An optimal controller design method for ultra-low frequency vibration isolation system

Abstract

The present invention proposes an optimal controller design method for an ultra-low frequency vibration isolation system, the method comprising the following steps: establishing a dynamic model of the ultra-low frequency vibration isolation system in the present invention, deriving the state space equation of the ultra-low frequency vibration isolation system; allocating weight matrices Q and R according to the objective function of the optimal quadratic algorithm and the control target of the vibration isolation system; deriving the control force model of the optimal controller according to the weight matrix Q and R and in combination with the state space equation of the vibration isolation system, substituting the control force into the control loop to obtain a complete optimal controller and obtaining a better control effect by optimizing the weight allocation of the Q and R matrices. Thus, the active control performance of the ultra-low frequency vibration isolation system is improved. The method is low-cost, simple and easy to use, and effectively improves the vibration isolation performance of the ultra-low frequency vibration isolation system.

Description

Technical Field

[0001] The invention relates to the technical field of vibration isolation systems, and in particular to an optimal controller design method for an ultra-low frequency vibration isolation system. Background Art

[0002] Vibration isolation technology is a key supporting technology for large-scale ultra-precision processing, manufacturing and measurement equipment, and is also one of the iconic technologies for a country's advanced equipment processing, manufacturing and measurement capabilities. At present, the research and development of high-end precision active vibration isolation systems in my country is still blank. There is no domestic commercial high-performance vibration isolation system, and it mainly relies on imports, and the channels are very limited.

[0003] In ultra-low frequency vibration isolation systems, it is difficult to find the global optimal solution using traditional active control methods. Therefore, an optimal controller design method for ultra-low frequency vibration isolation systems is designed. The optimal control algorithm allows the problem to be solved to be nonlinear and discontinuous, and can find the global optimal solution and suboptimal solution from the entire feasible solution space, avoiding only obtaining the local optimal solution, and effectively improving the vibration isolation performance of the ultra-low frequency vibration isolation system. Summary of the invention

[0004] The purpose of the present invention is to solve the problems in the prior art and to provide an optimal controller design method for an ultra-low frequency vibration isolation system.

[0005] The present invention is implemented by the following technical scheme. The present invention proposes an optimal controller design method for an ultra-low frequency vibration isolation system, and the method comprises the following steps:

[0006] Step 1: Establish a dynamic model of the ultra-low frequency vibration isolation system;

[0007] Step 2: derive the state space equation of the vibration isolation system according to the ultra-low frequency vibration isolation system dynamics model established in step 1;

[0008] Step 3, assigning weight matrices Q and R according to the objective function of the optimal quadratic algorithm and the control objective of the vibration isolation system;

[0009] Step 4: Derivation of the control force model of the optimal controller based on the weight matrices Q and R obtained in step 3 and the state space equation of the vibration isolation system in step 2;

[0010] Step 5: According to the control force model obtained in step 4, the control force is substituted into the control loop to obtain a complete optimal controller and a better control effect is obtained by optimizing the weight distribution of the Q and R matrices.

[0011] Furthermore, in the step 1, according to the force condition of the vibration isolation system, as well as the stiffness and damping of the vibration isolation system, the dynamic equation is listed to obtain the dynamic model;

[0012] The kinetic model formula is as follows:

[0013]

[0014] In the model, m s is the load mass of the isolator, m w is the quality of the basic framework, k s and c s are the stiffness and damping of the vibration isolation system, k t and c t are the stiffness and damping of the base frame, respectively, x s and x w are the isolator load and the displacement of the base frame, u is the active control force, x g Input for the vibration isolation foundation displacement.

[0015] Furthermore, in the step 2, the state variables of the state equation are respectively selected as the displacement and velocity of the load of the vibration isolation system, the input is the external disturbance and the control output of the controller, and the output is selected as the velocity of the load of the vibration isolation system;

[0016] The state space equation is as follows:

[0017]

[0018] y=Cx+Du+GW

[0019] in:

[0020]

[0021]

[0022] Furthermore, in step three, the weight matrices Q and R are associated with the control speed and energy consumption of the system respectively, the Q matrix is ​​allocated according to the degree of concern for the state variables, and the size of the R matrix is ​​determined by the degree of concern for the control energy consumption.

[0023] Furthermore, the objective function of the optimal quadratic algorithm is as follows:

[0024]

[0025] Where Q = C T Q y C is a non-negative definite symmetric matrix, R = D T Q y D+R y is a positive definite gain matrix, N = C T Q y D is a positive definite gain matrix.

[0026] Furthermore, in the step 4, the optimal controller gain matrix K is derived according to the weight matrix Q, R and the state space equation of the system, thereby obtaining a control force model;

[0027] The control force model of the optimal controller is as follows:

[0028] u(t)=Kx(t);

[0029] in:

[0030] u(t)=Kx(t)=-R -1 (B T P+N T )x(t)

[0031] P is the solution of the Riccati equation:

[0032] PA+A T P-(PB+N)R -1 (B T P+N T )+Q=0.

[0033] Furthermore, in step five, the gain matrix K obtained in step four is brought into the feedback loop to form an optimal controller, and the weight distribution of the Q and R matrices is optimized according to the control effect to obtain the optimized gain matrix K, thereby obtaining a better control effect.

[0034] The present invention has the following beneficial effects:

[0035] 1. The method described in the present invention systematically and comprehensively introduces the optimal controller design method of the ultra-low frequency vibration isolation system, establishes the dynamic model of the vibration isolation system, and derives the state space equation of the vibration isolation system and the control force model of the optimal controller.

[0036] 2. The control method of the present invention allows the problem to be solved to be nonlinear and discontinuous, and can find the global optimal solution and suboptimal solution from the entire feasible solution space, avoiding only obtaining the local optimal solution, and effectively improving the vibration isolation performance of the ultra-low frequency vibration isolation system. BRIEF DESCRIPTION OF THE DRAWINGS

[0037] Figure 1 It is a structural schematic diagram of the ultra-low frequency vibration isolation system of the present invention.

[0038] Figure 2 Design flow chart of the optimal controller. DETAILED DESCRIPTION

[0039] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0040] See also Figure 1 to Figure 2 , the structure of ultra-low frequency vibration isolation system is as follows Figure 1 As shown, based on the ultra-low frequency vibration isolation system, the present invention proposes an optimal controller design method for the ultra-low frequency vibration isolation system, and the method comprises the following steps:

[0041] Step 1: Establish a dynamic model of the ultra-low frequency vibration isolation system;

[0042] Step 2: derive the state space equation of the vibration isolation system according to the ultra-low frequency vibration isolation system dynamics model established in step 1;

[0043] Step 3: assign weight matrices Q and R according to the objective function of the optimal quadratic algorithm and the control objective of the vibration isolation system;

[0044] Step 4: Derivation of the control force model of the optimal controller based on the weight matrix Q, R obtained in step 3 and the state space equation of the vibration isolation system in step 2;

[0045] Step 5: According to the control force model obtained in step 4, the control force is substituted into the control loop to obtain a complete optimal controller and a better control effect is obtained by optimizing the weight distribution of the Q and R matrices.

[0046] In the step 1, according to the stress condition of the vibration isolation system, as well as the stiffness and damping of the vibration isolation system, the dynamic equations are listed to obtain the dynamic model.

[0047] The kinetic model formula is as follows:

[0048]

[0049] In the model, m s is the load mass of the isolator, m w is the quality of the basic framework, k s and c s are the stiffness and damping of the vibration isolation system, k t and c t are the stiffness and damping of the base frame, respectively, x s and x w are the isolator load and the displacement of the base frame, u is the active control force, x g Input for the vibration isolation foundation displacement.

[0050] In step 2, the state space equation of the vibration isolation system is derived according to the dynamic model obtained in step 1. The state variables of the state equation are respectively selected as the displacement and velocity of the vibration isolation system load, the input is the external disturbance and the control output of the controller, and the output is selected as the velocity of the vibration isolation system load.

[0051] Furthermore, the state space equation is as follows:

[0052]

[0053] y=Cx+Du+GW

[0054] in:

[0055]

[0056]

[0057] In the step three, weight matrices Q and R are allocated according to the objective function of the optimal quadratic algorithm and the control objective of the vibration isolation system; the weight matrices Q and R are respectively associated with the control speed and energy consumption of the system, the Q matrix is ​​allocated according to the degree of concern for the state variables, and the size of the R matrix is ​​determined by the degree of concern for the control energy consumption.

[0058] Furthermore, the objective function of the optimal quadratic algorithm is as follows:

[0059]

[0060] Where Q = C T Q y C is a non-negative definite symmetric matrix, R = D T Q y D+R y is a positive definite gain matrix, N = C T Q y D is a positive definite gain matrix. Because the ultra-low frequency vibration isolation system is more concerned with the vibration velocity of the vibration isolation system load and the control speed of the entire vibration isolation system, the vibration isolation load velocity distribution ratio in the Q matrix should be as large as possible to achieve better vibration isolation effect.

[0061] In the step 4, the control force model of the optimal controller is derived based on the weight matrix Q, R obtained in step 3 and the state space equation of the vibration isolation system in step 2; specifically: the optimal controller gain matrix K is derived based on the weight matrix Q, R and the state space equation of the system, thereby obtaining the control force model.

[0062] Furthermore, the control force model of the optimal controller is as follows:

[0063] u(t)=Kx(t);

[0064] in:

[0065] u(t)=Kx(t)=-R -1 (B T P+N T )x(t)

[0066] P is the solution of the Riccati equation:

[0067] PA+A T P-(PB+N)R -1 (B T P+N T )+Q=0.

[0068] In the step 5, the control force model obtained in step 4 is used to substitute the control force into the control loop to obtain a complete optimal controller and obtain a better control effect by optimizing the weight distribution of the Q and R matrices. Specifically, the gain matrix K obtained in step 4 is brought into the feedback loop to form an optimal controller, and the weight distribution of the Q and R matrices is optimized according to the control effect to obtain an optimized gain matrix K, thereby obtaining a better control effect. In the simulation software, the linear quadratic optimal function can be directly called to calculate the gain matrix.

[0069] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the aforementioned embodiments, it is still possible for a person skilled in the art to modify the technical solutions described in the aforementioned embodiments, or to replace some of the technical features therein by equivalents. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. An optimal controller design method for an ultra-low frequency vibration isolation system, characterized in that: The method comprises the following steps: Step 1: Establish a dynamic model of the ultra-low frequency vibration isolation system; Step 2: derive the state space equation of the vibration isolation system according to the ultra-low frequency vibration isolation system dynamics model established in step 1; Step 3, assigning weight matrices Q and R according to the objective function of the optimal quadratic algorithm and the control objective of the vibration isolation system; Step 4: Derivation of the control force model of the optimal controller based on the weight matrices Q and R obtained in step 3 and the state space equation of the vibration isolation system in step 2; Step 5: According to the control force model obtained in step 4, the control force is substituted into the control loop to obtain a complete optimal controller and a better control effect is obtained by optimizing the weight distribution of the Q and R matrices; In the step 1, according to the stress condition of the vibration isolation system, as well as the stiffness and damping of the vibration isolation system, the dynamic equation is listed to obtain the dynamic model; The kinetic model formula is as follows: In the model, m s is the load mass of the vibration isolation system, m w is the quality of the basic framework, k s and c s are the stiffness and damping of the vibration isolation system, k t and c t are the stiffness and damping of the base frame, respectively, x s and x w are the load of the vibration isolation system and the displacement of the base frame, u is the control force of the controller, x g is the displacement input of the vibration isolation foundation; In the step 2, the state variables of the state space equation are respectively selected as the displacement and velocity of the vibration isolation system load, the input is the external disturbance and the control force of the controller, and the output is selected as the velocity of the vibration isolation system load; The state space equation is as follows: y=Cx+Du+GW in: In the step 3, the weight matrices Q and R are associated with the control speed and energy consumption of the system respectively, the Q matrix is ​​allocated according to the degree of concern for the state variables, and the size of the R matrix is ​​determined by the degree of concern for the control energy consumption; The objective function of the optimal quadratic algorithm is as follows: Where Q = C T Q y C is a non-negative definite symmetric matrix, R = D T Q y D+R y is a positive definite gain matrix, N = C T Q y D is the positive definite gain matrix; In the step 4, the optimal controller gain matrix K is derived according to the weight matrix Q, R and the state space equation of the system, thereby obtaining a control force model; The control force model of the optimal controller is as follows: u(t)=Kx(t); in: u(t)=Kx(t)=-R -1 (B T P+N T )x(t) P is the solution of the Riccati equation: PA+A T P-(PB+N)R -1 (B T P+N T )+Q=0; In the step 5, the gain matrix K obtained in the step 4 is brought into the feedback loop to form an optimal controller, and the weight distribution of the Q and R matrices is optimized according to the control effect to obtain the optimized gain matrix K, thereby obtaining a better control effect.

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