows:

[0057]

[0058] In the expression, a=C1 / M1, b=K1 / M1, u represents the control variable, where f is called generalized disturbance or disturbance because it includes the unknown internal dynamics of the active vibration isolation system. and external disturbance w;

[0059] Step 2: Based on the dynamic equation, derive the state space equation of the vibration isolation system;

[0060] In step 2, the state space equation of the vibration isolation system can be obtained by converting the dynamic equation into a state space form. This usually includes defining the state variables of the vibration isolation system and substituting them into the original dynamic equation to form a set of linear differential equations, thus converting the physical model into a mathematical model for easy computer processing and simulation.

[0061] The state space equation of the vibration isolation system is written as follows:

[0062]

[0063] In the expression x1=q1, x3=f is added as an augmented state, As an unknown disturbance;

[0064] Step 3: Design a state space observer for the ultra-low frequency vibration isolation system;

[0065] In step 3, for ultra-low frequency vibration isolation systems, an extended state observer (ESO) or linear extended state observer (LESO) can be used. These observers can handle unknown disturbances (disturbances) and parameter uncertainties and provide fast and accurate state estimation. Real-time estimation of system states ensures the accuracy of feedback control.

[0066] The total disturbance f is estimated using a state observer based on state space. The state observer is written as follows:

[0067]

[0068] y=Cz

[0069] in

[0070]

[0071] Step 4: Express the state space observer as a linear extended state observer;

[0072] In step 4, by adjusting parameters and introducing additional observation gains, the above observer can be transformed into a linear extended state observer (LESO), which improves the observation accuracy and robustness and adapts to the needs of complex systems.

[0073] The state space observer in step 3 is expressed as a linear extended state observer, which is constructed as follows:

[0074]

[0075] In the expression, L is the observer gain vector L = [-3ω0 -3ω0 2 -ω0 3 ] T .

[0076] With a properly designed observer, the controller can be given by:

[0077]

[0078] Ignoring the estimation error in z3, the model can be simplified to a unity-gain double integrator:

[0079]

[0080] Step 5: Design a proportional-derivative controller for the model after the linear extended state observer compensation;

[0081] In step 5, after obtaining the output of the linear extended state observer, the PID controller (proportional differential controller) calculates the control signal according to the current error and its rate of change, and adjusts the vibration isolation system behavior to achieve the desired goal based on the estimated state information;

[0082] The proportional-derivative controller can be given by:

[0083] u0=k p (r-z1)-k d z2

[0084] In the expression, r is the reference signal, k p =ω c 2 , k d =2ξω c ,ω c and ξ are the closed-loop natural frequency and damping ratio of the desired system, ξ is chosen to avoid any oscillation;

[0085] This method does not rely on accurate system parameters because it has a linearly extended state observer, which has stronger anti-interference ability. At the same time, this method is simple and easy to use, and effectively improves the vibration isolation effect of the ultra-low frequency air floating vibration isolation system.

[0086] In terms of robust control, model dependence is low: LADRC (Linear Active Disturbance Rejection Control) does not rely on the precise mathematical model of the controlled object. Its extended state observer (LESO) can estimate the disturbance and uncertainty of the system in real time, making the system more adaptable to model errors and unknown disturbances.

[0087] In terms of anti-interference capability, LADRC effectively enhances the system's anti-interference capability by estimating and compensating for disturbances in real time. Compared with traditional robust control methods (such as H∞ control and sliding mode control), LADRC can suppress disturbances over a wider frequency range, significantly improving the robustness of the system.

[0088] Easier to design and adjust: The LADRC controller design is relatively simple, usually requiring only a small number of parameters to be adjusted (such as the bandwidth of the LESO and the bandwidth of the controller), rather than the complex parameter design and optimization required by traditional robust control. Therefore, the present invention is easier to implement and debug.

[0089] In terms of dynamic performance, it has good dynamic performance: LADRC can effectively handle system uncertainties and external disturbances, and excels in achieving fast system response and small overshoot. This makes the present invention outstanding in control scenarios that require fast response and precise tracking.

[0090] In terms of versatility, LADRC is applicable to various types of control systems, including linear systems, nonlinear systems, multiple-input multiple-output (MIMO) systems, etc., and is applied to vibration isolation systems, which makes the present invention highly versatile.

[0091] It should be noted that in the above embodiments, as long as the technical solutions are not contradictory, they can be permuted and combined. Those skilled in the art can exhaust all possibilities based on the mathematical knowledge of permutations and combinations. Therefore, the present invention will no longer describe the technical solutions after permutations and combinations one by one, but it should be understood that the technical solutions after permutations and combinations have been disclosed by the present invention.

[0092] The foregoing description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention are intended to be within the scope of protection of the present invention.

Claims

1. A design method for an active disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system, characterized by: The following steps are involved: Step 1: Establish the dynamic equation of the ultra-low frequency air flotation vibration isolation system; In step 1, the dynamic equation of the vibration isolation system is established based on the knowledge of rigid body dynamics as follows: In the expression 、 、 are the mass, damping and stiffness of the vibration isolation system respectively; 、 are the displacement of the base frame and the displacement of the vibration isolation platform, Provides power to the vibration isolation system actuator; The dynamic equation of the vibration isolation system is rewritten as follows: In the expression , , , , ; Represents the control quantity, here It is called a generalized disturbance or perturbation because it includes the unknown internal dynamics of the active vibration isolation system. and external disturbances ; Step 2: Based on the dynamic equation, derive the state space equation of the vibration isolation system; In step 2, the state space equation of the vibration isolation system is written as follows: In the expression Added as augmented state, As an unknown disturbance; Step 3: Design a state space observer for the ultra-low frequency vibration isolation system; In step 3, the total disturbance is estimated using a state space based state observer , the state observer is written as follows: in Step 4: Express the state space observer as a linear extended state observer; In step 4, the state space observer in step 3 is expressed as a linear extended state observer, which is constructed as follows: In the expression is the observer gain vector ; The controller can be given by: neglect The estimated error in , the model is simplified to a unity gain double integrator: Step 5: Design a proportional-derivative controller for the model after linear extended state observer compensation.

2. The method for designing an active disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system according to claim 1, characterized in that: In step 5, the proportional-derivative controller can be given by the following formula: In the expression is the reference signal, , , and is the closed-loop natural frequency and damping ratio of the desired system, select to avoid any oscillations.

3. The method for designing an active disturbance rejection feedback controller for an ultra-low frequency air-floating vibration isolation system according to any one of claims 1 to 2, characterized in that: Design of a linear active disturbance rejection feedback controller for ultra-low frequency air-floating vibration isolation system.

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