Global finite time ship trajectory tracking control method considering input and output limitation

Abstract

The invention discloses a global finite time ship trajectory tracking control method considering input and output limitation. The global finite time ship trajectory tracking control method comprises the following steps: S1, establishing an under-actuated ship mathematical model; s2, obtaining a trajectory tracking error based on the actual trajectory and the expected trajectory, limiting the trajectory tracking error to obtain a new error, and designing an expected longitudinal velocity and an expected transverse velocity based on the new error; s3, designing a first finite time sliding mode surface and a second finite time sliding mode surface; s4, longitudinal control force and yawing control torque are designed; and S5, the finite time sliding mode surface is adjusted to be converged through the longitudinal control force and the yawing control moment, then trajectory tracking errors are converged, and finally ship trajectory tracking is achieved. The under-actuated ship mathematical model is established under the condition of considering the input limitation, and the trajectory tracking error is limited under the condition of considering the output limitation, so that the transient and steady-state performance is ensured, the navigation trajectory is within a preset range, and the ship navigation safety is ensured.

Description

Technical Field

[0001] The present invention relates to the technical field of ship trajectory control, and in particular to a global finite-time ship trajectory tracking control method considering input and output constraints. Background Art

[0002] In the past 20 years, with the widespread use of underactuated ships in environmental surveys, ocean sampling and maritime rescue, the tracking control of underactuated ships has received increasing attention in the fields of control engineering and marine engineering. In order to complete the above tasks, it is usually necessary to accurately control the ship and improve its control performance. Since underactuated ships have no control input in the lateral direction, and the external disturbances they face are time-varying disturbances, and the problem of model uncertainty increases the difficulty of accurate control of ships. The current control methods used to solve the above problems do not meet the actual control conditions because they do not consider the input and output restrictions, resulting in insufficient control accuracy. At the same time, they do not consider accelerating the error convergence rate, which will cause the error convergence rate to slow down and affect navigation safety. Therefore, how to effectively accelerate the error convergence rate under the premise of considering the input and output restrictions, effectively resist the influence of unknown interference, and improve the trajectory tracking rate and accuracy of the ship is an urgent problem to be solved. Summary of the invention

[0003] The present invention provides a global finite-time ship trajectory tracking control method considering input and output limitations, so as to overcome the technical problems that the prior art does not consider the input and output limitations, does not conform to the actual control situation, leads to insufficient control accuracy, and does not consider accelerating the error convergence rate, thus slowing down the error convergence rate and affecting navigation safety.

[0004] In order to achieve the above object, the technical solution of the present invention is:

[0005] A global finite-time ship trajectory tracking control method considering input and output constraints, the specific steps include:

[0006] S1: Establish a mathematical model of underactuated ship under the condition of input limitation;

[0007] S2: obtaining a trajectory tracking error based on the actual trajectory and the expected trajectory of the mathematical model of the underactuated ship, and then limiting the trajectory tracking error in combination with a preset performance function and an obstacle Lyapunov function under the condition of considering output limitation to obtain a new error, and designing an expected longitudinal speed and an expected lateral speed based on the new error;

[0008] S3: Design a first linear anti-saturation compensator and a second linear anti-saturation compensator, and design a first finite-time sliding surface based on the actual longitudinal velocity, the expected longitudinal velocity and the first state variable in the first linear anti-saturation compensator of the mathematical model of the underactuated ship in combination with the finite-time convergence theory; design a second finite-time sliding surface based on the actual lateral velocity, the expected lateral velocity and the second state variable in the second linear anti-saturation compensator of the mathematical model of the underactuated ship;

[0009] S4: designing a first disturbance observer and a second disturbance observer, obtaining a first total disturbance estimate based on the first disturbance observer, and obtaining a second total disturbance estimate based on the second disturbance observer; designing a longitudinal control force based on the first finite time sliding surface and the first total disturbance estimate; designing a yaw control torque based on the second finite time sliding surface and the second total disturbance estimate;

[0010] S5: The first finite time sliding mode surface and the second finite time sliding mode surface are adjusted by the longitudinal control force and the yaw control torque, so that the first finite time sliding mode surface and the second finite time sliding mode surface converge, and then the trajectory tracking error converges, and finally the ship trajectory tracking is realized.

[0011] Furthermore, in S1, under the condition of input limitation, the mathematical model of the underactuated ship is established as follows:

[0012]

[0013] in

[0014]

[0015] Where x, y, ψ are the forward displacement, drift displacement and heading angle of the ship respectively; u, v, r are the longitudinal velocity, transverse velocity and bow angular velocity of the ship in the appendage coordinate system respectively; τ usat is the actual longitudinal control force of the ship after considering the limitation, τ rsat is the actual yaw control moment after considering the limitation; m 11 ,m 22 ,m 33 are inertia parameters respectively; d 11 ,d 22 ,d 33 are the hydrodynamic parameters respectively; Δf u ,Δf v ,Δf r are the uncertainties of the ship model; τ wu ,τ wv ,τ wr are external unknown disturbances respectively; τ im is the upper bound of the control input; τ uc is the ideal longitudinal control force, τrc is the ideal bow control torque; Δτ u is the difference between the actual longitudinal control force and the ideal longitudinal control force, Δτ r It is the difference between the actual yaw control torque and the ideal yaw control torque.

[0016] Further, in S2, the trajectory tracking error is obtained based on the actual trajectory and the expected trajectory of the mathematical model of the underactuated ship, and then the trajectory tracking error is limited by combining the preset performance function and the obstacle Lyapunov function under the condition of considering the output limitation to obtain a new error. The process of designing the expected longitudinal speed and the expected lateral speed based on the new error is as follows:

[0017] S21: The trajectory tracking error is obtained based on the actual trajectory and expected trajectory of the mathematical model of the underactuated ship, which is expressed as:

[0018]

[0019] In the formula, x d and d are the ordinate and abscissa of the expected trajectory of the ship respectively; e and e are the longitudinal trajectory tracking error and the lateral trajectory tracking error respectively; x, y are the forward displacement and the lateral drift displacement of the ship respectively, i.e., the output;

[0020] S22: Under the condition of considering the output limitation, the trajectory tracking error is limited by combining the preset performance function and the obstacle Lyapunov function, and the obtained new error is expressed as:

[0021]

[0022] In the formula, α2, T are positive parameters that need to be designed; α1, p, All are positive parameters that need to be designed;

[0023] S23: The expected longitudinal speed and expected lateral speed designed based on the new error are expressed as:

[0024]

[0025] Furthermore, in S3, the designed first linear anti-saturation compensator is expressed as:

[0026]

[0027] Actual longitudinal velocity u and expected longitudinal velocity u based on mathematical model of underactuated ship d And the process of designing the first finite-time sliding surface with the first state variable ξ1 in the first linear anti-saturation compensator is:

[0028] make

[0029] u e =uu d -ξ1 (22)

[0030] The designed first finite-time sliding surface s1 is shown in formula (23):

[0031]

[0032] In the formula, λ1,λ2 are positive parameters that need to be designed; ξ1 is the first state variable;

[0033] By taking the derivative of formula (23) and combining formula (1) and formula (22), we can obtain:

[0034]

[0035] The designed second linear anti-saturation compensator is expressed as:

[0036]

[0037] Where ξ2 is the second state variable;

[0038] Based on the actual lateral velocity v, the expected lateral velocity v d And the process of designing the second finite-time sliding mode surface with the second state variable ξ2 in the second linear anti-saturation compensator is:

[0039] make

[0040]

[0041] The designed second finite-time sliding surface s2 is shown in formula (27):

[0042]

[0043] make

[0044]

[0045] but

[0046]

[0047] Using dynamic surface technology and introducing a new variable f r As the output of the first-order low-pass filter of f, in order to avoid the differential explosion problem caused by the derivative of f, the expression of the dynamic surface technology is as follows:

[0048]

[0049] Where, T r is the design parameter, and Tr >0;

[0050] By taking the derivative of formula (27) and combining formula (1) and formula (26), we can obtain:

[0051]

[0052] In the formula, λ3 and λ4 are positive parameters that need to be designed.

[0053] Furthermore, in S4, the designed first disturbance observer is shown in formula (32):

[0054]

[0055] Where ω1 is a positive design parameter, β1 is the state variable of the first disturbance observer; μ1 = τ wu -Δf u , is the estimated value of μ1;

[0056] The process of designing the longitudinal control force based on the first finite-time sliding surface and the first total disturbance estimate is:

[0057] Combining equations (21), (24) and (32), τ is designed as:

[0058]

[0059] In the formula, c5 and c6 are both positive design parameters;

[0060] The designed second disturbance observer is shown in formula (34):

[0061]

[0062] Where ω2 is a positive design parameter, β2 is the state variable of the disturbance observer, and μ2 = τ wr -Δf r , is the estimated value of μ2;

[0063] The process of designing the yaw control torque based on the second finite time sliding surface and the second total disturbance estimate is:

[0064] Combining equations (25), (31) and (34), we can rc Designed for:

[0065]

[0066] Where c7 and c8 are both positive design parameters.

[0067] Beneficial effects: The present invention establishes an underactuated ship mathematical model under the condition of input limitation, obtains a trajectory tracking error based on the actual trajectory and expected trajectory of the underactuated ship mathematical model, and then limits the trajectory tracking error in combination with a preset performance function and an obstacle Lyapunov function under the condition of output limitation, thereby ensuring its transient and steady-state performance, so that the navigation trajectory is always within a preset range, and ensuring the navigation safety of the ship; a first disturbance observer and a second disturbance observer are designed to observe the total disturbance formed by internal and external disturbances, and add it to the design of the longitudinal control force and the bow control torque to offset the influence of the disturbance; for the problem of slow convergence speed, the first finite-time sliding surface and the second finite-time sliding surface are designed in combination with the finite-time convergence theory to achieve global finite-time convergence, so that the error can achieve finite-time convergence whether in the approach stage or the sliding stage. BRIEF DESCRIPTION OF THE DRAWINGS

[0068] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative labor.

[0069] Figure 1 It is a flow chart of a global finite-time ship trajectory tracking control method considering input and output constraints in the present invention;

[0070] Figure 2 It is a design flow chart of a global finite-time ship trajectory tracking control method considering input and output constraints in the present invention;

[0071] Figure 3 A comparison diagram of the expected trajectory and the actual trajectory of a ship in a plane in an embodiment of the present invention;

[0072] Figure 4 This is a comparison diagram of the effect of lateral position error in an embodiment of the present invention;

[0073] Figure 5 This is a comparison diagram of the longitudinal position error effect in an embodiment of the present invention;

[0074] Figure 6 A comparison diagram of lateral velocity errors in an embodiment of the present invention;

[0075] Figure 7 A longitudinal velocity error comparison diagram in an embodiment of the present invention;

[0076] Figure 8 is a graph of the longitudinal control force input in an embodiment of the present invention;

[0077] Fig. 9 is a graph showing the yaw control torque input curve in an embodiment of the present invention;

[0078] Fig.10 This is a disturbance observer estimation curve diagram in an embodiment of the present invention. DETAILED DESCRIPTION

[0079] In order to make the purpose, technical solution and advantages of the embodiments of the present invention clearer, the technical solution in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are 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.

[0080] When a ship is sailing on the sea, it will face various internal and external disturbances. External disturbances are usually caused by the wind, waves and currents acting on the ship, such as deviation, speed change and shaking; internal disturbances are caused by the model uncertainty caused by changes in ship parameters due to various consumptions such as cargo loading and unloading, fuel and food. Therefore, internal and external disturbances should be considered when designing a ship controller; at the same time, making the ship quickly track to the reference trajectory from any initial position has always been a key research content in the field of ship motion control, but the increase in convergence rate will inevitably increase the control input. Due to the inherent characteristics of the ship's actuator, there will be certain limits on the control input. Therefore, input restrictions must be taken into account when designing a ship controller, otherwise it will be out of touch with reality. In addition to the above issues that need to be paid attention to, when a ship is performing a certain actual task, it often needs to accurately track the reference trajectory, requiring the ship to have good transient and steady-state performance under the action of the controller.

[0081] Based on the above situation, this embodiment provides a global finite-time ship trajectory tracking control method considering input and output constraints, such as Figure 1 and Figure 2 As shown, the specific steps include:

[0082] S1: Establish a mathematical model of underactuated ship under the condition of input limitation;

[0083] S2: obtaining a trajectory tracking error based on the actual trajectory and the expected trajectory of the mathematical model of the underactuated ship, and then limiting the trajectory tracking error in combination with a preset performance function and an obstacle Lyapunov function under the condition of considering output limitation to obtain a new error, and designing an expected longitudinal speed and an expected lateral speed based on the new error;

[0084] S3: Design a first linear anti-saturation compensator and a second linear anti-saturation compensator, and design a first finite-time sliding surface based on the actual longitudinal velocity, the expected longitudinal velocity and the first state variable in the first linear anti-saturation compensator of the mathematical model of the underactuated ship in combination with the finite-time convergence theory; design a second finite-time sliding surface based on the actual lateral velocity, the expected lateral velocity and the second state variable in the second linear anti-saturation compensator of the mathematical model of the underactuated ship;

[0085] S4: designing a first disturbance observer and a second disturbance observer, obtaining a first total disturbance estimate based on the first disturbance observer, and obtaining a second total disturbance estimate based on the second disturbance observer; designing a longitudinal control force based on the first finite time sliding surface and the first total disturbance estimate; designing a yaw control torque based on the second finite time sliding surface and the second total disturbance estimate;

[0086] S5: The first finite time sliding mode surface and the second finite time sliding mode surface are adjusted by the longitudinal control force and the yaw control torque, so that the first finite time sliding mode surface and the second finite time sliding mode surface converge, and then the trajectory tracking error converges, and finally the ship trajectory tracking is realized.

[0087] In a specific embodiment, in S1, under the condition of input limitation, the mathematical model of the underactuated ship is established as follows:

[0088]

[0089] in

[0090]

[0091] Where x, y, ψ are the forward displacement, drift displacement and heading angle of the ship respectively; u, v, r are the longitudinal velocity, transverse velocity and bow angular velocity of the ship in the appendage coordinate system respectively; τ usat is the actual longitudinal control force of the ship after considering the limitation, τ rsat is the actual yaw control moment after considering the limitation; m 11 ,m 22 ,m 33 are inertia parameters respectively; d 11 ,d 22 ,d 33 are the hydrodynamic parameters respectively; Δf u ,Δf v ,Δf r are the uncertainties of the ship model; τ wu ,τ wv ,τ wr are external unknown disturbances respectively; τ im To control the upper limit of the input, that is, the maximum value allowed for the input, in this embodiment, by τ imLimit the input to a maximum value; τ uc is the ideal longitudinal control force, τ rc is the ideal bow control torque; Δτ u is the difference between the actual longitudinal control force and the ideal longitudinal control force, Δτ r It is the difference between the actual yaw control torque and the ideal yaw control torque.

[0092] In a specific embodiment, in S2, a trajectory tracking error is obtained based on the actual trajectory and the expected trajectory of the mathematical model of the underactuated ship, and then the trajectory tracking error is limited by combining the preset performance function and the obstacle Lyapunov function under the condition of considering the output limitation to obtain a new error. The process of designing the expected longitudinal speed and the expected lateral speed based on the new error is as follows:

[0093] S21: The trajectory tracking error is obtained based on the actual trajectory and expected trajectory of the mathematical model of the underactuated ship, which is expressed as:

[0094]

[0095] In the formula, x d and d are the ordinate and abscissa of the expected trajectory of the ship respectively; e and e are longitudinal trajectory tracking error and lateral trajectory tracking error respectively;

[0096] Specifically, in this embodiment, it is assumed that the expected trajectory of the ship (x d ,y d ) is smooth and has first- and second-order derivatives.

[0097] S22: Under the condition of considering the output limitation, the trajectory tracking error is limited by combining the preset performance function and the obstacle Lyapunov function, and the obtained new error is expressed as:

[0098]

[0099] In the formula, α2, T are positive parameters that need to be designed; α1, p, All are positive parameters that need to be designed;

[0100] Specifically, in formula (3), x is the output. In this embodiment, by calculating x in the trajectory tracking error, e is restricted, and because x d is bounded, thereby achieving the restriction on the output x.

[0101] S23: The expected longitudinal speed and expected lateral speed designed based on the new error are expressed as:

[0102]

[0103] In this embodiment, since the output needs to be limited, according to formula (4), if η zx and η zy If convergence can be achieved, then η zx and η zy Bounded, if η zx and η zy is bounded, then -ρ1k z <x e <ρ1k z , -ρ2k z <y e <ρ2k z , the new error converges, and the output constraints can be satisfied.

[0104] By taking the derivative of formula (4), we can get:

[0105]

[0106] According to formula (8), if we want to make η zx and η zy tend to 0, then and Designed for:

[0107]

[0108] In the formula, c1, c2, c3, c4 are all positive parameters that need to be designed;

[0109] Combining equation (1), equation (3) and equation (9), we can get:

[0110]

[0111] When u approaches u d , v approaches v d When u e and v e When both tend to 0, formula (8) can be simplified as:

[0112]

[0113] In this embodiment, Lemma 1 is given: For any ε>0, the following inequality exists:

[0114] 0≤|s|-stanh(s / ε)≤0.2785ε (12)

[0115] Where s is an arbitrary variable.

[0116] Lemma 2: Consider a system x = f(x), f(0) = 0, x∈Rn , where f(x) is a continuous function. Assume that there exists a continuous Lyapunov function V:D→R that satisfies the following conditions: V is a positive definite function, and there exists a real number θ l > 0 and an open-loop neighborhood near the origin satisfies Then the system is practically finite-time stable, and the stability time T r satisfy:

[0117]

[0118] Among them, V(0) is the initial value of the system V(x).

[0119] Construct the Lyapunov function as shown in equation (13) to prove that η zx and η zy Can achieve convergence:

[0120]

[0121] Deriving formula (13) and combining formula (11) with Lemma 1, we can obtain:

[0122]

[0123] in

[0124]

[0125] From Lemma 2, we can see that V1 can achieve finite time convergence, and the convergence time T r1 satisfy:

[0126]

[0127] By taking the derivative and simplifying formula (7), we can get:

[0128]

[0129] In a specific embodiment, considering that in the actual ship maneuvering process, the output of the controller is often limited by the actuator, and excessive control input will cause the propeller and rudder angle of the ship to fail to execute, so in order to solve this problem, this embodiment uses a linear anti-saturation compensator to handle input saturation. The general expression of the linear anti-saturation compensator is shown in formula (20):

[0130]

[0131] In the formula, k1 and k2 are the parameters to be designed, and ξ is the state variable of the compensator.

[0132] In a specific embodiment, based on the general expression of a linear anti-saturation compensator, in S3, the designed first linear anti-saturation compensator is expressed as:

[0133]

[0134] Actual longitudinal velocity u and expected longitudinal velocity u based on mathematical model of underactuated ship d And the process of designing the first finite-time sliding surface with the first state variable ξ1 in the first linear anti-saturation compensator is:

[0135] make

[0136] u e =uu d -ξ1 (22)

[0137] The designed first finite-time sliding surface s1 is shown in formula (23):

[0138]

[0139] Where λ1 and λ2 are positive parameters to be designed; ξ1 is the first state variable, the purpose of which is to eliminate the nonlinear term Δτ caused by input saturation. u ;

[0140] By taking the derivative of formula (23) and combining formula (1) and formula (22), we can obtain:

[0141]

[0142] The designed second linear anti-saturation compensator is expressed as:

[0143]

[0144] Where ξ2 is the second state variable, the purpose of which is to eliminate the nonlinear term Δτ caused by input saturation. r ;

[0145] Based on the actual lateral velocity v, the expected lateral velocity v d And the process of designing the second finite-time sliding mode surface with the second state variable ξ2 in the second linear anti-saturation compensator is:

[0146] make

[0147]

[0148] The designed second finite-time sliding surface s2 is shown in formula (27):

[0149]

[0150] For the convenience of subsequent writing,

[0151]

[0152] but

[0153]

[0154] Specifically, in order to avoid the differential explosion phenomenon when directly derivation of f, in this embodiment, the dynamic surface technology is used. Since the output of the first-order filter can avoid directly derivation of f, a new variable f is introduced. r As the output of the first-order low-pass filter of f, in order to avoid the differential explosion problem caused by the derivative of f, the expression of the dynamic surface technology is as follows:

[0155]

[0156] Where, T r is the design parameter, and T r >0;

[0157] By taking the derivative of formula (27) and combining formula (1) and formula (26), we can obtain:

[0158]

[0159] In the formula, λ3 and λ4 are positive parameters that need to be designed.

[0160] In this embodiment, it is assumed that the change rates of the external disturbance and internal disturbance to the ship are bounded and satisfy:

[0161]

[0162] Among them, D1 and D2 are both greater than 0.

[0163] Based on the above assumptions, in a specific embodiment, in S4, the designed first disturbance observer is shown in formula (32):

[0164]

[0165] Where ω1 is a positive design parameter, β1 is the state variable of the disturbance observer; μ1 = τ wu -Δf u , is the estimated value of μ1;

[0166] In this embodiment, the first disturbance observer is used to observe the total disturbance μ1, and the observed value is Add to τ uc To compensate for the impact of disturbances during the design process;

[0167] The process of designing the longitudinal control force based on the first finite-time sliding surface and the first total disturbance estimate is:

[0168] Combining equations (21), (24) and (32), we can get τ uc Designed for:

[0169]

[0170] In the formula, c5 and c6 are both positive design parameters;

[0171] The designed second disturbance observer is shown in formula (34):

[0172]

[0173] Where ω2 is a positive design parameter, β2 is the state variable of the disturbance observer, and μ2 = τ wr -Δf r , is the observed value of μ2;

[0174] The process of designing the yaw control torque based on the second finite time sliding surface and the second total disturbance estimate is:

[0175] Combining equations (25), (31) and (34), τ is designed as:

[0176]

[0177] Where c7 and c8 are positive design parameters.

[0178] In this embodiment, in order to verify the stability of the designed controller, the Lyapunov function shown in formula (37) is defined:

[0179]

[0180] In the formula,

[0181] Taking the derivative of both sides of equation (37) and substituting them into equations (24), (33), (31) and (35), we can obtain:

[0182]

[0183] In formula (38),

[0184]

[0185] because

[0186]

[0187] Combining equations (32), (34), (39) and (40), equation (38) can be simplified as follows:

[0188]

[0189] in

[0190]

[0191] According to formula (41), we can get:

[0192]

[0193] in, when When Therefore, according to Lemma 2, V2 can converge to the set region in a finite time. And time T r2 satisfy:

[0194]

[0195] Since V2 is bounded, s1, s2, are bounded, so u e , v e is also bounded, and because μ1 and μ2 are bounded, so is also bounded, thus ensuring that all error signals in the underactuated ship trajectory tracking control system are consistent and ultimately bounded.

[0196] In this embodiment, the trajectory tracking error u e and v e Perform a finite time stability analysis as follows:

[0197] Since s1 and s2 are bounded, the derivatives of s1 and s2 are also bounded, assuming that they are both less than Δ s ,but

[0198]

[0199] Define the following Lyapunov function, as shown in formula (48):

[0200]

[0201] By taking the derivative of formula (48) and combining it with formula (47), we can obtain:

[0202]

[0203] According to Lemma 1 and Young's inequality, equation (49) can be transformed into:

[0204]

[0205] in

[0206]

[0207] θ3=Δ s 2 +0.2785λ2+0.2785λ4 (23)

[0208] According to Lemma 2, V3 can achieve finite time convergence, and time T r3 satisfy:

[0209]

[0210] In summary, the trajectory tracking error u e and v e Finite-time convergence can be achieved in both the approaching stage and the sliding stage, thus achieving the goal of global finite-time convergence.

[0211] In order to verify the effectiveness of the method proposed in this paper, a computer simulation experiment was conducted with a patrol ship named BAY CLASS of Singapore Customs as the simulation object. The ship is 38m long and has a mass of m=1.18×10 5 kg, other parameters are m 11 =1.2×10 5 kg, m 22 =1.779×10 5 kg, m 33 =6.36×10 7 kg, d 11 =2.15×10 4 kg / s,d 22 =1.47×10 5 kg / s,d 33 =8.02×10 6 kg / s.

[0212] In order to verify the effect of the ship under the control strategy proposed in this invention, this simulation experiment is divided into two groups: the first group does not use the finite time sliding surface and does not consider the output limit. d With v d , sliding surface, control law, parameters The adaptive law is shown in equations (55), (56), (57) and (58); the second group is the effect of the control strategy of the present invention, and the effect of the first group is set to the first group, and the effect of the second group is set to the second group.

[0213]

[0214] In order to simulate the sea conditions, the external time-varying disturbance is expressed by equation (59):

[0215]

[0216] Set the parameters c1 = 0.9, c2 = 0.6, c3 = 4, c4 = 0.4, c5 = 5 × 10 5 , c6=2×10 7 , c7=5×10 2 , c8=5×10 4 , k1=2, k2=0.2, k3=1, k4=1, λ1=10, λ2=2, λ3=5, λ4=3, ε1=ε3=0.1, ε2=ε4=0.05, α1=15, α2=0.5, T=20,T r =0.1, p=5, ω1=2, ω2=4, η1=5×10 4 ,η2=5×10 3 ,η3=5×10 3 ,η4=5.6×10 3 ,σ1=2×10 -7 ,σ2=1.7×10 -6 ,σ3=1×10 -1 ,σ4=1.2×10 -7 .

[0217] Let the uncertainty of the model be Δf u =0.2d 11 u 2 +0.1d 11 u 3 , Δf v =0.2d 22 v 2 +0.1d 22 v 3 , Δf r =0.2d 33 r 2 +0.2d 33 r 3 . Set the input upper limit τ um =6.2×10 6 N, τ rm =4×10 8 (N·m).

[0218] Set the expected sine trajectory to x d =300sin(0.03t),y d =10t, the initial position and velocity of the ship are selected as (x0; y0; ψ0; u0; v0; r0) = (40; 0; 0; 0.1; 0; 0).

[0219] Specifically, the simulation results are as follows: Figures 3 to 10 As shown, from Figure 3 It can be seen that under the action of different control strategies, the ship can complete trajectory tracking when the input is restricted.

[0220] First, from Figure 4 and Figure 5 It can be seen that under the control of the second group of control strategies, the convergence rate of the lateral position error and the longitudinal position error of the ship is faster than that of the first group, and it has better transient and steady-state performance. Figure 6 and Figure 7 It can be seen from the figure that the speed tracking error of the ship can converge to near 0 under the two different control strategies. Figure 6 It can be seen that the first group of control strategies fluctuated around 115s, while the second group had almost no fluctuation. Figure 8 and Fig. 9 It can be seen that the longitudinal control forces of the two control strategies are almost the same, but when the control force is limited, the bow control torque of the first group fluctuates violently, while the second group also has almost no change, which reflects the superiority of the control strategy of the present invention. Fig.10 It can be seen that the disturbance observer can better estimate the sum of the external disturbance and the uncertainty of the internal model μ i , i = 1, 2, and add this to the design of the controller to compensate for the influence of the disturbance and improve the tracking accuracy of the control method of the present invention.

[0221] Through the above comparison simulation experiment results, the beneficial effects brought by the present invention are summarized into the following three points:

[0222] (1) In view of unknown external disturbances, model uncertainties of ships and differential explosion problems, a disturbance observer is used to estimate the total disturbance caused by external disturbances and internal model uncertainties, and dynamic surface technology is used to solve the differential explosion problem.

[0223] (2) To address the convergence speed issue, the finite-time convergence theory is combined to design the controller and the finite-time sliding surface is used to achieve global finite-time convergence. First, by adding a finite-time term in the approach phase, the velocity tracking error achieves finite-time convergence in the approach phase; second, by using a finite-time sliding surface, when the error reaches the sliding surface or near it, the velocity error on the sliding surface will also achieve finite-time convergence in the sliding phase due to the action of the special sliding surface. At the same time, in response to the input saturation problem, this solution uses a linear anti-saturation compensator to solve the nonlinear term caused by input saturation.

[0224] (3) In order to solve the problem of ship navigation safety, the output is restricted by combining the preset performance function and the obstacle Lyapunov function so that the system has better transient and steady-state performance and the navigation trajectory is always within the preset range.

[0225] The global finite-time ship trajectory tracking control method considering input and output constraints proposed in the present invention can effectively solve the problem of ships being under unknown external disturbances, model uncertainties and input and output constraints, and accelerate the convergence rate of errors. Through Lyapunov theory, it is proved that all signals of the ship system are bounded, and the tracking error can converge within a finite time; after comparison with the existing control strategy, it is verified that the control strategy proposed in this embodiment can achieve output limitation, so that the navigation trajectory is always within the pre-set limit, and can achieve finite time convergence in both the approach stage and the sliding stage, achieving global finite time convergence. Theoretical analysis and simulation experimental results show that the feasibility of the global finite-time ship trajectory tracking control method considering input and output constraints proposed in the present invention is confirmed.

[0226] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or replace some or all of the technical features therein with equivalents. However, these modifications or replacements do not cause the essence of the corresponding technical solutions to deviate from the scope of the technical solutions of the embodiments of the present invention.

Claims

1. A global finite-time ship trajectory tracking control method considering input and output constraints, characterized in that: The specific steps include: S1: Establish a mathematical model of underactuated ship under the condition of input limitation; S2: obtaining a trajectory tracking error based on the actual trajectory and the expected trajectory of the mathematical model of the underactuated ship, and then limiting the trajectory tracking error in combination with a preset performance function and an obstacle Lyapunov function under the condition of considering output limitation to obtain a new error, and designing an expected longitudinal speed and an expected lateral speed based on the new error; S3: Design a first linear anti-saturation compensator and a second linear anti-saturation compensator, and design a first finite-time sliding surface based on the actual longitudinal velocity, the expected longitudinal velocity and the first state variable in the first linear anti-saturation compensator of the mathematical model of the underactuated ship in combination with the finite-time convergence theory; design a second finite-time sliding surface based on the actual lateral velocity, the expected lateral velocity and the second state variable in the second linear anti-saturation compensator of the mathematical model of the underactuated ship; S4: designing a first disturbance observer and a second disturbance observer, obtaining a first total disturbance estimate based on the first disturbance observer, and obtaining a second total disturbance estimate based on the second disturbance observer; designing a longitudinal control force based on the first finite time sliding surface and the first total disturbance estimate; designing a yaw control torque based on the second finite time sliding surface and the second total disturbance estimate; S5: The first finite time sliding mode surface and the second finite time sliding mode surface are adjusted by the longitudinal control force and the yaw control torque, so that the first finite time sliding mode surface and the second finite time sliding mode surface converge, and then the trajectory tracking error converges, and finally the ship trajectory tracking is realized.

2. The global finite-time ship trajectory tracking control method considering input and output constraints according to claim 1 is characterized in that: In S1, under the condition of input limitation, the mathematical model of the underactuated ship is established as follows: in Where x, y, ψ are the forward displacement, drift displacement and heading angle of the ship respectively; u, v, r are the longitudinal velocity, transverse velocity and bow angular velocity of the ship in the appendage coordinate system respectively; τ usat is the actual longitudinal control force of the ship after considering the limitation, τ rsat is the actual yaw control moment after considering the limitation; m 11 ,m 22 ,m 33 are inertia parameters respectively; d 11 ,d 22 ,d 33 are the hydrodynamic parameters respectively; Δf u ,Δf v ,Δf r are the uncertainties of the ship model; τ wu ,τ wv ,τ wr are external unknown disturbances respectively; τ im is the upper bound of the control input; τ uc is the ideal longitudinal control force, τ rc is the ideal bow control torque; Δτ u is the difference between the actual longitudinal control force and the ideal longitudinal control force, Δτ r It is the difference between the actual yaw control torque and the ideal yaw control torque.

3. The global finite-time ship trajectory tracking control method considering input and output constraints according to claim 2 is characterized in that: In S2, the trajectory tracking error is obtained based on the actual trajectory and the expected trajectory of the mathematical model of the underactuated ship. Then, under the condition of considering the output limitation, the trajectory tracking error is limited by combining the preset performance function and the obstacle Lyapunov function to obtain a new error. The process of designing the expected longitudinal speed and the expected lateral speed based on the new error is as follows: S21: The trajectory tracking error is obtained based on the actual trajectory and expected trajectory of the mathematical model of the underactuated ship, which is expressed as: In the formula, x d and d are the ordinate and abscissa of the expected trajectory of the ship respectively; e and e are the longitudinal trajectory tracking error and the lateral trajectory tracking error respectively; x, y are the forward displacement and the lateral drift displacement of the ship respectively, i.e., the output; S22: Under the condition of considering the output limitation, the trajectory tracking error is limited by combining the preset performance function and the obstacle Lyapunov function, and the obtained new error is expressed as: k z =α1z(t) p +θ (5) In the formula, α2, T are positive parameters that need to be designed; α1, p, θ are positive parameters that need to be designed; S23: The expected longitudinal speed and expected lateral speed designed based on the new error are expressed as:

4. The global finite-time ship trajectory tracking control method considering input and output constraints according to claim 3 is characterized in that: In S3, the designed first linear anti-saturation compensator is expressed as: Actual longitudinal velocity u and expected longitudinal velocity u based on mathematical model of underactuated ship d And the process of designing the first finite-time sliding surface with the first state variable ξ1 in the first linear anti-saturation compensator is: make in e =uuu d -ξ1 (22) The designed first finite-time sliding surface s1 is shown in formula (23): In the formula, λ1,λ2 are positive parameters that need to be designed; ξ1 is the first state variable; By taking the derivative of formula (23) and combining formula (1) and formula (22), we can obtain: The designed second linear anti-saturation compensator is expressed as: Where ξ2 is the second state variable; Based on the actual lateral velocity v, the expected lateral velocity v d And the process of designing the second finite-time sliding mode surface with the second state variable ξ2 in the second linear anti-saturation compensator is: make The designed second finite-time sliding surface s2 is shown in formula (27): make but Using dynamic surface technology and introducing a new variable f r As the output of the first-order low-pass filter of f, in order to avoid the differential explosion problem caused by the derivative of f, the expression of the dynamic surface technology is as follows: Where, T r is the design parameter, and T r >0; By taking the derivative of formula (27) and combining formula (1) and formula (26), we can obtain: In the formula, λ3 and λ4 are positive parameters that need to be designed.

5. The global finite-time ship trajectory tracking control method considering input and output constraints according to claim 4 is characterized in that: In S4, the designed first disturbance observer is shown in formula (32): Where ω1 is a positive design parameter, β1 is the state variable of the first disturbance observer; μ1 = τ wu -Δf u , is the estimated value of μ1; The process of designing the longitudinal control force based on the first finite-time sliding surface and the first total disturbance estimate is: Combining equations (21), (24) and (32), we can get τ uc Designed for: In the formula, c5 and c6 are both positive design parameters; The designed second disturbance observer is shown in formula (34): Where ω2 is a positive design parameter, β2 is the state variable of the disturbance observer, and μ2 = τ wr -Δf r , is the estimated value of μ2; The process of designing the yaw control torque based on the second finite time sliding surface and the second total disturbance estimate is: Combining equations (25), (31) and (34), τ is designed as: rc Where c7 and c8 are both positive design parameters.

Images