A bridge crane sliding mode control method based on unknown input observer technology

Abstract

A kind of bridge crane sliding mode control method based on unknown input observer technology, the dynamic model of bridge crane system is established, system state and control parameter are initialized;Unknown input observer is designed;Double-layer hyperbolic approach law is designed;Based on unknown input observer and double-layer hyperbolic approach law, sliding mode controller is designed.The present application can effectively solve the problem of rapid positioning of bridge crane under different persistent non-matching disturbance conditions, effectively suppress the residual swing of the load of bridge crane, and ensure the smoothness of control input.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of control of underactuated bridge cranes, and proposes a bridge crane sliding mode control method based on unknown input observer technology, in particular to a bridge crane sliding mode control method containing different types of non-matching disturbances. BACKGROUND

[0002] An underactuated system is a special system, and its specific meaning is that the number of independent control inputs of the system is less than the degrees of freedom of the mechanism of the system. A crane is a typical underactuated system, which is widely used in ports, construction sites, offshore drilling platforms and other places, and plays an irreplaceable role in modern industrial applications.

[0003] However, in actual production application, the crane will be affected by external disturbances such as wind, and these uncertain factors can easily cause serious safety accidents to the operator. In recent years, many domestic and foreign scholars have devoted themselves to the research work of crane sway-free positioning control, and have achieved many fruitful results. However, most of the existing research results are a series of control methods proposed under the condition that the crane is in the absence of disturbance or in the presence of matching disturbance (including friction force, etc.). There are few relevant research results that can well handle the influence of non-matching disturbance (including wind, air resistance, etc.) on the control performance of the crane. Therefore, it is very meaningful to study effective methods to suppress the non-matching disturbance existing in the crane system.

[0004] Non-matching disturbance exists in many control systems, and many scholars have proposed new control theories and methods to suppress such disturbance, mainly including linear control methods based on generalized extended state observer, hierarchical sliding mode control methods, backstepping method, fuzzy control method and sliding mode control method based on nonlinear disturbance observer, etc. These methods can effectively suppress the influence of non-matching disturbance on control performance. At present, the research on suppression of non-matching disturbance existing in underactuated cranes is still in its infancy, and many existing research works are only for non-matching disturbance under specific types or non-matching disturbance under non-continuous action. Therefore, it is very necessary to find a control method that can be quickly implemented and has good suppression effect for different types of non-matching disturbance under continuous action in the crane system, and it is also of great significance to the sway-free positioning performance of the crane. SUMMARY

[0005] In order to reduce the influence of non-matching disturbances under different types of sustained action on the anti-swing positioning of the crane, and make the crane reach the steady state quickly and have good robustness, the present application proposes a bridge crane sliding mode control method based on unknown input observer technology, which estimates the matching disturbance term and non-matching disturbance term existing in the crane system by using unknown input observer technology, and then designs a double-layer hyperbolic reaching law, and designs a sliding mode controller based on the estimated matching disturbance term and non-matching disturbance term and the double-layer hyperbolic reaching law. The design can effectively suppress the influence of matching disturbances and non-matching disturbances under different types of sustained action on the anti-swing positioning performance of the crane, and ensure that the crane system has fast state convergence ability and smooth control output effect.

[0006] In order to solve the above technical problems, the technical scheme proposed is as follows:

[0007] A bridge crane sliding mode control method based on unknown input observer technology, comprising the following steps:

[0008] Step 1, establish the dynamic model of the bridge crane system, initialize the state and control parameters of the system, the process is as follows:

[0009] 1.1 The dynamics of the bridge crane is represented as:

[0010]

[0011] Where M and m represent the mass of the trolley and the mass of the load, respectively; The acceleration of the trolley motion is represented as l, and the length of the rope is represented as θ, The angle, angular velocity and angular acceleration of the load swing are represented as F, respectively; d x The matching disturbance lumped term is represented as d θ The non-matching disturbance lumped term is represented as g, and the gravitational acceleration is represented as g;

[0012] 1.2 The acceleration expression of the trolley motion is obtained by combining equations (1)-(2) as:

[0013]

[0014] 1.3 Substitute equation (3) into equation (2), and after mathematical transformation, the control input F is obtained as:

[0015]

[0016] Where,

[0017] 1.4 In order to facilitate the design of the subsequent controller, an auxiliary control variable v is defined as

[0018]

[0019] Substitute equation (5) into equation (4) and re-transform to get:

[0020]

[0021] 1.5 Define the following state variables:

[0022]

[0023] where x d , are the reference trajectory and velocity of the trolley motion, respectively; sec(·) represents the inverse of the cosine function;

[0024] 1.6 Combine equation (7) to construct the following form of state equation:

[0025]

[0026] where are the first-order derivatives of in equation (7), respectively; λ i , i = 1, 2, 3 are the defined auxiliary variables; are the construction functions of state variables ; their specific expressions are as follows:

[0027]

[0028] 1.7 In order to facilitate the design of the unknown input observer later, equation (8) is transformed into the following form:

[0029]

[0030] where d m = λ2d x - λ3d θ

[0031] 1.8 For the actual application scenario of the crane, the angle of load swing satisfies Therefore, the auxiliary variables λ1, λ2, λ3 defined by equation (9) are bounded, and the construction functions are also bounded by the same reasoning;

[0032] Step 2, design the unknown input observer to estimate the matched and unmatched disturbances in the crane system, the process is as follows:

[0033] 2.1 Assume that the matched and unmatched disturbances, as well as their first-order derivatives, are bounded, and define as follows:​

[0034]

[0035] where, denote the upper bounds of the matched and unmatched disturbances, respectively, denote the upper bounds of the first-order derivatives of the matched and unmatched disturbances, respectively;

[0036] 2.2 The transformed crane system model (10) is first-order low-pass filtered to design the unknown input observer as follows:

[0037]

[0038] where k > 0 denotes the filter coefficient; define [·] / (ks+1) = (·) in (12) f ; (12) is transformed into:

[0039]

[0040] where, denote the original state variables and the first-order derivatives of the low-pass filtered results of mmf f mf denote the first-order derivatives of mm m

[0041] 2.3 Each state variable in (13) has the following properties:

[0042]

[0043] The expressions of mmf mf are obtained from (13)-(14):

[0044]

[0045] The disturbance observer is designed in combination with (15) as follows:

[0046]

[0047] where, denote the estimated values of the disturbance terms mm m

[0048] 2.4 It is found from (16) that the disturbance observer has only one filter coefficient k that can be adjusted, thus avoiding the noise amplification problem caused by system derivation; ​​​​​​​​

[0049] Step 3, design of double-layer hyperbolic reaching law, the process is as follows:

[0050] Design double-layer hyperbolic reaching law, as shown below:

[0051]

[0052] Where, Indicates the first derivative of the designed sliding mode surface; γ1> 0, γ2> 0, k1> 0, k2> 0, p represents a positive odd number; tanh(k1s) represents the hyperbolic tangent function, the expression is as follows:

[0053]

[0054] Indicates the inverse hyperbolic sine function, the expression is as follows:

[0055]

[0056] Step 4, design of sliding mode control law based on unknown input observer technology, the process is as follows:

[0057] 4.1 Combined with equation (7), define the sliding mode surface as follows:

[0058]

[0059] Where, a > 0, b > 0, c > 0;

[0060] 4.2 Derive equation (20) combined with equation (10) to get:

[0061]

[0062] 4.3 Design the sliding mode controller of equation (10) based on double-layer hyperbolic reaching law combined with equations (16)-(17) as follows:

[0063]

[0064] 4.4 Select the following Lyapunov function:

[0065]

[0066] Derive the above equation, and substitute equation (22) into it to get That is, the system is stable;

[0067] 4.5 Substitute the auxiliary control law designed by equation (22) into equation (6) to get the expression of F as follows:

[0068]

[0069] Formula (24) represents the actual control law in the crane system.

[0070] The application designs a sliding mode control method for the bridge crane based on the unknown input observer technology, solves the influence of the matching disturbance and the non-matching disturbance on the swing elimination positioning performance of the crane system, effectively improves the convergence speed of the system state through the double-layer double hyperbolic reaching law, and realizes good control of the bridge crane system.

[0071] The technical concept of the application: for the bridge crane system with matching disturbance and non-matching disturbance under the action of different types, the first-order low-pass filtering method in the unknown input observer technology is used to estimate the matching and non-matching disturbance terms in the crane system, and a sliding mode controller based on double-layer double hyperbolic reaching law is designed according to the estimated value, which can effectively suppress the influence of the matching disturbance and the non-matching disturbance on the crane system, and ensure that the state of the crane system can be quickly stabilized. The application proposes a sliding mode control method based on the unknown input observer technology, which can effectively suppress the matching disturbance and the non-matching disturbance under the action of different types, realize the rapid positioning and swing elimination of the crane, and ensure that the bridge crane system can achieve good control effect.

[0072] The effective effect of the application: realize the rapid positioning of the bridge crane, suppress the residual swing of the load of the bridge crane, and ensure the smoothness of the control input. BRIEF DESCRIPTION OF DRAWINGS

[0073] Figure 1 The control flowchart of the application;

[0074] Figure 2 The state trajectory of x when the nonlinear disturbance observer is added in the constant non-matching disturbance;

[0075] Figure 3 The state curve of theta when the nonlinear disturbance observer is added in the constant non-matching disturbance;

[0076] Figure 4 The signal curve of F when the nonlinear disturbance observer is added in the constant non-matching disturbance;

[0077] Figure 5 The state trajectory of x when the generalized extended state observer is added in the constant non-matching disturbance;

[0078] Figure 6 The state curve of theta when the generalized extended state observer is added in the constant non-matching disturbance;

[0079] Figure 7 The signal curve of F when the generalized extended state observer is added in the constant non-matching disturbance;

[0080] Figure 8State trajectory of x when the sliding mode controller of the present application is added in constant non-matching disturbance

[0081] Figure 9 State curve of θ when the sliding mode controller of the present application is added in constant non-matching disturbance

[0082] Figure 10 Signal curve of F when the sliding mode controller of the present application is added in constant non-matching disturbance

[0083] Figure 11 State trajectory of x when the nonlinear disturbance observer is added in periodic non-matching disturbance

[0084] Figure 12 State curve of θ when the nonlinear disturbance observer is added in periodic non-matching disturbance

[0085] Figure 13 Signal curve of F when the nonlinear disturbance observer is added in periodic non-matching disturbance

[0086] Figure 14 State trajectory of x when the generalized extended state observer is added in periodic non-matching disturbance

[0087] Figure 15 State curve of θ when the generalized extended state observer is added in periodic non-matching disturbance

[0088] Figure 16 Signal curve of F when the generalized extended state observer is added in periodic non-matching disturbance

[0089] Figure 17 State trajectory of x when the sliding mode controller of the present application is added in periodic non-matching disturbance

[0090] Figure 18 State curve of θ when the sliding mode controller of the present application is added in periodic non-matching disturbance

[0091] Figure 19 Signal curve of F when the sliding mode controller of the present application is added in periodic non-matching disturbance DETAILED DESCRIPTION

[0092] The present application is further described below in conjunction with the accompanying drawings.

[0093] Reference Figures 1-19 A bridge crane sliding mode control method based on unknown input observer technology, comprising the following steps:

[0094] Step 1, establish the dynamic model of the bridge crane system, initialize the state and control parameters of the system, the process is as follows:

[0095] 1.1 The dynamics of the overhead crane is given by:

[0096]

[0097] where M, m represent the mass of the trolley and the mass of the load, respectively; a represents the acceleration of the trolley; l represents the length of the rope; θ, represent the angle, angular velocity and angular acceleration of the load swing, respectively; F represents the control input; d x represents the matched disturbance lumped term, including friction, unmodeled dynamics, etc.; d θ represents the unmatched disturbance lumped term, including air resistance, friction, etc.; g represents the acceleration of gravity;

[0098] 1.2 The acceleration of the trolley when moving is given by combining equations (1)-(2):

[0099]

[0100] 1.3 Substituting equation (3) into equation (2), and after mathematical transformation, the control input F is:

[0101]

[0102] where,

[0103] 1.4 In order to facilitate the design of the subsequent controller, an auxiliary control variable v is defined:

[0104]

[0105] Substituting equation (5) into equation (4) and retransforming gives:

[0106]

[0107] 1.5 The following state variables are defined:

[0108]

[0109] where, x d , are the reference trajectory and speed of the trolley movement, respectively; sec(·) represents the inverse of the cosine function;

[0110] 1.6 The following state equation of the form is constructed in combination with equation (7):

[0111]

[0112] where, represent the first-order derivatives of in equation (7), respectively; λi i = 1,2,3 represent defined auxiliary variables; represent state variables The constructor of them are as follows:

[0113]

[0114] 1.7 For the convenience of the design of unknown input observer, transform equation (8) into the following form:

[0115]

[0116] where, d m = λ2d x - λ3d θ ;

[0117] 1.8 For the actual application scenario of the crane, the angle of load swing satisfies Therefore, the auxiliary variables λ1, λ2, λ3 defined by equation (9) are bounded, and the constructor is also bounded by the same reason;

[0118] Step 2, design the unknown input estimator to estimate the matched and unmatched disturbances in the crane system, the process is as follows:

[0119] 2.1 Assume that the matched disturbance and the unmatched disturbance, and their first-order derivatives are bounded, defined as follows:

[0120]

[0121] where, and represent the upper bound of the matched disturbance and the unmatched disturbance, respectively, represent the upper bound of the first-order derivative of the matched disturbance and the unmatched disturbance;

[0122] 2.2 Perform first-order low-pass filtering transformation on the transformed crane system model equation (10), and design the unknown input observer as follows:

[0123]

[0124] where k > 0 represents the filtering coefficient; define [·] / (ks + 1) = (·) in equation (12) f ; equation (12) is transformed into:

[0125]

[0126] where, represent the original state variable after low-pass filtering transformation the first derivative of d mmf f mf mm m the result after low-pass filtering;

[0127] 2.3 Each state variable in formula (13) has the following properties:

[0128]

[0129] The expression of d mmf mf is obtained from formula (13)-(14):

[0130]

[0131] The disturbance observer is designed in combination with formula (15), as follows:

[0132]

[0133] wherein, represents the estimated value of the disturbance term d mm m ;

[0134] 2.4 It is found from formula (16) that the disturbance observer has only one filter coefficient k that can be adjusted, thereby avoiding the noise amplification problem caused by system derivation;

[0135] Step 3, design of double-layer hyperbolic approaching law, the process is as follows:

[0136] The double-layer hyperbolic approaching law is designed, as shown below:

[0137]

[0138] wherein, represents the first derivative of the sliding mode surface to be designed; γ1>0, γ2>0, k1>0, k2>0, and p represents a positive odd number; tanh(k1s) represents the hyperbolic tangent function, and the expression is as follows:

[0139]

[0140] asinh(k2s p ) represents the inverse hyperbolic sine function, and the expression is as follows:

[0141]

[0142] Step 4, design of sliding mode control law based on unknown input observer technology, the process is as follows:

[0143] ​​​​​​4.1 In combination with formula (7), the sliding surface is defined as follows:

[0144]

[0145] Wherein, a>0, b>0, c>0;

[0146] 4.2 In combination with formula (10), the derivation of formula (20) is as follows:

[0147]

[0148] 4.3 In combination with formula (16)-(17), the sliding mode controller of formula (10) based on double-layer hyperbolic reaching law is designed as:

[0149]

[0150] 4.4 The following Lyapunov function is selected:

[0151]

[0152] The derivation of the above formula is carried out, and formula (22) is substituted into it, to obtain That is, the system is stable;

[0153] 4.5 The auxiliary control law designed by formula (22) is substituted into formula (6), and the F expression is obtained as:

[0154]

[0155] Formula (24) represents the actual control law in the crane system.

[0156] In order to verify the effectiveness of the method, the control effect of the bridge crane sliding mode controller based on unknown input observer technology shown in formula (24) is simulated, and the control effect of the traditional sliding mode controller based on nonlinear disturbance observer and the state feedback controller based on generalized extended state observer are compared. The initial conditions and control parameters in the experiment are set as follows: the sampling step is 0.001s; the system parameters are selected as M=5kg, m=10kg, l=6m, g=9.8m / s 2 ; the controller parameters are selected as a=0.44, b=1.74, c=2.28, γ1=28, k1=15, γ2=15, k2=10, p=3; the expected positioning target is x d =10m; the filter coefficient k in the unknown input observer is 0.01; the matching disturbance is d x =0.5sin(t); the constant non-matching disturbance is d θ =2, the periodic non-matching disturbance is d θ =sin(t); and the initial state of the system is

[0157] Figures 2-19 is a comparison chart of simulation results of the bridge crane system with different types of persistent non-matching disturbance. Figures 2-10 is a comparison chart of simulation results of the system (8) with persistent constant non-matching disturbance. Figures 2-4 is a state curve chart x, θ, u of the system (8) with a traditional sliding mode controller based on a nonlinear disturbance observer, from which it can be seen that x enters a stable state at about 18 seconds, θ enters a stable state at 24 seconds and there is a certain jitter problem, and F has obvious input jitter problem. Figures 5-7 is a state curve chart x, θ, u of the system (8) with a feedback controller based on a generalized extended state observer, from which it can be seen that x enters a stable state at about 17 seconds, θ enters a stable state at about 21 seconds, and F enters a stable range at about 20 seconds, but there is obvious up and down fluctuation, which is not stable enough. Figures 8-10 is a state curve chart x, θ, u of the system (8) with a sliding mode controller based on unknown input observer technology of the application, from which it can be seen that x enters a stable state at about 13 seconds, θ enters a stable state at about 15 seconds, and the load does not have residual swing phenomenon, F enters a stable state at about 2 seconds, and there is no obvious chattering or fluctuation phenomenon. Figures 11-19 is a comparison chart of simulation results of the system (8) with persistent periodic non-matching disturbance. Figures 11-13 is a state curve chart x, θ, u of the system (8) with a traditional sliding mode controller based on a nonlinear disturbance observer, from which it can be seen that x enters a stable state at about 18 seconds, θ enters a stable range at 22 seconds but the load has obvious residual swing phenomenon, and F has severe chattering phenomenon. Figures 14-16 is a state curve chart x, θ, u of the system (8) with a feedback controller based on a generalized extended state observer, from which it can be seen that x enters a stable state at about 18 seconds, θ enters a stable range at about 21 seconds but the load also has obvious residual swing phenomenon, and F enters a stable range at about 22 seconds, but there is obvious up and down fluctuation, which is not stable enough. Figures 17-19The state curve x, θ, u of the system (8) is added with the sliding mode controller based on the unknown input observer technology of the application, and it can be found from the graph that x enters the stable state at about 15 seconds, θ enters the stable state at about 16 seconds, and there is no obvious residual swing phenomenon of the load, F enters the stable state at about 5 seconds, and there is no obvious chattering or fluctuation phenomenon, and the output is smooth. The simulation comparison results of the three methods can be obtained, compared with the traditional sliding mode controller based on the nonlinear disturbance observer and the feedback controller based on the generalized extended state observer, the sliding mode controller based on the unknown input observer technology of the application can ensure that the bridge crane system state quickly reaches the stable range, and has good control effect on different types of persistent non-matching disturbances. In summary, the sliding mode control method of the bridge crane based on the unknown input observer technology can effectively solve the suppression problem of the bridge crane facing persistent non-matching disturbances, and reduce the transition time of the system state to reach the steady state, so that the bridge crane has good positioning anti-swing performance.

[0158] The simulation comparison experiments given by the application show the superiority of the designed method, and obviously, the application is not limited to the above examples, and various deformations can be implemented without deviating from the basic spirit of the application and without exceeding the scope of the essential content of the application. The control method designed by the application has good positioning anti-swing control effect on the bridge crane with different persistent non-matching disturbances, and can effectively improve the working efficiency of the crane.

Claims

1. A bridge crane sliding mode control method based on unknown input observer technique, characterized in that: the method comprises the following steps: Step 1, establish the dynamic model of the bridge crane system, initialize the state and control parameters of the system, the process is as follows: 1.1 The dynamics of the overhead crane is expressed as: Where M and m represent the mass of the trolley and the mass of the load, respectively; The θ represents the acceleration of the trolley during its motion; l represents the length of the suspension rope; θ, These represent the angle, angular velocity, and angular acceleration of the load swing, respectively; F represents the control input; d x The lumped term representing the matched perturbation includes friction and unmodeled dynamics; d θ represents the lumped term of the unmatched disturbance, including air resistance and friction; g represents the acceleration due to gravity; 1.2 The acceleration of the trolley motion can be obtained by combining equations (1)-(2) as: 1.3 Substitute equation (3) into equation (2), and after mathematical transformation, the control input F is obtained as: wherein, 1.4 Define an auxiliary control variable v as: Substitute equation (5) into equation (4) and re-transform to get: 1.5 Define the following state variables: wherein, x d , respectively the reference trajectory and velocity of the trolley motion; sec(·) denotes the inverse cosine function; 1.6 Combine equation (7) to construct the following form of state equation: where denote the first derivative of respectively; λ i , i = 1, 2, 3 denote defined auxiliary variables; denote the state variables the constructor of the state variables; their concrete expressions are as follows: 1.7 Transform equation (8) into the following form: wherein d m = λ2d x - λ3d θ ; 1.8 For the practical application scenario of the crane, the angle of load swing satisfies Therefore, the auxiliary variables λ1, λ2, λ3 defined by formula (9) are bounded, and the constructor is also bounded. Step 2, design the unknown input estimator to estimate the matched and unmatched disturbances in the crane system, the process is as follows: 2.1 Assume that the matched and unmatched disturbances, as well as their first-order derivatives, are bounded, and define as follows: wherein, respectively denote upper bounds for the matched and unmatched perturbations, respectively denote upper bounds for the first derivatives of the matched and unmatched perturbations; 2.2 Perform first-order low-pass filtering transformation on the transformed crane system model equation (10), and design the unknown input observer as follows: where k > 0 denotes a filter coefficient; defining [•] / (ks+1) = (•) in equation (12) f ; equation (12) transforms into: wherein represents the original state variable a first derivative of the low-pass filtered variable d mmf v f d mf represents d mm v, d m the result of the low-pass filtering; 2.3 The state variables in equation (13) have the following properties: d is obtained from equations (13) - (14) mmf d mf the expression: Combine equation (15) to design the disturbance observer as follows: wherein represents the disturbance term d mm , d m an estimate of the disturbance term d 2.4 From equation (16), it is found that the disturbance observer has only one filter coefficient k that can be adjusted, thus avoiding the noise amplification problem caused by system derivation; Step 3, design of double-layer hyperbolic reaching law, the process is as follows: Design the double-layer hyperbolic reaching law as follows: wherein denotes the first derivative of the sliding surface to be designed; γ1>0, γ2>0, k1>0, k2>0, p denotes a positive odd number; tanh(k1s) denotes the hyperbolic tangent function, which is expressed as follows: asinh(k2s p ) denotes the inverse hyperbolic sine function, expressed as follows: Step 4, design of sliding mode control law based on unknown input observer technology, the process is as follows: 4.1 Combine equation (7) to define the sliding surface as: Where a>0, b>0, c>0; 4.2 Derive equation (20) with respect to equation (10) to get: 4.3 Design the sliding mode controller of equation (10) based on double-layer hyperbolic reaching law by combining equations (16)-(17) as: 4.4 Select the following Lyapunov function: Taking the derivative of the above equation and substituting equation (22) into it, we obtain i.e. the system is stable; 4.5 Substitute the auxiliary control law designed in equation (22) into equation (6) to get the expression of F as: Equation (24) represents the actual control law in the crane system.

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