Fixed time anti-interference sliding mode tracking control method for under-actuated bridge crane

Abstract

A fixed time anti-interference sliding mode tracking control method for an under-actuated bridge crane comprises the following steps: firstly, designing a disturbance observer which can converge in fixed time and is used for accurately estimating disturbance in a system so as to provide effective disturbance compensation information for a subsequent control strategy; secondly, in combination with the fixed time disturbance observer, a sliding mode controller is provided to ensure that the crane can accurately track a given trajectory and enable a tracking error to converge to a sliding mode surface within a fixed time. According to the invention, accurate positioning of the trolley can be realized, and load swing can be effectively inhibited in a complex environment. According to the method, the control precision and robustness of the under-actuated bridge crane in a disturbed environment can be remarkably improved, and an effective solution is provided for crane control under complex working conditions.

Description

Technical Field

[0001] The present invention relates to the field of underactuated robot control, and more specifically to a fixed-time anti-disturbance sliding mode tracking control method for an underactuated bridge crane. The present invention provides a fixed-time anti-disturbance sliding mode tracking control method for an underactuated bridge crane. Background Art

[0002] The bridge crane is a typical underactuated system, which is widely used in industry, logistics, warehousing, construction and other fields. In order to improve its production efficiency, it is necessary to be able to deliver the goods to the designated location as quickly as possible under the constraint of the smallest possible swing angle. At the same time, it will be subject to various external disturbances and uncertainties in the process. Therefore, it is another urgent need for the crane control method to have satisfactory robustness to these sudden factors.

[0003] At present, there are many methods to propose solutions to these problems. Sun Ning constructed a three-stage acceleration sway elimination trajectory with an analytical expression based on the change law of the phase trajectory and combined with constraints. Li Gang et al. proposed a time-optimal trajectory planning method, which can obtain a trajectory with global time optimality and sway elimination capability. However, these methods do not complete closed-loop control and cannot ensure that the crane accurately tracks the designed trajectory. In order to solve this problem, Ouyang Huimin et al. proposed a sliding mode control algorithm containing a nonlinear sliding surface, which can accurately track a given trajectory. Chen H et al. designed a nonlinear tracking control law based on feedback linearization technology to achieve accurate tracking of a given trajectory. Although these methods can achieve closed-loop control, they do not fully consider the impact of external disturbances and system uncertainties on control performance.

[0004] In order to solve the robustness problem of crane control, Sun Ning et al. designed a new nonlinear control law, which enables the crane to have a certain robustness while tracking the planned trajectory. Lu B et al. combined the method of extended disturbance observer to compensate for most of the disturbances, and Wu X et al. combined the finite time disturbance observer to further improve the convergence speed of the control system. However, the convergence time of existing control strategies is mostly infinite or determined by the initial conditions, and the long adjustment process of the observer will have an adverse effect on the performance of the control system. Since the convergence time is fixed and independent of the initial conditions, the fixed time control strategy has received widespread attention in recent years. At present, the research on the fixed time convergence characteristics mainly includes the design of the controller, the design of the differentiator and the design of the state observer, which is widely used in various systems, such as robotic arm systems, underactuated ground vehicles, reusable launch vehicles and multi-agent systems. However, the fixed time convergence method is currently less used in the control of underactuated bridge cranes. Summary of the invention

[0005] In order to overcome the shortcomings of the prior art, the present invention proposes a fixed-time anti-disturbance sliding mode tracking control method for an under-actuated bridge crane system affected by disturbances. This control strategy can make the tracking error converge to the sliding mode surface within a fixed time and effectively suppress the influence of external disturbances and system uncertainties on the control performance.

[0006] In order to solve the above technical problems, the present invention proposes the following technical solutions:

[0007] A fixed-time anti-disturbance sliding mode tracking control method for an underactuated bridge crane comprises the following steps:

[0008] Step 1, problem description, the dynamic equation of the two-dimensional bridge crane is as follows:

[0009]

[0010] In equations (1) and (2), M represents the mass of the trolley, m represents the mass of the weight, x represents the displacement of the trolley, θ represents the load swing angle, L and g represent the rope length and gravity acceleration respectively, and F a represents the driving force of the car, δ x represents the sum of uncertainties and external disturbances;

[0011] In practical applications, the transportation process of bridge cranes is often disturbed by the friction between the track and the trolley. However, for bridge cranes, it is difficult to accurately measure and obtain the actual friction, and it is not realistic to identify the change of friction in real time during the operation process. Therefore, the present invention includes friction and other similar factors, such as uncertain parameters and unmodeled dynamic effects, into δ x In this case, this processing not only makes the model expression more concise, but also more reasonable in practical applications. In addition, assuming that in practice, δ x and its derivative is bounded

[0012]

[0013] In formula (3): represents a priori known constant;

[0014] Step 2, using auxiliary variables to transform the crane dynamics model into a quasi-integrator chain structure;

[0015] Define the following auxiliary signals

[0016]

[0017] Using auxiliary variables (4), the crane dynamics model is transformed into a quasi-integrator chain structure.

[0018]

[0019] In formula (5), is defined as a new disturbance term, μ is the control input, and observing the second equation in (5), we find Since the weight swing angle is usually less than π / 18 rad in practice, therefore Formula (5) can be written as follows:

[0020]

[0021] It can be found that equation (6) is still highly nonlinear and only one high-order negligible term is excluded. Therefore, the system shown in equation (6) can well maintain the characteristics of the original crane dynamic equations (1) and (2);

[0022] Step 3, track trajectory selection. In practice, the crane swing angle will not exceed 10°. Therefore, by observing the auxiliary signal, we can find that X1 = x + Lln (secθ + tanθ) ≈ x + Lθ can be regarded as the horizontal position of the weight, and χ3 = -gtanθ ≈ -gθ can be regarded as the swing angle multiplied by -g. Considering the high efficiency and safety of the crane during operation, χ1 needs to track a pre-planned trajectory χ 1r

[0023] Reach the designated location smoothly and safely dx , and in the process it can be robust to external disturbances, friction and uncertainty, so χ 1r The following conditions must be met:

[0024] 3.1) The car arrives and stops at the target position p within the specified time dx During operation, the speed, acceleration and jerk are kept within the set range;

[0025] 3.2) The weight swing angle is always kept within a reasonable range during operation, and there is no residual swing when the trolley reaches the target position;

[0026] Step 4, fixed-time nonlinear disturbance observer design and analysis, design the following fixed-time nonlinear disturbance observer to estimate the disturbance term d x :

[0027]

[0028] In formula (7),

[0029] φ1(x)=「x」 1-α +「x」 1+α

[0030] φ2(x)=「x」 1-2α +「x」 2α+1

[0031] 「x」 α =|x| α sign(x), sign(x) represents the sign function, z1 and z2 represent the effect of χ4 and the disturbance term d respectively. x Observation value, α∈(0,0.5), k1, k2>0, ε>0;

[0032] For the system shown in equation (6), the disturbance observer (7) can make z1 and z2 affect χ4 and d x The observation error at a fixed time T DO Converges to the neighborhood of the origin, if the observation error e is defined d =d x -z2, then when t≥T DO When d Converges to a small area, max|e d |≤γ d ;

[0033] Step 5: Design and analysis of tracking controller. Further, the process of step 5 is:

[0034] First, define the error function

[0035]

[0036] It is found that the error function satisfies

[0037]

[0038] When the car reaches the target position Therefore, the control target of the designed controller is η i →0(i=1,2,3,4);

[0039] Next, define the sliding surface

[0040] s=η1+α2η2+α3η3+α4η4 (10)

[0041] Among them, α2, α3, and α4 can make 1+α2s+α3s 2 +α4s 3 =0 is a Hurwitz polynomial and α4>0, so if s→0 can be made, then η1, η2, η3, η4→0 can be deduced;

[0042] Define the Lyapunov function as V2 = s 2 , and we can derive

[0043]

[0044] Design μ to be:

[0045]

[0046] Among them, k3>0, k4>α4γ d ,β∈(0,1), sign(s) represents the sign function;

[0047] Substituting equation (9) and equation (12) into equation (11), we can obtain

[0048]

[0049] It is easy to prove that the sliding mode variable s converges to a region of the origin in a fixed time, then η i It will converge to the origin along the sliding surface, i=1,2,3,4.

[0050] The beneficial effects of the present invention compared with the prior art are as follows:

[0051] (1) The present invention designs a disturbance observer that can converge within a fixed time and can quickly and accurately estimate the disturbance in the system, thereby providing effective disturbance compensation information for subsequent control strategies.

[0052] (2) Combined with the fixed-time observer, the present invention adopts a sliding mode controller that can compensate for disturbance effects in real time and make the tracking error converge to the sliding mode surface within a fixed time. By introducing the disturbance compensation mechanism, the sliding mode controller can effectively reduce the system's sensitivity to external disturbances and improve the system's robustness. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is the system flow chart;

[0054] Figure 2 For bridge crane systems;

[0055] Figure 3 are x, θ and F without external disturbance a The simulation results of

[0056] Figure 4 In order to avoid interference with the observer, a sinusoidal disturbance term with an amplitude of 40 and a period of 0.5 s is added between 9 and 15 s, that is, δ x =40sin(4πt) x, θ and F a The simulation results of

[0057] Figure 5 To add a disturbance observer, a sinusoidal disturbance term with an amplitude of 40 and a period of 0.5s is added between 9 and 15s, that is, δ x=40sin(4πt) when x, θ, F a , z2 and d x The simulation results are shown in Figure 2. DETAILED DESCRIPTION

[0058] The present invention will be further described below in conjunction with the accompanying drawings.

[0059] Reference Figure 1 to Figure 5 , a fixed-time anti-disturbance sliding mode tracking control method for an underactuated bridge crane, comprising the following steps:

[0060] Step 1, problem description, the dynamic equation of the two-dimensional bridge crane is as follows:

[0061]

[0062] In equations (1) and (2), M represents the mass of the trolley, m represents the mass of the weight, x represents the displacement of the trolley, θ represents the load swing angle, L and g represent the rope length and gravity acceleration respectively, and F a represents the driving force of the car, δ x represents the sum of uncertainties and external disturbances;

[0063] In practical applications, the transportation process of a bridge crane is often disturbed by the friction between the track and the trolley. However, for a bridge crane, it is difficult to accurately measure and obtain the actual friction, and it is not realistic to identify the change of friction in real time during the operation. Therefore, this embodiment includes friction and other similar factors, such as uncertain parameters and unmodeled dynamic effects, into δ x This treatment not only makes the model expression more concise, but also more reasonable in practical applications. In addition, assuming that in practice, δ x and its derivative is bounded

[0064]

[0065] In formula (3): represents a priori known constant;

[0066] Step 2: Use auxiliary variables to transform the crane dynamics model into a quasi-integrator chain structure and define the following auxiliary signals:

[0067]

[0068] Using auxiliary variables (4), the crane dynamics model is transformed into a quasi-integrator chain structure.

[0069]

[0070] In formula (5), is defined as a new disturbance term, μ is the control input. Observing the second equation in (5), we find Since the weight swing angle is usually less than π / 18 rad in practice, therefore Formula (5) can be written as follows:

[0071]

[0072] It can be found that equation (6) is still highly nonlinear and only one high-order negligible term is excluded; therefore, the system shown in equation (6) can well maintain the characteristics of the original crane dynamic equations (1) and (2);

[0073] Step 3, track selection,

[0074] In practice, the crane's swing angle will not exceed 10°, so by observing the auxiliary signal, we can find that χ1=x+Lln(secθ+tanθ)≈x+Lθ can be regarded as the horizontal position of the weight, and χ3=-gtanθ≈-gθ can be regarded as the swing angle multiplied by -g. Considering the high efficiency and safety of the crane during operation, χ1 needs to track a pre-planned trajectory χ 1r Reach the designated location smoothly and safely dx , and in the process it can be robust to external disturbances, friction and uncertainty, so χ 1r The following conditions must be met:

[0075] 3.1) The car arrives and stops at the target position p within the specified time dx During operation, the speed, acceleration and jerk are kept within a certain range.

[0076] 3.2) The weight swing angle is always kept within a reasonable range during operation, and there is no residual swing when the trolley reaches the target position;

[0077] Step 4, fixed-time nonlinear disturbance observer design and analysis, design the following fixed-time nonlinear disturbance observer to estimate the disturbance term d x :

[0078]

[0079] In formula (7),

[0080] φ1(x)=「x」 1-α +「x」 1+α

[0081] φ2(x)=「x」 1-2α +「x」 2α+1

[0082] 「x」 α =|x| α sign(x), sign(x) represents the sign function, z1 and z2 represent the effect of χ4 and the disturbance term d respectively. x Observation value, α∈(0,0.5), k1, k2>0, ε>0;

[0083] For the system shown in equation (6), the disturbance observer (7) makes z1 and z2 affect χ4 and d x The observation error at a fixed time T DO Converges to the neighborhood of the origin, if the observation error e is defined d =d x -z2, then when t≥T DO When d Converges to a small area, max|e d |≤γ d ;

[0084] Step 5: Design and analysis of tracking controller. First, define the error function

[0085]

[0086] It is found that the error function satisfies

[0087]

[0088] When the car reaches the target position Therefore, the control target of the designed controller is η i →0, i=1,2,3,4;

[0089] Next, define the sliding surface

[0090] s=η1+α2η2+α3η3+α4η4 (10)

[0091] Among them, α2, α3, and α4 can make 1+α2s+α3s 2 +α4s 3 =0 is a Hurwitz polynomial and α4>0, so if s→0 can be made, then η1, η2, η3, η4→0 can be deduced;

[0092] Define the Lyapunov function as V2 = s 2 , and we can derive

[0093]

[0094] Design μ to be:

[0095]

[0096] Among them, k3>0, k4>α4γ d ,β∈(0,1), sign(s) represents the sign function;

[0097] Substituting equation (9) and equation (12) into equation (11), we can obtain

[0098]

[0099] It is easy to prove that the sliding mode variable s converges to a region of the origin in a fixed time, then η i It will converge to the origin along the sliding surface, i=1,2,3,4.

[0100] Step 6: Simulation and result analysis

[0101] In order to prove the effectiveness of the method proposed in the present invention, simulation is carried out in the MATLAB environment to verify the performance of the proposed method. The relevant parameters in the simulation are set as follows:

[0102] The physical parameters of the trolley are set as m = 20 kg, M = 30 kg, L = 2 m, g = 9.8 m / s 2 , the target position is set to p dx =0.6, X that meets the conditions 1r Select as

[0103]

[0104] Where a0 = -487 / 8, a1 = 227 / 4, a2 = 55 / 2, a3 = -215 / 8, a4 = 11 / 8, a5 = 17 / 8, a6 = 1 / 2

[0105] b1=88, b2=-115 / 2, b3=0, b4=15 / 2, b5=0, τ=πt / 2T*, T*=3.9505.

[0106] The relevant parameters of the disturbance observer are k1=2, k2=1.5, ε=0.002, α=0.2, and the relevant parameters in the controller are α2=3, α3=3, α4=1, k3=1.5, k4=0.9, β=0.1.

[0107] For comparative analysis, the present invention performs simulations in the following three situations to verify the performance of the proposed control method:

[0108] (1) Case 1: No external disturbances.

[0109] (2) Case 2: No disturbance observer is added, and a sinusoidal disturbance term is added between 9 and 15 seconds, with an amplitude of 40 and a period of 0.5 seconds, that is, δ x=40sin(4πt).

[0110] (3) Case 3: Add a disturbance observer and add a sinusoidal disturbance term between 9 and 15 seconds, with an amplitude of 40 and a period of 0.5 seconds, that is, δ x =40sin(4πt).

[0111] from Figure 3 It can be seen that without considering external disturbances, the designed trajectory can reach the desired position as quickly as possible without the swing amplitude exceeding the preset limit, and there is no residual swing when the trolley reaches the destination. The overall control performance is good, which shows that the proposed method can work well in some cases where the swing assumption is not strictly considered.

[0112] from Figure 4 , Figure 5 It can be seen that after the disturbance is added, the displacement and swing angle of the system are disturbed, but after the disturbance observer is added, the displacement and swing angle are not affected by the disturbance, and z2 can observe d well. x , which shows that the proposed disturbance observer can well estimate and eliminate the disturbance imposed on the system. These results to some extent show that the proposed control method not only has good tracking control performance, but also has strong robustness to disturbances.

[0113] The above description is about the excellent effect shown by an embodiment of the present invention. Obviously, the present invention is not limited to the above embodiment, and various modifications can be made to it without departing from the basic spirit of the present invention and without exceeding the scope of the essential content of the present invention.

Claims

1. A fixed-time anti-disturbance sliding mode tracking control method for an underactuated bridge crane, characterized in that: The method comprises the following steps: Step 1, Problem Description The dynamic equation of a two-dimensional bridge crane is as follows: In equations (1) and (2), M represents the mass of the trolley, m represents the mass of the weight, x represents the displacement of the trolley, θ represents the load swing angle, L and g represent the rope length and gravity acceleration respectively, and F a represents the driving force of the car, δ x represents the sum of uncertainties and external disturbances; Assume that in practice, δ x and its derivative is bounded In formula (3): represents a priori known constant; Step 2: Use auxiliary variables to transform the crane dynamics model into a quasi-integrator chain structure and define the following auxiliary signals: Using auxiliary variables (4), the crane dynamics model is transformed into a quasi-integrator chain structure. In formula (5), is defined as a new disturbance term, μ is the control input, and observing the second equation in (5), we find Since the weight swing angle is less than π / 18 rad in practice, therefore Formula (5) can be written as follows: Step 3, track trajectory selection, χ1=x+Lln(secθ+tanθ)≈x+Lθ, regarded as the horizontal position of the weight, χ3=-gtanθ≈-gθ regarded as the swing angle multiplied by -g, χ1 tracks a pre-planned trajectory χ 1r Reach the designated location smoothly and safely dx , and in the process it can be robust to external disturbances, friction and uncertainty factors, so χ 1r The following conditions must be met: 3.1) The car arrives and stops at the target position p within the specified time dx During operation, the speed, acceleration and jerk are kept within the set range; 3.2) The weight swing angle is always kept within a reasonable range during operation, and there is no residual swing when the trolley reaches the target position; Step 4, fixed-time nonlinear disturbance observer design and analysis, design the following fixed-time nonlinear disturbance observer to estimate the disturbance term d x : In formula (7), φ1(x)=「x」 1-α +「x」 1+α φ2(x)=「x」 1-2α +「x」 2α+1 「x」 α =|x| α sign(x), sign(x) represents the sign function, z1 and z2 represent the effect of χ4 and the disturbance term d respectively. x Observation value, α∈(0,0.5), k1, k2>0, ε>0; For the system shown in equation (6), the disturbance observer (7) makes z1 and z2 affect χ4 and d x The observation error at a fixed time T DO Converges to the neighborhood of the origin, if the observation error e is defined d =d x -z2, then when t≥T DO When d Converges to a small area, max|e d |≤γ d ; Step 5: Design and analysis of tracking controller.

2. The fixed-time anti-disturbance sliding mode tracking control method for an underactuated bridge crane according to claim 1, characterized in that: The process of step 5 is: First, define the error function It is found that the error function satisfies When the car reaches the target position Therefore, the control target of the designed controller is η i →0, i=1,2,3,4; Next, define the sliding surface s=η1+α2η2+α3η3+α4η4 (10) Among them, α2, α3, α4 make 1+α2s+α3s 2 +α4s 3 =0 is a Hurwitz polynomial and α4>0, so if s→0 can be made, then η1, η2, η3, η4→0 can be deduced; Define the Lyapunov function as V2 = s 2 , and we can derive Design μ to be: Among them, k3>0, k4>α4γ d ,β∈(0,1), sign(s) represents the sign function; Substituting equation (9) and equation (12) into equation (11), we can obtain It is easy to prove that the sliding mode variable s converges to a region of the origin in a fixed time, then η i It will converge to the origin along the sliding surface, i=1,2,3,4.

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