Boundary nested saturation control method for flexible crane system

Abstract

The invention belongs to the technical field of flexible crane system control, and particularly relates to an actuator saturation constraint-oriented control method. According to the technical scheme, the method specifically comprises the following steps: firstly, analyzing the conservative property of a flexible crane system, namely, keeping the energy of the system constant under the condition of no control input; then, based on the characteristic, designing a boundary nested saturation controller which only depends on the position of the trolley and a speed signal, so as to realize effective control and avoid supersaturation of an actuator; thirdly, proving the suitability of the constructed closed-loop system by utilizing a Lipschitz perturbation theory; on the basis, a weak Lyapunov functional is constructed to prove the Lyapunov stability of the closed-loop system; and finally, proving the asymptotic stability of the closed-loop system by further constructing a new Lyapunov functional. According to the method, the blank of boundary control of the flexible crane system under the saturation constraint of the actuator is filled up theoretically, and a control solution which is efficient, feasible, stable in operation, safe and reliable is provided in engineering practice.

Description

Technical Field

[0001] The present invention belongs to the technical field of flexible crane system control, and particularly relates to a control method for actuator saturation constraints. Background Art

[0002] Flexible crane systems are a type of mechanical equipment widely used in industrial production, warehousing logistics, port handling and other scenarios, mainly for object handling and position adjustment, to reduce the intensity of manual labor and significantly improve operation efficiency. Its basic structure is as Figure 1 shown, usually including a top trolley, a flexible rope and a bottom load. Among them, the top trolley is driven by an actuator such as a motor and can move left and right along the guide rail; the heavy object at the bottom is suspended and towed by a flexible steel cable to achieve the handling and precise positioning of the heavy object.

[0003] Due to the introduction of the flexible rope, the system has significant distributed parameter characteristics and can be modeled as a dynamic system coupled with partial differential equations (PDEs) and ordinary differential equations (ODEs). This coupling characteristic makes the system control somewhat challenging, especially in the actual controller design, issues such as the physical limitations of the actuator need to be considered. Especially in industrial applications, to ensure the safety and stability of the equipment, saturation constraints are usually imposed on the controller output, that is, the control input cannot exceed the set maximum value. Otherwise, the out-of-range control signal will be forcibly truncated, thus affecting the system performance and even leading to instability.

[0004] In this context, boundary control technology has become an attractive control means in flexible crane systems due to its advantages such as simple control structure and easy docking with actual measurement signals. Especially when the system can only obtain limited boundary signals (such as trolley position and speed), boundary control is easier to deploy and implement in hardware compared to distributed control. By applying a control input at the boundary of one end of the trolley, not only can the effective regulation of the overall dynamics of the system be achieved, but also the layout of sensors and actuators can be simplified, improving the maintainability and reliability of the system.

[0005] However, existing research has mostly focused on the boundary control design under ideal conditions and has not fully considered the actual problem of actuator saturation limits. In the presence of saturation constraints, traditional linear controllers may lead to a decline in system performance and even cause closed-loop instability.

[0006] Therefore, designing a boundary control method that can take into account saturation limits and system stability has become an important topic in the current research on flexible crane system control. Summary of the Invention

[0007] To solve the problems existing in the prior art, the present invention provides a boundary proportional derivative (PD) nested saturation control method for a flexible crane system, aiming to solve the system stability problem when the controller is limited by saturation during actual operation. Based on the measurable boundary signals of the system, namely the trolley position and speed information, a boundary nested saturation controller combining the proportional and derivative terms of the trolley position and speed signals is designed. Under the condition of only using two boundary measurement information (trolley position and speed), the controller ensures that the control input is always limited within the allowable range, ensuring that the trolley can stay stably at the target position, thereby achieving the asymptotic stable control objective of the closed-loop system.

[0008] The first aspect of the present invention provides a boundary nested saturation control method for a flexible crane system, and the boundary nested saturation control method mainly includes the following steps:

[0009] S1. For the dynamic model of the flexible crane system, construct its energy function and analyze the conservative characteristics of the flexible crane system;

[0010] S2. Based on the above conservative characteristics, design a boundary nested saturation controller combining proportional (P) and derivative (D) terms;

[0011] S3. Under the action of the controller designed in step S2, analyze and prove the well-posedness of the closed-loop system based on the Lipschitz perturbation theory;

[0012] S4. Construct a weak Lyapunov functional to prove the Lyapunov stability of the closed-loop system;

[0013] S5. On the basis of the Lyapunov stability, further construct a new Lyapunov functional to prove the asymptotic stability of the closed-loop system.

[0014] Further, step S1 includes:

[0015] The first step is to model the flexible crane system using Hamilton's principle as follows:

[0016]

[0017] where is the lateral displacement of the rope with length L at height x at time t, y x (x, t) and y t(x, t) are the first-order partial derivatives of the rope with respect to the x variable and the t variable, ρ and M are the mass per unit length of the rope and the mass of the trolley respectively, u(t) is the control force applied at the trolley end to be designed, S(x) = mg + ρg(L - x) > 0 is the tension of the rope at the x position, m is the mass of the load, g is the acceleration due to gravity, and γ0 represents the saturation upper limit of the control input (primary amplitude limit);

[0018] In the second step, consider the following energy function:

[0019]

[0020] The derivative of this energy function with respect to time is expressed as:

[0021]

[0022] where E(t) = E(y, y t , v, w), when the control input u(t) = 0, it indicates that the system is a conservative system.

[0023] Furthermore, step S2 includes: Based on the conservative characteristics of the system, design the following boundary nested saturation controller with proportional (P) and derivative (D) terms:

[0024]

[0025] where k d > 0 and k p > 0 are the controller gains, γ1 is the saturation upper limit (secondary limit amplitude), the proportional (P) term is k p y(0, t), and the derivative (D) term is k d v(t).

[0026] Furthermore, based on the above boundary nested saturation controller, use the Lipschitz perturbation theory to prove the well-posedness of the closed-loop system. Step S3 specifically includes:

[0027] In the first step, based on the above controller, define the following Hilbert space

[0028]

[0029] Then, define the inner product on this Hilbert space as follows:

[0030]

[0031] In the second step, based on the controller, the following closed-loop system can be obtained:

[0032]

[0033] Among them, the operator and the operator

[0034]

[0035] Among them, the operator is defined as follows:

[0036]

[0037] Among them, the operator has the following domain:

[0038]

[0039] In the third step, prove that the operator is an m-dissipative operator, is Lipschitz bounded, and use the Lipschitz perturbation theory to obtain that the closed-loop system is well-posed.

[0040] Furthermore, step S4 includes: constructing a weak Lyapunov functional and proving the Lyapunov stability of the closed-loop system, specifically including:

[0041] In the first step, construct the following Lyapunov functional:

[0042] V(η) = E(η) + U(y(0)),

[0043]

[0044] Among them,

[0045] In the second step, take the derivative of V(η) with respect to time t, and the saturation limit value needs to satisfy γ0 > γ1, and we can get:

[0046]

[0047] From this, it can be obtained that the closed-loop system is Lyapunov stable.

[0048] Furthermore, step S5 includes: on the basis of Lyapunov stability, further construct a new Lyapunov functional and prove the asymptotic stability of the closed-loop system;

[0049] In the first step, construct a new Lyapunov functional as follows:

[0050]

[0051] where \(c\) is a positive constant, \(t\geq0\), and \(s\) is a positive constant;

[0052] In the second step, taking the partial derivative of the new Lyapunov functional with respect to time \(t\), we can obtain

[0053]

[0054] From the above results, it can be seen that the solution trajectory \(\{\eta(t),t\geq0\}\) of the closed-loop system is precompact in the Hilbert space ;

[0055] In the third step, by using the Lasalle invariance principle, it can be obtained that the closed-loop system is asymptotically stable.

[0056] Beneficial effects:

[0057] The boundary nested saturation control method proposed by the present invention fully considers the structural characteristics of the flexible crane system and the saturation constraints of the actuator. The designed controller has a simple structure and only depends on two boundary signals, the position and velocity of the trolley end, which is easy to implement in engineering. By introducing the nested saturation function, it can effectively limit the control input within an acceptable range and ensure the safe operation of the system. This method has a rigorous theoretical basis and can ensure the well-posedness and asymptotic stability of the closed-loop system, and has good practicability and popularization value.

[0058] Compared with the prior art, its significant advantages are as follows:

[0059] (1) Taking into account both system structural characteristics and actual constraint conditions: Aiming at the actuator saturation limitation problem commonly existing in the flexible crane system, the present invention fully considers the dynamic characteristics of the system modeled by PDE-ODE coupling and the physical constraint of the limited control input amplitude, and achieves an effective balance between theoretical analysis and practical feasibility.

[0060] (2) The controller has a simple structure and is easy to implement: The designed boundary controller adopts a PD structure, and the control law only contains the position and velocity signals of the trolley end that are easy to measure, without observing or estimating the internal state of the system, which greatly reduces the complexity of sensor configuration and signal acquisition and is convenient for integration and implementation in industrial scenarios.

[0061] (3) Using nested saturation functions to ensure input limitation: The design of nested saturation functions is introduced into the control law, which can naturally limit the control input within a feasible safety interval, avoid the actuator from being overloaded due to too large controller output, and thus improve the safety and robustness of the system operation. Description of the Drawings

[0062] The drawings are used to provide a further understanding of the present application and constitute a part of the specification. They are used together with the embodiments of the present application to explain the present application and do not constitute a limitation to the present application. In the drawings:

[0063] Figure 1 It is a schematic diagram of the structure of a flexible crane system;

[0064] Figure 2 It is a flowchart of the control method of the present invention;

[0065] Figure 3 Three-dimensional displacement diagram of the crane system;

[0066] Figure 4 Displacement diagram of the trolley;

[0067] Figure 5 Controller Change. Detailed implementation manner

[0068] In order to make the above objects, features and advantages of the present invention more obvious and understandable, the following detailed description of the specific implementation manner of the present invention will be given in conjunction with the embodiments of the specification. Many specific details are set forth in the following description in order to fully understand the present invention, but the present invention can also be implemented in other ways different from those described herein. Those skilled in the art can make similar generalizations without departing from the connotation of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below.

[0069] The present invention proposes a boundary nested saturation control method for a flexible crane system, and the specific idea is as follows:

[0070] Step 1, for the dynamic model of the flexible crane system, construct its energy function and analyze the conservative characteristics of the flexible crane system;

[0071] Furthermore, the flexible crane system is modeled using Hamilton's principle as follows:

[0072]

[0073] In the formula, is the lateral displacement of the rope with length L at height x at time t, y x (x, t) and y t (x, t) are the first-order partial derivatives of the rope with respect to the x variable and the t variable respectively, ρ and M are the mass per unit length of the rope and the mass of the trolley respectively, u(t) is the control force to be designed applied to the trolley end, S(x) = mg + ρg(L - x) > 0 is the tension of the rope at position x, m is the mass of the load, g is the acceleration due to gravity, and γ0 represents the saturation upper limit of the control input (primary amplitude limit).

[0074] First, we consider the following energy function:

[0075]

[0076] The derivative of the energy function with respect to time is expressed as:

[0077]

[0078] where \(E(t)=E(y,y t ,v,w)\). When the control input \(u(t) = 0\), it indicates that the system is a conservative system.

[0079] Step 2: Based on the above conservative characteristics, design a boundary PD nested saturation controller;

[0080] Furthermore, based on the conservative characteristics of the above system, design the following PD boundary nested saturation controller:

[0081]

[0082] where \(k d >0\) and \(k p >0\) are controller gains, \(\gamma_1\) is the saturation upper limit (secondary limit value), the proportional (P) term is \(k p y(0,t)\), and the derivative (D) term is \(k d v(t)\).

[0083] Step 3: Based on the designed controller, use the Lipschitz perturbation theory to prove the well-posedness of the closed-loop system;

[0084] Furthermore, define the following Hilbert space

[0085]

[0086] Then, define the inner product on this Hilbert space as follows:

[0087]

[0088] According to the designed boundary controller (1), the following closed-loop system can be obtained:

[0089]

[0090] where the operator and the operator

[0091]

[0092] where the operator is defined as follows:

[0093]

[0094] Among them, the operator has the following domain:

[0095]

[0096] Prove that the operator is an m-dissipative operator, is Lipschitz bounded, and by using the Lipschitz perturbation theory, it is obtained that the closed-loop system (2) is well-posed.

[0097] Step 4, construct a weak Lyapunov functional to prove the Lyapunov stability of the closed-loop system;

[0098] Furthermore, construct the following weak Lyapunov functional:

[0099]

[0100] Among them,

[0101] Differentiate equation (3) with respect to time t, and the saturation limit value needs to satisfy γ0 > γ1, then we can get:

[0102]

[0103] From this, it can be obtained that the closed-loop system (2) is Lyapunov stable.

[0104] Step 5, on the basis of Lyapunov stability, further construct a new Lyapunov functional to prove the asymptotic stability of the closed-loop system.

[0105] Furthermore, construct a new Lyapunov functional as follows:

[0106]

[0107] Among them, c is a positive constant, t ≥ 0, s is a positive constant. Take the partial derivative of the new Lyapunov functional with respect to time t, and we can get

[0108]

[0109] From the above results, it can be obtained that the solution trajectory {η(t), t ≥ 0} of the closed-loop system (2) is precompact in the Hilbert space Then, by using the Lasalle invariance principle, it can be obtained that the closed-loop system is asymptotically stable.

[0110] Next, the effectiveness of the proposed method is illustrated by specific parameters.

[0111] First, the system parameter values are set as follows:

[0112] The mass of the top trolley is M = 2.1 kg, the length of the flexible rope is L = 1 m, the unit mass of the rope is ρ = 0.2 kg / m, the mass of the bottom load is m = 2.1 kg, and the acceleration due to gravity is g = 9.8 N / kg. The two-stage saturation limit values are γ0 = 0.5 and γ1 = 0.4 respectively, and the initial values of the system are y(x, 0) = 1 m, y t (x, 0) = 0.

[0113] Secondly, the controller parameters are selected as follows:

[0114] k p = 19, k d = 8.

[0115] Figure 3 Shows the three-dimensional displacement of the crane system under the action of the proposed boundary controller (1).

[0116] Figure 4 Presents the change in the position of the trolley under the action of the proposed boundary controller (1).

[0117] Figure 5 Shows under the proposed boundary controller (1), the change of.

[0118] In summary, the present invention discloses a boundary nested saturation control method for a flexible crane system. For a flexible crane system modeled by the coupling of partial differential equations and ordinary differential equations, considering the actual situation of the saturation constraint of the system actuator, a nested saturation boundary controller is designed only using two boundary signals of the measurable trolley end position and speed to ensure that the control input is always limited within a preset range, thereby realizing the asymptotic stability of the closed-loop system. The specific technical solutions of the present invention include the following aspects: First, analyze the conservativeness of the flexible crane system, that is, under the condition of no control input, the energy of the system remains constant; subsequently, based on this characteristic, design a boundary nested saturation controller that only depends on the trolley position and speed signals to achieve effective control and avoid actuator over-saturation; then, use the Lipschitz perturbation theory to prove the well-posedness of the constructed closed-loop system; on this basis, construct a weak Lyapunov functional to prove the Lyapunov stability of the closed-loop system; finally, by further constructing a new Lyapunov functional, prove the asymptotic stability of the closed-loop system. The present invention not only fills the gap in the boundary control of the flexible crane system under the saturation constraint of the actuator theoretically, but also provides an efficient, feasible, stable and safe control solution in engineering practice.

[0119] It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit them. Although the present invention has been described in detail with reference to the preferred embodiments, those of ordinary skill in the art should understand that the technical solutions of the present invention can be modified or equivalently replaced without departing from the spirit and scope of the technical solutions of the present invention, and all of them should be covered within the scope of the present invention.

Claims

1. A boundary nested saturation control method for a flexible crane system, characterized in that, It includes the following steps: S1. For the dynamic model of the flexible crane system, construct its energy function and analyze the conservative characteristics of the flexible crane system; S2. Based on the above conservative characteristics, design a boundary nested saturation controller combining proportional (P) and derivative (D) terms; S3. Under the action of the controller designed in step S2, analyze and prove the well-posedness of the closed-loop system based on the Lipschitz perturbation theory; S4. Construct a weak Lyapunov functional and prove the Lyapunov stability of the closed-loop system; S5. On the basis of Lyapunov stability, further construct a new Lyapunov functional and prove the asymptotic stability of the closed-loop system.

2. The boundary nested saturation control method for a flexible crane system according to claim 1, wherein Step S1 includes: The first step is to model the flexible crane system using Hamilton's principle as follows: wherein, is the lateral displacement of a rope with length L at height x at time t, y x (x, t) and y t (x, t) are the first-order partial derivatives of the rope with respect to the x variable and the t variable respectively, ρ and M are the mass per unit length of the rope and the mass of the trolley respectively, u(t) is the control force applied to the trolley end to be designed, S(x) = mg + ρg(L - x) > 0 is the tension of the rope at position x, m is the mass of the load, g is the acceleration due to gravity, and γ0 represents the saturation upper limit of the control input (primary amplitude limit); The second step is to consider the following energy function: The derivative of this energy function with respect to time is expressed as: Among them, E(t) = E(y, y t , v, w), when the control input u(t) = 0, it indicates that the system is a conservative system.

3. The boundary nested saturation control method for a flexible crane system according to claim 2, characterized in that, Step S2 includes: Based on the conservative characteristics of the system, design the following boundary nested saturation controller with proportional (P) and derivative (D) terms: where k d > 0 and k p > 0 are controller gains, γ1 is the saturation upper limit (two-level limit value), the proportional (P) term is k p y(0, t), and the derivative (D) term is k d v(t).

4. The boundary nested saturation control method for a flexible crane system according to claim 3, characterized in that, Based on the above boundary nested saturation controller, use the Lipschitz perturbation theory to prove the well-posedness of the closed-loop system. Step S3 specifically includes: First step, based on the above-mentioned controller, define the following Hilbert space Then, define the inner product on this Hilbert space as follows: The second step is that based on the controller, the following closed-loop system can be obtained: Among them, the operator and the operator Among them, the operator is defined as follows: Among them, the operator has the following domain of definition: Step 3: Prove that the operator is an m-dissipative operator, is Lipschitz bounded, and use the Lipschitz perturbation theory to obtain that the closed-loop system is well-posed.

5. The boundary nested saturation control method for a flexible crane system according to claim 4, characterized in that Step S4 includes: Construct a weak Lyapunov functional and prove the Lyapunov stability of the closed-loop system, specifically including: The first step is to construct the following Lyapunov functional: V(η) = E(η) + U(y(0)), Among them, The second step is to take the derivative of V(η) with respect to time t, and the saturation limit value needs to satisfy γ0 > γ1, and we can get: It can be obtained therefrom that the closed-loop system is Lyapunov stable.

6. A boundary nested saturation control method for a flexible crane system according to claim 5, characterized in that On the basis of Lyapunov stability, further construct a new Lyapunov functional and prove the asymptotic stability of the closed-loop system; The first step is to construct a new Lyapunov functional as follows: where c is a positive constant, t ≥ 0, and s is a positive constant; The second step is to take the partial derivative of the new Lyapunov functional with respect to time t, and we can get From the above results, the solution trajectory {η(t), t ≥ 0} of the closed-loop system is precompact in the Hilbert space ; The third step is to use LaSalle's invariance principle to obtain that the closed-loop system is asymptotically stable.

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