A control method for belt conveyors based on the generalized UK theory

Abstract

The invention provides a control method for a belt conveyor based on the generalized UK theory, which relates to the technical field of belt conveyor control and specifically includes the following steps: establishing a dynamic model of the belt conveyor system that simultaneously considers equality and inequality constraints; performing state transformation based on the diffeomorphism theory, and transforming the original state of the belt conveyor system speed into a new state by limiting the integral difference; regarding the speed of the control target as a constraint, and proposing a robust control design in combination with the dynamic model; simulating the belt conveyor system to verify the robust control. The technical solution of the invention overcomes the problem in the prior art that the conveyor system cannot be stably started.

Description

Technical Field

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

[0002] Due to its structural characteristics, there are many uncertainties in the engineering practice of long-distance belt conveyors, resulting in large fluctuations in the conveyor belt speed and tension. During the entire control process, the control in the starting stage is relatively difficult, and inappropriate driving forces may cause accidents such as belt stacking and belt breakage. Therefore, accurately controlling 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 is restricted to be within a certain range. The speed difference is reflected by the speed error, and the difference range is controlled by inequality constraints, thereby realizing the starting speed constraint control of the long-distance belt conveyor.

[0003] Therefore, there is a need for a belt conveyor control method based on the generalized UK theory that can control the stable starting 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 in the prior art that the conveyor system cannot be stably started.

[0005] To achieve the above purpose, the present invention provides a belt conveyor control method based on the generalized UK theory, which specifically includes the following steps:

[0006] S1. Establish a dynamic model of the belt conveyor system that simultaneously considers equality and inequality constraints.

[0007] S2. Based on the diffeomorphism theory, perform state transformation, and transform the original state of the belt conveyor system speed into a new state by limiting the integral difference.

[0008] S3. Regard the speed of the control target as a constraint, and combine the dynamic model to propose a robust control design.

[0009] S4. Simulate the belt conveyor system to verify the robust control.

[0010] Further, step S1 specifically includes the following steps:

[0011] S1.1. Consider the dynamic model of the belt conveyor system with equality and inequality constraints as:

[0012] (1);

[0013] Wherein, denotes the inertia matrix, and the dimension of the inertia matrix is , is an integer, denotes the belt acceleration, denotes the inherent force, and respectively denote the externally applied forces based on equality constraints and inequality constraints.

[0014] S1.2. Consider the equality constraints. The first-order form of the equality constraints is expressed as:

[0015] (2);

[0016] where denotes the constraint matrix, and the dimension of the constraint matrix is , is an integer and there is , is the first-order constraint matrix.

[0017] Taking the derivative of Equation (2) with respect to time , the second-order form of the equality constraints is obtained:

[0018] (3);

[0019] where is the second-order constraint matrix, is the generalized coordinate of the belt, denotes the belt speed.

[0020] S1.3. According to the Udwadia-Kalaba method, the externally applied force based on the equality constraints is obtained:

[0021] (4);

[0022] where denotes the Moore-Penrose generalized inverse.

[0023] S1.4. Consider the inequality constraints:

[0024] (5);

[0025] where denotes the lower bound of denotes the upper bound of is the th component of , and

[0026] S1.5, External force based on inequality constraints It is designed as:

[0027] (6);

[0028] Among them, represents the identity matrix, is a value to be determined.

[0029] Furthermore, step S2 specifically includes the following steps:

[0030] S2.1, To handle the inequality constraints, perform a transformation through diffeomorphism:

[0031] (7);

[0032] Among them, is the coordinate of the belt conveyor belt subsystem after transformation, is a diffeomorphism in the form of the tangent function, represents continuous n - order differentiability.

[0033] S2.2, Expand and , and write equation (7) in matrix form:

[0034] (8);

[0035] Among them, and ; and are the expanded and respectively. Take the integral and derivative of equation (8) respectively, and we get:

[0036] (9);

[0037] (10);

[0038] Among them, represents the integration constant.

[0039] Consider the estimated value function as the estimated value of the original function . Assume that the estimated value function has the following properties:

[0040] (11);

[0041] Among them, represents related to The relevant time-varying function.

[0042] Consider the existence of a bounded time-varying function such that:

[0043] (12).

[0044] S2.3. Consider the following dynamic belt conveyor system with uncertainties:

[0045] (13);

[0046] where, represents the uncertain parameter, is a compact set representing the region where the uncertain parameter is located, is the inertia matrix, is the centrifugal force matrix, is the gravity matrix.

[0047] Perform a state transformation on Equation (13) to obtain:

[0048] (14);

[0049] where, represents the estimated value of, is the control force of the belt conveyor system after state transformation, is the derivative of.

[0050] Furthermore, Step S2 also includes the following steps:

[0051] S2.4. To simplify Equation (14), let , , ; then the equation of the transformed belt conveyor system is:

[0052] (15);

[0053] Thus, the state equation is obtained:

[0054] (16).

[0055] S2.5. Let represent the nominal part of , and , the function is a continuous function;

[0056] Let:

[0057] ,

[0058] ,

[0059] ; then there is:

[0060] (17).

[0061] For the unique solution of the given Riccati equation , , let:

[0062] (18).

[0063] There exists a constant such that for all there is:

[0064] (19).

[0065] S2.6, consider any function , there exists a vector and a function such that for all , there is:

[0066] (20);

[0067] wherein, is a constant vector, and the function is the framework of the uncertainty boundary.

[0068] Furthermore, step S3 specifically includes the following steps:

[0069] S3.1, consider the control force of the belt conveyor system as:

[0070] (21);

[0071] wherein, represents the controller when uncertainties are not considered, represents a term used in the controller to compensate for the incompatibility problem of the system initial conditions.

[0072] Let:

[0073] (22);

[0074] wherein, is a scalar design parameter; for simplicity of the formula, let , then let:

[0075] (23);

[0076] wherein, is the scalar design parameter, is a real number.

[0077] S3.2, consider controlling for the estimated value:

[0078] (24);

[0079] Consider that there exists a bounded function , and , such that:

[0080] (25);

[0081] wherein, , represents and the maximum value of the difference norm of

[0082] S3.3, consider the functional differential equation under the control , wherein represents the solution at the moment of , assume that there exists a continuous differential function and four continuous and monotonically increasing functions , , and with the property , represents the independent variable, when then , and when then , such that:

[0083] (26);

[0084] For the functional differential equation there is:

[0085] (27);

[0086] wherein, represents the Hamiltonian operator, , , that is, the solution of the belt conveyor system satisfies uniform boundedness and ultimate uniform boundedness, is a Lyapunov function.

[0087] Furthermore, step S3 further includes the following steps:

[0088] S3.4, select the Lyapunov function as:

[0089] (28);

[0090] where is the unique solution of the Riccati equation, , ; Take the first derivative of (28) with respect to , and we can get:

[0091] (29).

[0092] S3.5, in order to simplify formula (29), from equations (17) and (21) and let and , for all there is:

[0093] (30);

[0094] where is a scalar design parameter:

[0095] (31).

[0096] From equation (26), the uniform boundedness formula is:

[0097] (32);

[0098] (33);

[0099] where represents the boundary of uniform boundedness, is a parameter reflecting the boundary state, represents the convergence radius, , , and respectively represent 's minimum and maximum eigenvalues; for any , represents 's maximum value, represents 's minimum value.

[0100] The final uniform boundedness is as follows:

[0101] (34);

[0102] (35).

[0103] Among them, represents the ultimately uniformly bounded boundary; to simplify formula (35), let:

[0104] (36);

[0105] Among them, represents the smallest positive integer, represents the time-varying function of the maximum and minimum values, such that

[0106] (37);

[0107] Among them,

[0108] (38);

[0109] That is, the belt conveyor system represented by formula (15) has uniform boundedness and ultimate uniform boundedness.

[0110] The present invention has the following beneficial effects:

[0111] 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 the generalized Udwadia-Kalaba theory and the diffeomorphism theory is proposed to handle inequality constraints, and the nominal system is controlled and the uncertainties are compensated. Finally, it is proved that the belt conveyor system is uniformly bounded and ultimately uniformly bounded.

[0112] The present invention proposes a method of using the diffeomorphism method to transform the system state to solve the modeling problem, and transforms the system inequality constraint handling problem into a system stability problem. The present invention establishes a speed inequality constraint handling problem based on the generalized Udwadia-Kalaba (i.e., GUK) theory, and designs a robust control for the transformed system to achieve the purpose of controlling the stable start of the conveyor system. Brief Description of the Drawings

[0113] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the following will briefly introduce the drawings required for the description of the specific embodiments or the prior art. Obviously, the drawings in the following description are some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can also be obtained based on these drawings. In the drawings:

[0114] Figure 1 Fig. shows the flowchart of a belt conveyor control method based on the generalized UK theory of the present invention.

[0115] Figure 2 Fig. shows the schematic diagram of the belt conveyor system model structure in the embodiment of the present invention.

[0116] Figure 3 Fig. shows the schematic diagram of the robust control verification comparison of the present invention.

[0117] Figure 4 Fig. shows Figure 3 The enlarged detail view of part A.

[0118] Figure 5 Fig. shows Figure 3 The enlarged detail view of part B.

[0119] Figure 6 Fig. shows the schematic diagram of the cumulative error verification comparison of the present invention. Specific embodiments

[0120] The following will clearly and completely describe the technical solutions of the present invention with reference to the drawings. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present invention.

[0121] A belt conveyor control method based on the generalized UK theory specifically includes the following steps:

[0122] S1. Establish a dynamic model of the belt conveyor system that simultaneously considers equality and inequality constraints.

[0123] S2. Perform state transformation based on the diffeomorphism theory, and transform the original state of the belt conveyor system speed into a new state by limiting the integral difference.

[0124] S3. Regard the speed of the control target as a constraint, and propose a robust control design in combination with the dynamic model.

[0125] S4. Simulate the belt conveyor system to verify the robust control.

[0126] AsFigure 1 As shown, the present invention first establishes a system dynamics model that simultaneously includes equality and inequality constraints, and then performs a diffeomorphic transformation on the speed of the belt conveyor system. By defining the integral difference, the original state of the belt conveyor system speed is transformed into a new state, and the control target speed is regarded as a constraint. Combining with the dynamic equation, a control design is proposed. The belt conveyor system is simulated to verify the robust control until the control requirements are met, and then the control parameters are output.

[0127] Specifically, step S1 specifically includes the following steps:

[0128] S1.1, considering the dynamic model of the belt conveyor system with equality and inequality constraints as:

[0129] (1);

[0130] Wherein, represents the inertia matrix, and the dimension of the inertia matrix is , is an integer, represents the belt acceleration, represents the inherent force, and respectively represent the external forces based on equality and inequality constraints.

[0131] S1.2, considering the equality constraint, the first-order form of the equality constraint is expressed as:

[0132] (2);

[0133] Wherein, represents the constraint matrix, and the dimension of the constraint matrix is , is an integer and there is , is the first-order constraint matrix.

[0134] Taking the derivative of equation (2) with respect to time , the second-order form of the equality constraint is obtained:

[0135] (3);

[0136] Wherein, is the second-order constraint matrix, is the generalized coordinate of the belt, represents the belt speed.

[0137] S1.3, according to the Udwadia-Kalaba method, the external force based on the equality constraint is obtained:

[0138] (4);

[0139] Among them, represents the Moore - Penrose generalized inverse.

[0140] S1.4. Consider the inequality constraint:

[0141] (5);

[0142] Among them, represents the lower bound of, represents the upper bound of, is the th component of, and .

[0143] S1.5. The external force based on the inequality constraint is designed as:

[0144] (6);

[0145] Among them, represents the identity matrix, is a value to be determined.

[0146] Given the form of in (6), the acceleration of the belt conveyor system still satisfies all equality constraints, which indicates that the inequality constraint will not affect the equality constraint . Solving the simultaneous equations (1), (4) and (6) can obtain:

[0147] (7);

[0148] Among them represents the belt acceleration based on the equality constraint, represents the belt acceleration based on the equality constraint.

[0149] For there is,

[0150] (8);

[0151] For there is,

[0152] (9).

[0153] Considering equations (8) and (9), it can be known that

[0154] (10);

[0155] As can be seen from Equation (10), and both have values of . Therefore, it is feasible to consider both equality constraints and inequality constraints in the system.

[0156] Specifically, step S2 specifically includes the following steps:

[0157] S2.1, assume there exists a function , where , , represents the original state variables of a certain mechanical system, represents the new state variables after the state transformation of the mechanical system, represents the n-dimensional real number set, represents the m-dimensional real number set. If it is smooth, it is called a diffeomorphism, and its inverse function exists and is also smooth. To handle the inequality constraints, perform a transformation through diffeomorphism:

[0158] (11);

[0159] where, is the coordinate of the belt conveyor belt subsystem after transformation, is a diffeomorphism in the form of the tangent function, represents continuous nth-order differentiability.

[0160] The state transformation function needs to satisfy the following conditions:

[0161] (i) The domain of the transformation function should be , and its range should be ;

[0162] (ii) The transformation function is an increasing function and continuous within the domain.

[0163] From the above conditions, it can be obtained that when , there is . If the transformed state is bounded, then the original state of the belt conveyor belt subsystem speed can satisfy the inequality constraint . Thus, the inequality constraint problem is transformed into a system stability problem.

[0164] S2.2, expand and , Equation (11) is written in matrix form as:

[0165] (12);

[0166] Where, And ; And Are the extended And respectively. Integrating and differentiating Equation (12) respectively, we get:

[0167] (13);

[0168] (14);

[0169] Where, represents the integration constant.

[0170] In the calculation, for with respect to the antiderivative of the integral is difficult to obtain. Therefore, consider the estimated value function as the estimated value of the antiderivative . Assume that the estimated value function has the following properties:

[0171] (15);

[0172] Where, represents a time-varying function related to .

[0173] Consider the existence of a bounded time-varying function such that:

[0174] (16);

[0175] Since the state is continuous, regardless of whether the antiderivative is unknown or known, for any value of the distance between the original state and the estimated state value

[0176] S2.3. Consider the following dynamic belt conveyor system with uncertainties:

[0177] (17);

[0178] Where, represents the uncertain parameter, is a compact set representing the uncertain parameter The area where is the inertia matrix, is the Coriolis / centrifugal force matrix, and is the gravity matrix; and both have appropriate dimensions. The functions and and are all continuous and bounded.

[0179] Performing a state transformation on Equation (17), we get:

[0180] (18);

[0181] where represents 's estimated value, is the control force of the belt conveyor system after state transformation, is 's derivative.

[0182] Specifically, step S2 further includes the following steps:

[0183] S2.4, to simplify Equation (18), let and and ; then the transformed belt conveyor system equation is:

[0184] (19).

[0185] Thus, we obtain the state equation:

[0186] (20).

[0187] The present invention considers uncertainties when designing the control .

[0188] S2.5, let represent the nominal part of , and , the function is a continuous function; let:

[0189] and

[0190] and

[0191] ; then we have:

[0192] (21).

[0193] For a given Riccati equation the unique solution of , , let:

[0194] (22);

[0195] There exists a constant (possibly unknown) such that for all there is:

[0196] (23);

[0197] Due to the uncertain boundary, i.e., the compact set is unknown, so the constant is also unknown. When the special case occurs (i.e., there is no uncertainty), , , so one can choose . Therefore, through the assumption of continuity, the influence of uncertainty is imposed on the possible deviation between and to keep it within a certain threshold, and this threshold is one-way (i.e., not restricted in one direction).

[0198] S2.6. Consider any function , there exists a vector (known or unknown and possibly depending on ) and a function (independent of ) such that for all , there is:

[0199] (24);

[0200] where is a constant vector and the function is the framework of the uncertainty boundary. The constant vector may be related to the boundary set . Equation (24) parameterizes the worst-case scenario related to uncertainty. This is feasible in most applications because the functions , and are continuous.

[0201] Specifically, step S3 specifically includes the following steps:

[0202] S3.1, consider the control force of the belt conveyor system as:

[0203] (25);

[0204] where, represents the controller without considering uncertainties, i.e., the nominal control, represents a term in the controller used to compensate for the incompatibility problem of the system initial conditions, i.e., to compensate for uncertainties.

[0205] Based on the above assumptions, let:

[0206] (26);

[0207] where, is a scalar design parameter; for simplicity of the formula, let , then let:

[0208] (27);

[0209] where, is a scalar design parameter, is a real number.

[0210] S3.2, if the displacement state vector can be accurately calculated, then the above control and can be easily applied to the dynamic system. However, due to difficulties in integration, consider controlling the estimated value. Consider controlling the estimated value:

[0211] (28);

[0212] Consider that there exists a bounded function , and , such that:

[0213] (29);

[0214] where, , represents and the maximum value of the difference norm of. In particular, when , . The meaning of Equation (29) is and the difference, that is, it is bounded whether the exact integration is carried out or not.

[0215] S3.3, consider the functional differential equation under the control , where The solution at is assumed to exist a continuous differential function and four continuous and monotonically increasing functions , , and with the property , denotes positive real numbers, denotes the independent variable. When , and when , , such that:

[0216] (30);

[0217] For the functional differential equation there is:

[0218] (31);

[0219] wherein, denotes the Hamiltonian operator, , , that is, the solution of the belt conveyor system satisfies uniform boundedness and ultimate uniform boundedness, is the Lyapunov function. It can be seen that under the control , the uncertain system can also satisfy uniform boundedness and ultimate uniform boundedness.

[0220] S3.4, select the Lyapunov function as:

[0221] (32);

[0222] wherein, is the unique solution of the Riccati equation , , . Differentiating (32) with respect to yields:

[0223] (33).

[0224] S3.5, to simplify formula (33), from equations (21) and (25) and let and , for all there is:

[0225] (34);

[0226] wherein, is a scalar design parameter and can satisfy at any time:

[0227] (35).

[0228] From equation (30), the uniform boundedness formula is:

[0229] (36);

[0230] (37);

[0231] where represents the boundary of uniform boundedness, is a parameter reflecting the boundary state, represents the radius of convergence, , , and respectively represent 's minimum and maximum eigenvalues. Therefore, the unique solution on the time range can be extended to . For any , represents 's maximum value, represents 's minimum value, and the ultimate uniform boundedness is as follows:

[0232] (38);

[0233] (39).

[0234] where represents the boundary of ultimate uniform boundedness. To simplify equation (39), let:

[0235] (40);

[0236] where represents the smallest positive integer, represents the maximum and minimum values of the time-varying function such that

[0237] (41).

[0238] where

[0239] (42);

[0240] That is, the belt conveyor system represented by equation (19) has uniform boundedness and ultimate uniform boundedness.​

[0241] Step S4 is specifically as follows:

[0242] Figure 2 The belt conveyor system shown is the control object of the present invention. The belt conveyor system considers two driving wheels and one tensioning wheel, and divides the system into three subsystems. Among them, , , represent the masses of the three subsystems, , , represent the stiffnesses of the three subsystems, , , represent the dampings of the three subsystems, , , represent the speeds of the three subsystems, that is, the belt speed, and represent the driving forces provided by the two driving wheels, represents the tension force provided by the tensioning wheel. In order to achieve the energy-saving control of the belt conveyor system and reduce its accident rate, first, the generalized Udwadia-Kalaba theory is applied to establish a belt conveyor model to describe the uncertainties such as the load, stiffness, damping, and friction of the belt conveyor; secondly, the system state is transformed through the diffeomorphism theory to transform the velocity inequality constraint problem into a system stability problem; finally, a robust controller is designed for the nominal system and the uncertainties are compensated to make the system reach uniform boundedness and ultimate uniform boundedness, so that the performance of the controller system is more stable and the occurrence of accidents is reduced.

[0243] As Figure 3 shown, the robust control designed based on the generalized Udwadia-Kalaba (i.e., GUK) theory is compared with the robust control designed based on the Udwadia-Kalaba (i.e., UK) theory, SMC control (sliding mode control), and LQR control (linear quadratic regulator control). As Figure 4 shown, the time for the GUK-based robust control to reduce the error is shorter, and the exponential control using any control parameters can make the system uniformly bounded, and no matter what the uncertainties are, the system can finally be made uniformly bounded. At the same time, the jitter under the GUK-based robust control is smaller than that under the UK, SMC, and LQR controls, and the system is more stable. As Figure 5 shown, the speed under the GUK-based robust control more stably satisfies the inequality constraint range than the speeds under the UK, SMC, and LQR controls, and no matter for the subsystems or the overall system, when there are initial condition deviations, the displacement and acceleration can approach the expected values faster with lower errors.

[0244] As Figure 6 shown, according to the cumulative error graphs of the three subsystems during the simulation process, compared with UK, SMC, and LQR controls, GUK control is more precise in controlling speed, with smaller errors and smaller cumulative errors, and can more stably meet the system speed inequality constraints. Under GUK control, the system has higher safety and more precise control performance.

[0245] 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 those skilled in the art within the scope of the essence 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, simulate the belt conveyor system and verify the robust control; 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.

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 Related 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 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.

Images