A boundary control and output regulation method for a flexible gantry crane system

Abstract

The application discloses a boundary control and output regulation method for a flexible gantry crane system. The application comprises the following steps: based on a non-disturbance system, a preliminary state feedback controller is designed by using the boundary state of the system; based on the preliminary controller, the existence of an ISS-Lyapunov function is proved; based on the existence of the ISS-Lyapunov function, an integral regulation term is added to the preliminary controller; a new Lyapunov function is constructed by using a feedforward method; the well-posedness of the obtained closed-loop system is analyzed by using the Lumer-Phillips theorem, and the exponential stability of the closed-loop system is proved; the above analysis is extended to a system containing disturbance, and the output regulation problem is solved; the application can reduce the complexity of implementation while improving the control performance of the system, realizing accurate output regulation and system stability, and better adapting to the requirements of the actual working environment.

Description

TECHNICAL FIELD

[0001] The present application relates to a boundary control and output regulation method for a flexible gantry crane system, and belongs to the technical field of control of flexible gantry crane systems. BACKGROUND

[0002] A flexible gantry crane system is commonly used in modern industry and is a mechanical device for transporting goods, as shown in the structural schematic diagram Figure 1 It generally includes guide rails, top trolleys, flexible steel ropes, and bottom loads. The guide rails are used to support and guide the movement of the top trolleys, and the flexible steel ropes are used to connect between the top trolleys and the bottom loads. The wide application of the flexible gantry crane system has important significance for improving industrial production efficiency and automation level.

[0003] The flexible gantry crane system is a kind of distributed parameter system, and partial differential equations can be used to describe its complex dynamic behavior. The control of the distributed parameter system generally has in-domain control and boundary control. The boundary control is a method of designing a controller on the boundary of the control system, and the control of the system behavior is achieved by exerting a control force on the boundary of the system. This control method is easier to implement in practical applications.

[0004] Output regulation is a method of controlling a flexible gantry crane system. In output regulation, the designed controller can counteract or suppress the effects of these disturbances on the system output to achieve accurate tracking of the reference signal or precise achievement of the target value. This is helpful for achieving the target position of the load, because the position of the load can be achieved by controlling the position of the trolley. In some cases, if the position of the trolley cannot be accurately controlled, it is difficult to achieve the target position of the load. For gantry crane systems where the trolley end and the load end are affected by disturbances, there are some methods in the prior art. Entessari et al. proposed a method using three boundary controllers acting on the trolley end, the flexible rope and the bottom load to achieve accurate arrival of the trolley at the specified position (Reference: Entessari F., Ardekany A.N., Alasty A., "Exponential stabilization of flexural sway vibration of gantry crane via boundary control method," Journal of Vibration and Control 26(1-2) (2020) 36-55. doi: 10.1177 / 1077546319876147). In addition, He et al. also considered the system affected by disturbances at the trolley and load ends, and they used two boundary controllers acting on the trolley and load ends to ensure that the trolley accurately reaches the target position (Reference: Wei He, Shuzhi Sam Ge, "Cooperative control of a nonuniform gantry crane with constrained tension," Automatica, 66, (2016), 146-154. doi: https: / / doi.org / 10.1016 / j.automatica.2015.12.026). However, these methods have some limitations. First, they are relatively complex and difficult to implement, requiring the simultaneous control of multiple boundaries. Second, these methods only consider the system affected by boundary disturbances, and do not consider distributed disturbances, such as wind, which are common disturbance factors in actual systems. SUMMARY

[0005] The present application provides a boundary control and output regulation method for a flexible gantry crane system. The method uses measurable boundary signals in the system to design a boundary controller that only acts on the trolley end, achieving accurate adjustment of the trolley position according to the given value and exponential stability of the system under the influence of disturbances. The technical solution is as follows:

[0006] Step one: obtaining the boundary state of the flexible gantry crane system;

[0007] Step two: designing a preliminary state feedback controller without disturbance based on the boundary state;

[0008] Step three: proving the existence of ISS-Lyapunov function based on the preliminary state feedback controller;

[0009] Step four: adding integral adjustment term to the preliminary state feedback controller based on the existence of ISS-Lyapunov function to obtain a boundary controller;

[0010] Step five: constructing a new Lyapunov function for the boundary controller by using feedforward method;

[0011] Step six: analyzing the well-posedness of the closed-loop system by using Lumer-Phillips theorem and proving the exponential stability of the closed-loop system;

[0012] Step seven: realizing the control and output regulation of the flexible gantry crane system under unknown disturbance by using the stable boundary controller.

[0013] Optionally, the step two includes:

[0014] Step 21: modeling the gantry crane system with disturbance by using Hamilton principle to obtain the master control equation of the gantry crane system:

[0015] ρy tt (x, t) = (S(x)y x (x, t)) x + d(x),

[0016] and boundary conditions:

[0017] My tt (0, t) = u(t) + S(0)y x (0, t) + ω0,

[0018] y x (L, t) = 0,

[0019] wherein, is the transverse 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 x variable and t variable, respectively, p and M are the unit length mass of the rope and the mass of the trolley, respectively, d(x) is a constant unknown disturbance distributed along the whole rope, ω0 is a constant unknown disturbance at the end of the trolley, u(t) is the control force to be designed applied at the end of the trolley, S(x) = mg + pg(L - x) > 0 is the tension of the rope at the x position, m is the mass of the load, and g is the acceleration of gravity;

[0020] Step 22: Based on the master equation and the boundary conditions, considering the unknown disturbances ω0 = 0 and d(x) = 0 in the system, the gantry crane system is equivalent to:

[0021]

[0022] The boundary conditions are:

[0023]

[0024] y x (L, t) = 0.

[0025] Step 23: Introduce a linear operator Map the state variable t y to a new set of state variables (y, y a , ξ), and express as:

[0026]

[0027] where ξ = -k p y'(0) + k a y(0) + x, k p > 0 and k> 0 are adjustable parameters; based on the above coordinate transformation, the boundary conditions become:

[0028] y t (0, t) = ξ(t) + k a y x (0, t) - k p y(0, t),

[0029] y x (L, t) = 0.

[0030] In addition, define two Hilbert spaces:

[0031]

[0032]

[0033] The inner products induced by them are as follows:

[0034]

[0035]

[0036] where, is a bounded linear operator from to , i.e. and there exist two positive real numbers C such that the following inequality holds:

[0037]

[0038] Step 24: Design the preliminary state feedback controller as follows:

[0039] u p (t) = -k1ξ(t) - S(0)y x (0, t) + Mk a y xt (0, t) - Mk p y t (0, t),

[0040] where k1 > 0 is a controller parameter.

[0041] Optionally, the process of proving the existence of the ISS Lyapunov function in step three includes:

[0042] Step 31: Based on the preliminary state feedback controller, the closed-loop system is written in the form of the following operator:

[0043]

[0044] where the domain of the operator is:

[0045] Step 32: Prove the existence of the ISS Lyapunov function.

[0046] Consider the following system:

[0047]

[0048] where v is an external disturbance;

[0049] Step 33: Consider a candidate Lyapunov function:

[0050]

[0051] where is a bilinear form;​

[0052] It is proved that the considered Lyapunov function satisfies the following form:

[0053]

[0054] where λ, μ > 0 are constants,

[0055] Optionally, the step four adds an integral adjustment term in the preliminary controller to obtain the designed boundary controller as follows:

[0056] u(t) = u p (t) - k i η(t),

[0057]

[0058] where k i > 0 is a normal number, and the integral action is used to update the output error.

[0059] Optionally, the step five includes:

[0060] First, a new Hilbert space is defined based on the Hilbert space

[0061]

[0062] Its inner product is defined as:

[0063]

[0064] where and is a linear operator defined as the solution to the following equation:

[0065]

[0066] It has the following form:

[0067]

[0068] where n3∈L 2 (0, L);

[0069] Then, by solving the parameters n i , i = 1... 4, can be obtained as follows:

[0070]

[0071] According to the designed boundary controller, a closed loop system is obtained as follows:

[0072]

[0073] The definition domain of the operator is as follows:

[0074]

[0075] wherein,

[0076] Finally, a new Lyapunov function is calculated to satisfy the following formula:

[0077]

[0078] wherein ε>0.

[0079] Optionally, the step six utilizes the Lumer-Phillips theorem to analyze the well-posedness of the obtained closed loop system, and prove the exponential stability of the closed loop system, including:

[0080] proving the dissipative property of the operator and proving that there is a such that

[0081] Optionally, the step seven considers the output regulation goal and stability of the gantry crane system with unknown disturbance, and designs the controller as follows:

[0082] u(t) = u pn (t) - k i η n (t),

[0083]

[0084] u p n(t) = -k1ξ n (t) - S(0)y x (0, t) + Mk a y xt (0, t) - Mk p y t (0, t),

[0085] ξ n (t) = -k a y x (0, t) + k p (y(0, t) - y ref ) + y t (0, t).

[0086] Wherein, the controller parameters k1>0, k p >0 ， k a >0, k i >0, y ref is a constant reference value of the position y(0, t) of the gantry crane.

[0087] The present application has the following advantages:

[0088] The controller designed by the present application can effectively suppress the adverse effects of disturbances on system performance, realize output regulation and exponential stability of the system, fully considering the characteristics of the system affected by disturbances.

[0089] The controller used in the present application uses a boundary measurable signal to obtain the state quantity of the system, which is obtained in real time by a sensor, etc. In the case of considering the existence of distributed disturbances in the system, only one boundary controller is used to realize the output regulation of the trolley position, which can improve the control performance of the system, thereby better controlling the position of the load, ensuring the safety of the system and improving the transportation efficiency of the system.

[0090] In summary, the present application solves the problem of the gantry crane system being affected by disturbances in actual operation, uses a boundary measurable signal to more accurately perceive the state of the system, and only uses one boundary controller, compared with the prior art, the present application reduces the complexity of implementation while improving the control performance of the system, realizing accurate output regulation and system stability, and can better adapt to the requirements of the actual working environment. BRIEF DESCRIPTION OF DRAWINGS

[0091] In order to more clearly illustrate the technical solutions in the embodiments of the present application, the following will briefly introduce the drawings needed to be used in the embodiment description. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can also be obtained by those skilled in the art without creative labor.

[0092] Figure 1 is a schematic diagram of a bridge crane structure.

[0093] Figure 2 is a flow chart of the control method of the present application.

[0094] Figure 3 is a three-dimensional displacement diagram of the gantry crane system of the second embodiment of the present application.

[0095] Figure 4 is a trolley displacement diagram of the second embodiment of the present application.

[0096] Figure 5is the error graph of the trolley position and the reference position of embodiment two of the present application. DETAILED DESCRIPTION

[0097] To make the objects, technical solutions and advantages of the present application clearer, the embodiments of the present application will be further described in detail below with reference to the drawings.

[0098] Embodiment one:

[0099] The embodiment provides a boundary control and output adjustment method for a flexible gantry crane system, comprising:

[0100] Step one: obtaining the boundary state of the flexible gantry crane system;

[0101] Step two: designing a preliminary state feedback controller without disturbance based on the boundary state;

[0102] Step three: proving the existence of the ISS-Lyapunov function based on the preliminary state feedback controller;

[0103] Step four: adding an integral adjustment term in the preliminary state feedback controller based on the existence of the ISS-Lyapunov function to obtain a boundary controller;

[0104] Step five: adopting a feedforward method to construct a new Lyapunov function for the boundary controller;

[0105] Step six: analyzing the well-posedness of the closed-loop system obtained by using the Lumer-Phillips theorem and proving the exponential stability of the closed-loop system;

[0106] Step seven: realizing the control and output adjustment of the flexible gantry crane system under unknown disturbance by using the stable boundary controller.

[0107] Embodiment two:

[0108] The embodiment provides a boundary control and output adjustment method for a flexible gantry crane system, referring to Figure 1 , and the specific steps are as follows:

[0109] Step 1: designing a preliminary state feedback controller based on the boundary state of the system without disturbance;

[0110] Modeling the gantry crane system containing disturbance by using the Hamilton principle, and obtaining the main control equation of the system as:

[0111] ρy tt (x, t) = (S(x)y x (x, t)) x +d(x),

[0112] The boundary conditions are:

[0113] My tt (0, t) = u(t) + S(0)y x (0, t) + ω0,

[0114] y x (L, t) = 0,

[0115] where, is the transverse displacement of the rope of length L at height x and time t, y x (x, t) and y t (x, t) are the first order partial derivatives of the rope with respect to x and t, respectively, p and M are the unit length mass of the rope and the mass of the trolley, respectively, d(x) is a constant unknown disturbance (e.g. wind) distributed along the entire rope, ω0is a constant unknown disturbance at the end of the trolley, u(t) is the control force to be designed to be applied at the end of the trolley, S(x) = mg + pg(L - x) > 0 is the tension of the rope at position x, m is the mass of the load, and g is the acceleration due to gravity.

[0116] First, this embodiment considers the disturbances ω0= 0 and d(x) = 0 in the system, and the gantry crane system can be equivalently converted into the following system:

[0117]

[0118] The boundary conditions are:

[0119]

[0120] y x (L, t) = 0.

[0121] Now introduce a linear operator which maps the state variable to a new set of state variables (y, y t , ξ) and is represented as:

[0122]

[0123] where ξ = -k a y'(0) + k p y(0) + χ, k a > 0 and k p > 0 are adjustable parameters. Based on the above coordinate transformation, the boundary conditions become:

[0124] y t (0, t) = ξ(t) + k a y x (0, t) - k p y(0, t),

[0125] y x (L, t) = 0.

[0126] Moreover, define two Hilbert spaces:

[0127]

[0128]

[0129] They induce inner products as follows:

[0130]

[0131]

[0132] where,

[0133] The operator is a bounded linear operator from to , i.e. and there exist two positive real numbers C, satisfying the following inequalities:

[0134]

[0135] Further, the preliminary state feedback controller is designed as follows:

[0136] u p (t) = -k1ξ(t) - S(0)y x (0, t) + Mk a y xt (0, t) - Mk p y t (0, t),

[0137] where k1 > 0 is a controller parameter.

[0138] Step 2: Based on the preliminary controller, prove the existence of an ISS (Input-to-state stability) Lyapunov function;

[0139] Based on the preliminary state feedback controller, the closed-loop system can be written in the form of the following operator:

[0140]

[0141] where the domain of the operator is:

[0142] To use the feedforward approach, the existence of an ISS (Input-to-state stability) Lyapunov function is first proved in this embodiment. This embodiment considers the following system:

[0143]

[0144] where, is an external disturbance.

[0145] Further, this embodiment considers a candidate Lyapunov function:

[0146]

[0147] where, is a bilinear form.

[0148] Further, this embodiment proves that the considered Lyapunov function satisfies the following form:

[0149]

[0150] where, λ, μ > 0 are constants,

[0151] Step 3: Based on the existence of the ISS-Lyapunov function, an integral adjustment term is added to the preliminary controller to obtain the designed boundary controller;

[0152] Further, based on the preliminary controller, this embodiment considers a control law with integral action:

[0153] u(t) = u p (t) - k i η(t),

[0154]

[0155] where, k i > 0 is a normal number, and the integral action is used to update the output error.

[0156] Step 4: Using the feedforward approach, a new Lyapunov function is constructed;

[0157] Further, based on the aforementioned Hilbert space a new Hilbert space

[0158]

[0159] Its inner product is defined as:

[0160]

[0161] where and is a linear operator defined as the solution to the following equation:

[0162]

[0163] It has the form:

[0164]

[0165] where n3∈L 2 (0, L).

[0166] Further, solving the parameter n i , i = 1...4, can be obtained as follows:

[0167]

[0168] According to the designed boundary controller, the following closed-loop system can be obtained:

[0169]

[0170] The definition domain of the operator is as follows:

[0171]

[0172] where,

[0173] Further, the embodiment calculates a new Lyapunov function that satisfies the following formula:

[0174]

[0175] where ε > 0.

[0176] Step 5, the obtained closed-loop system is analyzed for well-posedness by using the Lumer-Phillips theorem, and the exponential stability of the closed-loop system is proved;

[0177] Further, the dissipative property of the operator is proved, and it is proved that there exists a such that

[0178] Step 6: extend the above analysis to the system containing disturbance, and solve the output regulation problem.

[0179] Considering the output regulation objective and stability of gantry crane system with unknown disturbance, while considering the position y(0, t) of the gantry crane should be regulated to a given constant reference value y ref , the controller is designed as follows:

[0180] u(t) = u pn (t) - k i η n (t),

[0181]

[0182] u pn (t) = -k1ξ n (t) - S(0)y x (0, t) + Mk a y xt (0, t) - Mk p y t (0, t),

[0183] ξ n (t) = -k a y x (0, t) + k p (y(0, t) - y ref ) + y t (0, t).

[0184] Wherein, the controller parameters k1>0, k p >0, k a >0, k i >0. The analysis method is as described in steps 1-5.

[0185] The effectiveness of the method proposed in this embodiment is explained below in combination with specific parameters.

[0186] First, the system parameter values are set as follows:

[0187] The top trolley mass is M = 2.1 kg, the flexible rope length is L = 1 m, the unit mass of the rope is p = 0.2 kg / m, the bottom load mass is m = 2.1 kg, the gravitational acceleration is g = 9.8 N / kg. The distributed disturbance d(x) = 0.5x, the trolley end constant disturbance w0 = 0.1, the reference value of the output y(0, t) is y ref = 0.5m, the initial value of the system is y(x, 0) = 0.1m, y t (x, 0) = 0.

[0188] Second, the controller parameters are selected as follows:

[0189] k1 = 100, k a = 60, k p= 2, k i = 100.

[0190] Figure 3 The three-dimensional displacement of the gantry crane system under the proposed boundary control is shown, and it can be observed from the figure that the displacement of the system gradually approaches the predetermined position and remains near the position.

[0191] Figure 4 The change of the trolley position under the proposed boundary control is presented, and it can be seen from the figure that the trolley is gradually affected by the boundary controller, and its position gradually approaches the target position over time.

[0192] Figure 5 The error between the trolley position and the reference position under the proposed boundary control is shown, and it can be seen from the figure that the error between the trolley position and the reference position gradually decreases and approaches zero.

[0193] In summary, the proposed boundary control method has significant advantages in the gantry crane system. This method can achieve the stability of the system, accurate output regulation, and maintain the position error close to zero. This makes the system have better robustness and control performance in the actual working environment, and improves the stability and transportation efficiency of the system.

[0194] Some steps in the embodiments of the present application can be implemented by software, and the corresponding software program can be stored in a readable storage medium, such as an optical disc or a hard disk.

[0195] The above description is only the preferred embodiment of the present application, and is not intended to limit the present application. Any modification, equivalent replacement, improvement, etc. made within the spirit and principles of the present application shall be included in the protection scope of the present application.

Claims

1. A boundary control and output adjustment method for a flexible gantry crane system, characterized in that, The method includes: Step 1: Obtain the boundary state of the flexible gantry crane system; Step 2: Design a preliminary state feedback controller under undisturbed conditions using the boundary states; Step 3: Based on the preliminary state feedback controller, prove the existence of the ISS-Lyapunov function; Step 4: Based on the existence of the ISS-Lyapunov function, add an integral adjustment term to the initial state feedback controller to obtain the boundary controller; Step 5: Using the feedforward method, construct a new Lyapunov function for the boundary controller; Step 6: Use the Lumer–Phillips theorem to analyze the well-posedness of the closed-loop system and prove the exponential stability of the closed-loop system; Step 7: Utilize a stable boundary controller to control and adjust the output of the flexible gantry crane system under unknown disturbances; Step two, the process of designing the preliminary state feedback controller, includes: Step 21: Model the gantry crane system containing disturbances using Hamilton's principle to obtain the main control equations of the gantry crane system: and boundary conditions: In the formula, It is a length of The rope at a height of Place, Lateral displacement at time t, and The rope is relative to variables and The first partial derivative of the variable, and These are the mass per unit length of the rope and the mass of the cart, respectively. For constant unknown perturbations distributed throughout the entire rope, It is a constant unknown disturbance at the vehicle end. The control force to be applied to the end of the vehicle is to be designed. It's the rope that's there. Tension at the location, It is the mass of the load. It is the acceleration due to gravity; Step 22: Based on the master equation and boundary conditions, consider the unknown disturbances in the system. and The gantry crane system is equivalently converted to: The boundary conditions are: Step 23: Introduce a linear operator , will state variables Mapped to a new set of state variables And expressed as: in, , and These are adjustable parameters; based on the above coordinate transformation, the boundary conditions become: In addition, two Hilbert spaces are defined: The inner products they induce are as follows: in, Operator From arrive Bounded linear operators, i.e. And there exist two positive real numbers. The following inequalities are satisfied: Step 24: Design the preliminary state feedback controller as follows: in, These are controller parameters.

2. The boundary control and output adjustment method for a flexible gantry crane system according to claim 1, characterized in that, The process of proving the existence of the ISS Lyapunov function in step three includes: Step 31: Based on the preliminary state feedback controller, the closed-loop system is written in the form of the following operator: Among them, the operator The domain is: ; Step 32: Prove the existence of the ISS Lyapunov function; Consider the following system: in, It is an external disturbance; Step 33: Consider a candidate Lyapunov function: in, It is a bilinear form; Prove that the Lyapunov function under consideration satisfies the following form: in, It is a constant. .

3. The boundary control and output adjustment method for a flexible gantry crane system according to claim 2, characterized in that, In step four, an integral adjustment term is added to the preliminary controller to obtain the designed boundary controller as follows: in, It is a positive constant, and its integral action is... Used to update output errors.

4. The boundary control and output adjustment method for a flexible gantry crane system according to claim 3, characterized in that, Step five includes: First, based on Hilbert space Define a new Hilbert space : Its inner product is defined as: in and It is a linear operator, defined as a solution to the following equation: Its form is as follows: in ; Then, solve Parameters ,as follows: Based on the designed boundary controller, the following closed-loop system is obtained: Operator The domain is as follows: in, ; Finally, the new Lyapunov function is calculated to satisfy the following equation: in .

5. The boundary control and output adjustment method for a flexible gantry crane system according to claim 4, characterized in that, The well-posedness of the closed-loop system obtained in step six, using the Lumer-Phillips theorem analysis, is proven to be exponentially stable, including: Proof Operator The dissipative properties and prove the existence of a , making .

6. The boundary control and output adjustment method for a flexible gantry crane system according to claim 5, characterized in that, Step seven considers the output regulation target and stability of the gantry crane system with unknown disturbances, and designs the controller as follows: Among them, controller parameters , Location of the gantry crane A constant reference value.

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