Belt conveyor control method based on generalized UK theory

Abstract

The invention provides a belt conveyor control method based on a generalized UK theory, and relates to the technical field of belt conveyor control, and the method specifically comprises the following steps: building a belt conveyor system dynamic model considering equality and inequality constraints at the same time; state conversion is carried out based on the differential homeomorphic theory, and the original state of the system speed of the belt conveyor is converted into a new state by limiting an integral difference value; taking the speed of a control target as a constraint, and providing a robust control design in combination with a kinetic model; and carrying out simulation on the belt conveyor system, and carrying out robust control verification. According to the technical scheme, the problem that a conveyor system cannot be controlled to be stably started in the prior art is solved.

Description

Technical Field

[0001] The invention relates to the technical field of belt conveyor control, and in particular to a belt conveyor control method based on generalized UK theory. Background Art

[0002] Due to its structural characteristics, long-distance belt conveyors have many uncertainties in engineering practice, resulting in large fluctuations in conveyor belt speed and tension. During the entire control process, the startup phase is more difficult to control, and inappropriate driving force may cause accidents such as belt overlap and belt breakage. Therefore, accurate control of the overall speed is necessary for the safety of the conveyor system. To avoid accidents, the conveyor system needs to be divided into different subsystems, and the speed difference between adjacent subsystems must be limited to a certain range. The speed difference is reflected by the speed error, and the difference range is controlled by the inequality constraint, thereby realizing the startup speed constraint control of the long-distance belt conveyor.

[0003] Therefore, there is a need for a belt conveyor control method based on generalized UK theory that can control the stable startup of the conveyor system. Summary of the invention

[0004] The main purpose of the present invention is to provide a belt conveyor control method based on the generalized UK theory to solve the problem that the conveyor system cannot be controlled to start stably in the prior art.

[0005] To achieve the above object, the present invention provides a belt conveyor control method based on generalized UK theory, which specifically comprises the following steps: S1, establish a belt conveyor system dynamics model that considers both equality and inequality constraints.

[0006] S2, based on the differential homeomorphism theory, the state transformation is carried out, and the original speed state of the belt conveyor system is transformed into a new state by limiting the integral difference.

[0007] S3, considers the speed of the control target as a constraint and proposes a robust control design in combination with the dynamic model.

[0008] S4, the belt conveyor system is simulated for robust control verification.

[0009] Furthermore, step S1 specifically includes the following steps: S1.1, the dynamic model of the belt conveyor system with equality constraints and inequality constraints is: (1); in, represents the inertia matrix, the dimension of the inertia matrix is , is an integer, is the belt acceleration, Indicates inherent force, and represent the external forces based on equality constraints and inequality constraints, respectively.

[0010] S1.2, consider the equality constraint, the first-order form of the equality constraint is expressed as: (2); in, represents the constraint matrix, the dimension of the constraint matrix is , is an integer and has , is a first-order constraint matrix.

[0011] For equation (2), find the The derivative of , gives the second-order form of the equality constraint: (3); in, is the second-order constraint matrix, is the generalized coordinate of the belt, Indicates belt speed.

[0012] S1.3, according to the Udwadia-Kalaba method, the external force based on the equality constraint is obtained : (4); in, represents the Moore–Penrose generalized inverse.

[0013] S1.4, consider the inequality constraints: (5); in, express The lower bound of express The upper bound of for No. Quantity, and .

[0014] S1.5, Applied forces based on inequality constraints Designed for: (6); in, represents the identity matrix, The value is to be determined.

[0015] Furthermore, step S2 specifically includes the following steps: S2.1, in order to handle inequality constraints, Transform via diffeomorphism: (7); in, is the coordinate of the belt conveyor belt subsystem after transformation, is a diffeomorphism in the form of a tangent function, Indicates continuous The order is differentiable.

[0016] S2.2, Extension and , write equation (7) in matrix form: (8); in, and ; and The expanded and , integrate and differentiate equation (8) respectively, and we get: (9); (10); in, Represents the integration constant.

[0017] Consider the estimated value function As the original function The estimated value of , assuming that the estimated value function Features include: (11); in, Representation and The time-varying function.

[0018] Consider a bounded time-varying function , so that: (12).

[0019] S2.3, Dynamics of belt conveyor system considering the following uncertainties: (13); in, Indicates uncertain parameters, is a compact set, indicating uncertain parameters Area, is the inertia matrix, is the centrifugal force matrix, is the gravity matrix.

[0020] Transforming the state of equation (13), we get: (14); in, express The estimated value of It is the control force of the belt conveyor system after the state transformation. for The derivative of .

[0021] Furthermore, step S2 further includes the following steps: S2.4, in order to simplify formula (14), let , , ; Then the transformed belt conveyor system equation is: (15); Thus, the state equation is obtained: (16).

[0022] S2.5, let express The nominal part of ,function is a continuous function; make: , , ; then: (17).

[0023] The unique solution to the Riccati equation is , ,make: (18).

[0024] There is a constant So that for all Both have: (19).

[0025] S2.6, consider any function , there exists a vector and a function , so that for all , have: (20); in, is a constant vector, the function A framework for uncertainty boundaries.

[0026] Furthermore, step S3 specifically includes the following steps: S3.1, consider the belt conveyor system control force as: (twenty one); in, represents the controller without considering uncertainty, Represents a term in the controller that compensates for inconsistencies in the initial conditions of the system.

[0027] make: (twenty two); in, is a scalar design parameter; to simplify the formula, let , then let: (twenty three); in, is a scalar design parameter, is a real number.

[0028] S3.2, Consider controls on estimates: (twenty four); Consider a bounded function ,and , so that: (25); in, , express and The maximum value of the difference norm.

[0029] S3.3, consider the functional differential equation In Control Below, among them Representatives The solution at time , assuming there is a continuous differential function And four continuous monotonically increasing functions , , and With attributes , represents the independent variable, when hour , and when hour , so that: (26); For functional differential equations have: (27); in, represents the Hamiltonian operator, , , that is, the solution of the belt conveyor system Satisfies uniform boundedness and eventually uniform boundedness, is the Lyapunov function.

[0030] Furthermore, step S3 further includes the following steps: S3.4, select the Lyapunov function as: (28); in, is the unique solution to the Riccati equation, , ; will (28) Taking the first-order derivative, we get: (29).

[0031] S3.5, in order to simplify formula (29), from formula (17) and (21) and let as well as , for all have: (30); in, Design parameters for a scalar: (31).

[0032] According to formula (26), the uniform boundedness formula is: (32); (33);

[0033] in, represents a uniformly bounded boundary, is a parameter reflecting the boundary state. represents the convergence radius, , , and Respectively The minimum and maximum eigenvalues ​​of , express The maximum value of express The minimum value of .

[0034] The eventual uniform boundedness is as follows: (34); (35).

[0035] in, represents the eventually uniformly bounded boundary; to simplify formula (35), let: (36);

[0036] in, represents the smallest positive integer, Represents a time-varying function The maximum value of (37); in, (38); That is, the belt conveyor system represented by formula (15) has uniform boundedness and uniform ultimate boundedness.

[0037] The present invention has the following beneficial effects: The present invention designs an effective controller to handle system inequality constraints and compensate for uncertainties, and proposes a class of robust constraint control based on Lyapunov stability analysis. In the invention, a robust control based on generalized Udwadia-Kalaba theory and differential homeomorphism theory is proposed to handle inequality constraints, control the nominal system and compensate for uncertainties, and finally proves that the belt conveyor system is uniformly bounded and uniformly ultimately bounded.

[0038] The present invention proposes a method to solve the modeling problem by transforming the system state using the differential homeomorphism method, and transforms the system inequality constraint processing problem into the system stability problem. The present invention establishes a speed inequality constraint processing problem based on the generalized Udwadia-Kalaba (GUK) theory, and achieves the purpose of controlling the stable startup of the conveyor system by designing robust control of the transformed system. BRIEF DESCRIPTION OF THE DRAWINGS

[0039] In order to more clearly illustrate the specific implementation of the present invention or the technical solutions in the prior art, the following will briefly introduce the drawings required for the specific implementation or the prior art description. Obviously, the drawings described below are some implementations of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without creative work. In the drawings: Figure 1 A flow chart of a belt conveyor control method based on generalized UK theory of the present invention is shown.

[0040] Figure 2 A schematic diagram of the structure of a belt conveyor system model in an embodiment of the present invention is shown.

[0041] Figure 3 A schematic diagram of the robust control verification comparison of the present invention is shown.

[0042] Figure 4 Shows Figure 3 A magnified detail of point A.

[0043] Figure 5 Shows Figure 3 Enlarged detail of point B.

[0044] Figure 6 A schematic diagram of cumulative error verification comparison of the present invention is shown. DETAILED DESCRIPTION

[0045] The technical solution of the present invention will be described clearly and completely below in conjunction with the accompanying drawings. 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.

[0046] A belt conveyor control method based on generalized UK theory specifically includes the following steps: S1, establish a belt conveyor system dynamics model that considers both equality and inequality constraints.

[0047] S2, based on the differential homeomorphism theory, the state transformation is carried out, and the original speed state of the belt conveyor system is transformed into a new state by limiting the integral difference.

[0048] S3, considers the speed of the control target as a constraint and proposes a robust control design in combination with the dynamic model.

[0049] S4, the belt conveyor system is simulated for robust control verification.

[0050] like Figure 1As shown, the present invention first establishes a system dynamics model that contains both equality and inequality constraints, then performs differential homeomorphism transformation on the belt conveyor system speed, and transforms the original state of the belt conveyor system speed into a new state by limiting the integral difference, regards the control target speed as a constraint, and proposes a control design in combination with the dynamic equation. The belt conveyor system is simulated and robust control verification is performed until the control requirements are met and the control parameters are output.

[0051] Specifically, step S1 includes the following steps:

[0052] S1.1, the dynamic model of the belt conveyor system with equality constraints and inequality constraints is: (1); in, represents the inertia matrix, the dimension of the inertia matrix is , is an integer, is the belt acceleration, Indicates inherent force, and represent the external forces based on equality constraints and inequality constraints, respectively.

[0053] S1.2, consider the equality constraint, the first-order form of the equality constraint is expressed as: (2); in, Represents the constraint matrix, the dimension of the constraint matrix is , is an integer and has , is a first-order constraint matrix.

[0054] For equation (2), find the The derivative of , gives the second-order form of the equality constraint: (3); in, is the second-order constraint matrix, is the generalized coordinate of the belt, Indicates belt speed.

[0055] S1.3, according to the Udwadia-Kalaba method, the external force based on the equality constraint is obtained : (4); in, represents the Moore–Penrose generalized inverse.

[0056] S1.4, consider the inequality constraints: (5); in, express The lower bound of express The upper bound of for No. Quantity, and .

[0057] S1.5, Applied forces based on inequality constraints Designed for: (6); in, represents the identity matrix, The value is to be determined.

[0058] Given (6) In the form of, the acceleration of the belt conveyor system still meets all This shows that the inequality constraints Does not affect equality constraints . Combining equations (1), (4) and (6) yields: (7); in represents the belt acceleration based on the equality constraint, represents the belt acceleration based on the equality constraint.

[0059] for have, (8); for have, (9).

[0060] Considering equations (8) and (9), we can see (10); According to formula (10), and The values ​​of Therefore, it is feasible to consider both equality constraints and inequality constraints in the system at the same time.

[0061] Specifically, step S2 includes the following steps: S2.1, suppose there exists a function ,in , , Represents the original state of a mechanical system. Represents the new state quantity after the state of the mechanical system is transformed. express dimensional real number set, express dimensional set of real numbers. If it is smooth, it is called a diffeomorphism, and its inverse function exists and is also smooth. To handle inequality constraints, Transform via diffeomorphism: (11); in, is the coordinate of the belt conveyor belt subsystem after transformation, is a diffeomorphism in the form of a tangent function, Indicates continuous The order is differentiable.

[0062] The state transition function needs to meet the following conditions: (i) The domain of the transformation function should be , its value range should be ; (ii) The transformation function is increasing and continuous in the domain.

[0063] From the above conditions, it can be concluded that Sometimes, there is If the conversion status If the belt conveyor belt subsystem speed is bounded, then the original state Then the inequality constraint can be satisfied . Thus, the inequality constraint problem is transformed into a system stability problem.

[0064] S2.2, Extension and , write equation (11) in matrix form: (12); in, and ; and The expanded and , integrate and differentiate equation (12) to obtain: (13); (14); in, Represents the integration constant.

[0065] In the calculation, about Primitive function of the integral It is difficult to obtain. Therefore, we consider the estimated value function As the original function The estimated value of , assuming that the estimated value function Features include: (15); in, Representation and The time-varying function.

[0066] Consider a bounded time-varying function , so that: (16); Since the state is continuous, no matter the original function Unknown or known, What is the value of the original state? With estimated state value The distances between them are bounded.

[0067] S2.3, Dynamics of belt conveyor system considering the following uncertainties: (17); in, Indicates uncertain parameters, is a compact set, indicating uncertain parameters Area, is the inertia matrix, is the Coriolis / centrifugal force matrix, is the gravity matrix; , and All have the appropriate size. Function , and They are all continuous and bounded.

[0068] Transform the state of formula (17) to obtain: (18); in, express The estimated value of It is the control force of the belt conveyor system after the state transformation. for The derivative of .

[0069] Specifically, step S2 also includes the following steps: S2.4, in order to simplify formula (18), let , , ; Then the transformed belt conveyor system equation is: (19).

[0070] Thus, the state equation is obtained: (20).

[0071] The present invention is designed to control Uncertainty was taken into account.

[0072] S2.5, let express The nominal part of ,function is a continuous function; let: , , ; then: (twenty one).

[0073] For the given Riccati equation The only solution , ,make: (twenty two); There exists a constant (possibly unknown) So that for all Both have: (twenty three); Due to the uncertain boundary, that is, the compact set is unknown, so the constant It is also unknown. When special circumstances When (i.e., without uncertainty), , , so you can choose Therefore, the uncertainty effect is imposed on and The possible deviation between them is within a certain threshold, and this threshold is unidirectional (i.e. not restricted in one direction).

[0074] S2.6, consider any function , there exists a vector (known or unknown, and may depend on ) and a function (and irrelevant), so that for all , have: (twenty four); in, is a constant vector, the function is the frame of uncertainty boundary. The constant vector Possible and bounding set Equation (24) parameterizes the worst case associated with uncertainty. This is feasible in most applications because the function , and is continuous.

[0075] Specifically, step S3 includes the following steps: S3.1, consider the belt conveyor system control force as: (25); in, represents the controller without considering uncertainty, that is, nominal control, Represents a term in the controller used to compensate for the inconsistency of the system's initial conditions, that is, compensation uncertainty.

[0076] Based on the above assumptions, let: (26); in, is a scalar design parameter; to simplify the formula, let , then let: (27); in, is a scalar design parameter, is a real number.

[0077] S3.2, if the displacement state vector can be accurately calculated, then the above control and It can be easily applied to dynamic systems. However, due to the difficulty of integration, we consider controlling the estimated value. Consider controlling the estimated value: (28); Consider a bounded function ,and , so that: (29); in, , express and The maximum value of the difference norm of . In particular, when hour, The meaning of formula (29) is and The difference between , that is, whether the integral is exact or not, is bounded.

[0078] S3.3, consider the functional differential equation In Control Below, among them Representatives The solution at time , assuming there is a continuous differential function And four continuous monotonically increasing functions , , and With attributes , represents a positive real number, represents the independent variable, when hour , and when hour , so that: (30); For functional differential equations have: (31);

[0079] in, represents the Hamiltonian operator, , , that is, the solution of the belt conveyor system Satisfies uniform boundedness and eventually uniform boundedness, is the Lyapunov function. It can be seen that in the control Under this condition, the uncertain system can also satisfy the uniform boundedness and eventually uniform boundedness.

[0080] S3.4, select the Lyapunov function as: (32); in, is the Riccati equation The only solution is , . Put (32) to Taking the first-order derivative, we get: (33).

[0081] S3.5, in order to simplify formula (33), from formula (21) and (25) and let as well as , for all have: (34); in, Design parameters for a scalar that always satisfy: (35).

[0082] According to formula (30), the uniform boundedness formula is: (36); (37); in, represents a uniformly bounded boundary, is a parameter reflecting the boundary state. represents the convergence radius, , , and Respectively The minimum and maximum characteristic values ​​of The unique solution on can be expanded to For any , express The maximum value of express The minimum value of , and the final uniform boundedness is as follows: (38); (39).

[0083] in, represents the eventually uniformly bounded boundary. To simplify formula (39), let: (40);

[0084] in, represents the smallest positive integer, Represents a time-varying function The maximum value of (41).

[0085] in, (42); That is, the belt conveyor system represented by formula (19) has uniform boundedness and uniform ultimate boundedness.

[0086] Step S4 is specifically as follows: Figure 2 The belt conveyor system shown is the control object of the present invention. The belt conveyor system takes into account two driving wheels and one tensioning wheel, and divides the system into three subsystems. Among them, , , represents the quality of the three subsystems, , , represents the stiffness of the three subsystems, , , represents the damping of the three subsystems, , , represents the speed of the three subsystems, namely the belt speed, and Represents the driving force provided by the two driving wheels, Represents the tension provided by the tensioning wheel. In order to achieve energy-saving control of the belt conveyor system and reduce its accident rate, the belt conveyor model is first established by applying the generalized Udwadia-Kalaba theory, and the uncertainties of the belt conveyor such as load, stiffness, damping and friction are described; secondly, the system state is transformed through the differential homeomorphism theory, and the speed inequality constraint problem is transformed into a system stability problem; finally, a robust controller is designed for the nominal system, and the uncertainty is compensated to make the system uniformly bounded and ultimately uniformly bounded, so that the controller system performance is more stable and the occurrence of accidents is reduced.

[0087] like Figure 3 As shown in Figure 1, the robust control designed based on the generalized Udwadia-Kalaba (GUK) theory is compared with the robust control designed based on the Udwadia-Kalaba (UK) theory, SMC control (sliding mode control) and LQR control (linear quadratic regulator control). Figure 4 As shown in the figure, the time to reduce the error based on GUK robust control is shorter, and the exponential control using arbitrary control parameters can make the system consistently bounded, and no matter what the uncertainty is, the system can eventually be consistently bounded. At the same time, the jitter under GUK robust control is smaller than that under UK, SMC and LQR control, and the system is more stable. Figure 5 As shown in the figure, the speed under GUK robust control satisfies the inequality constraint range more stably than the speed under UK, SMC and LQR control, and no matter the subsystem or the overall system, when there is an initial condition deviation, the displacement and acceleration can approach the expected value faster with a lower error.

[0088] like Figure 6 As shown in the figure, according to the cumulative error diagram of the three subsystems during the simulation process, compared with UK, SMC and LQR control, GUK control has more precise speed control, smaller error and cumulative error, and can more stably meet the system speed inequality constraints. Under GUK control, the system has higher safety and more precise control performance.

[0089] Of course, the above description is not a limitation of the present invention, and the present invention is not limited to the above examples. Changes, modifications, additions or substitutions made by technicians in this technical field within the essential scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A belt conveyor control method based on generalized UK theory, characterized in that: The specific steps include: S1, establish the belt conveyor system dynamics model considering both equality and inequality constraints; S2, based on the differential homeomorphism theory, the state transformation is carried out, and the original speed state of the belt conveyor system is transformed into a new state by limiting the integral difference; S3, considers the speed of the control target as a constraint and proposes a robust control design in combination with the dynamics model; S4, the belt conveyor system is simulated for robust control verification.

2. A belt conveyor control method based on generalized UK theory according to claim 1, characterized in that: Step S1 specifically includes the following steps: S1.1, the dynamic model of the belt conveyor system with equality constraints and inequality constraints is: （1）； in, represents the inertia matrix, the dimension of the inertia matrix is , is an integer, is the belt acceleration, Indicates inherent force, and represent the external forces based on equality constraints and inequality constraints respectively; S1.2, consider the equality constraint, the first-order form of the equality constraint is expressed as: （2）； in, Represents the constraint matrix, the dimension of the constraint matrix is , is an integer and has , is the first-order constraint matrix; For equation (2), find the The derivative of , gives the second-order form of the equality constraint: （3）； in, is the second-order constraint matrix, is the generalized coordinate of the belt, Indicates belt speed; S1.3, according to the Udwadia-Kalaba method, the external force based on the equality constraint is obtained : （4）； in, represents the Moore-Penrose generalized inverse; S1.4, consider the inequality constraints: （5）； in, express The lower bound of express The upper bound of for No. Quantity, and ; S1.5, Applied forces based on inequality constraints Designed for: （6）； in, represents the identity matrix, The value is to be determined.

3. The belt conveyor control method based on generalized UK theory according to claim 1 is characterized in that: Step S2 specifically includes the following steps: S2.1, in order to handle inequality constraints, Transform via diffeomorphism: （7）； in, is the coordinate of the belt conveyor belt subsystem after transformation, is a diffeomorphism in the form of a tangent function, Indicates continuous The order is differentiable; S2.2, Extension and , write equation (7) in matrix form: （8）； in, and ; and The expanded and , integrate and differentiate equation (8) respectively, and we get: （9）； （10）； in, represents the integration constant; Consider the estimated value function As the original function The estimated value of , assuming that the estimated value function Features include: （11）； in, Representation and The relevant time-varying functions; Consider a bounded time-varying function , so that: （12）； S2.3, Dynamics of belt conveyor system considering the following uncertainties: （13）； in, Indicates uncertain parameters, is a compact set, indicating uncertain parameters Area, is the inertia matrix, is the centrifugal force matrix, is the gravity matrix; Transform the state of equation (13) to obtain: （14）； in, express The estimated value of It is the control force of the belt conveyor system after the state transformation. for The derivative of .

4. A belt conveyor control method based on generalized UK theory according to claim 3, characterized in that: Step S2 also includes the following steps: S2.4, in order to simplify formula (14), let , , ; Then the transformed belt conveyor system equation is: （15）； Thus, the state equation is obtained: （16）； S2.5, let express The nominal part of ,function is a continuous function; make: ， ， ; then: （17）； The unique solution to the Riccati equation is , ,make: （18）； There is a constant So that for all Both have: （19）； S2.6, consider any function , there exists a vector and a function , so that for all , have: （20）； in, is a constant vector, the function A framework for uncertainty boundaries.

5. The belt conveyor control method based on generalized UK theory according to claim 1, characterized in that: Step S3 specifically includes the following steps: S3.1, consider the belt conveyor system control force as: （21）； in, represents the controller without considering uncertainty, represents a term in the controller used to compensate for the incompatibility of the system initial conditions; make: （22）； in, is a scalar design parameter; to simplify the formula, let , then let: （23）； in, is a scalar design parameter, is a real number; S3.2, Consider controls on estimates: （24）； Consider a bounded function ,and , so that: （25）； in, , express and The maximum value of the difference norm; S3.3, consider the functional differential equation In Control Below, among them Representatives in The solution at time , assuming there is a continuous differential function And four continuous monotonically increasing functions , , and With attributes , represents the independent variable, when hour , and when hour , so that: （26）； For functional differential equations have: （27）； in, represents the Hamiltonian operator, , , that is, the solution of the belt conveyor system Satisfies uniform boundedness and eventually uniform boundedness, is the Lyapunov function.

6. A belt conveyor control method based on generalized UK theory according to claim 5, characterized in that: Step S3 also includes the following steps: S3.4, select the Lyapunov function as: （28）； in, is the unique solution to the Riccati equation, , ; will (28) Taking the first-order derivative, we get: （29）； S3.5, in order to simplify formula (19), from formula (17) and (21) and let as well as , for all have: （30）； in, Design parameters for a scalar: （31）； According to formula (26), the uniform boundedness formula is: （32）； （33）； in, represents a uniformly bounded boundary, is a parameter reflecting the boundary state. represents the convergence radius, , , and Respectively The minimum and maximum eigenvalues ​​of , express The maximum value of express The minimum value of , and the final uniform boundedness is as follows: （34）； （35）； in, represents the eventually uniformly bounded boundary; to simplify formula (35), let: （36）； in, represents the smallest positive integer, Represents a time-varying function The maximum value of （37）； in, （38）； That is, the belt conveyor system represented by formula (15) has uniform boundedness and uniform ultimate boundedness.

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