A tracking control method for underwater helicopter with preset performance considering propeller anomalies

Abstract

The present invention discloses a preset performance tracking control method for an underwater helicopter taking into account abnormal propeller output, comprising: 1) constructing a dynamic model of the AUH, expressing the influence of propeller failure in the form of a thrust distribution matrix, and generating a dynamic equation of the AUH expressed in terms of the state variables of the control system; 2) establishing an improved performance function, and determining the limit of the trajectory tracking error through the improved performance function; 3) constructing an error transformation according to the limit of the trajectory tracking error; 4) introducing propeller output saturation, loss rate and deviation, and designing a propulsion controller. The present invention can effectively ensure that the AUH can achieve stable trajectory tracking under the simultaneous existence of external environmental disturbances and propeller failures, saturation, output loss and deviation; at the same time, in combination with the ESO, a suitable observer gain can be intuitively designed to estimate the unknown state and lumped uncertainty of the AUV without the need for a large amount of data, making the controller more concise.

Description

Technical Field

[0001] The present invention belongs to the field of underwater helicopter trajectory tracking control, and in particular relates to an underwater helicopter preset performance tracking control method taking into account propeller output abnormality. Background Art

[0002] Understanding, utilizing, and protecting the ocean are crucial issues in developing the marine economy and building a strong maritime nation. With the advancement of industrial systems and intelligent manufacturing, research into the intelligent and operational capabilities of underwater unmanned systems has entered a new phase. Autonomous underwater vehicles (AUVs), with their wide range, deep dive depth, reusability, strong autonomy, ease of maintenance, and high precision, are widely used in topographic mapping, patrol reconnaissance, maritime rescue, marine engineering, resource exploration, and biological research. While traditional AUVs, incorporating mechanical design, control, navigation, power, and data transmission technologies, have enabled significant breakthroughs in marine resource exploration and development, significant room for development remains in achieving ultra-maneuverable missions in complex environments. These include efficiently and flexibly connecting to base stations, implementing mobile seabed observation networks, transmitting underwater IoT data, and performing comprehensive missions in deep sea or polar regions. The demand for AUVs in this new era has driven the emergence of underwater helicopters. The autonomous underwater helicopter (AUH), a new type of AUV developed by a team from Zhejiang University, boasts exceptional maneuverability, including full-circle steering, precise hovering, and free takeoff and landing. It can accommodate large-scale deployments and carry a variety of observation payloads, offering new insights into AUV versatility. The AUH is disc-shaped and equipped with four vertical and four horizontal thrusters.

[0003] Underwater helicopters often need to adapt to complex environments and avoid collisions when docking with base stations and operating in clusters. This places high demands on their trajectory tracking control. Therefore, AUHs require a control algorithm with high precision and good dynamic performance, and the control scheme must also account for factors such as ocean current disturbances and thruster failures.

[0004] Existing control algorithms for AUV or AUH trajectory tracking primarily include PID control, adaptive control, sliding mode control, fuzzy control, and neural network control. Some approaches also combine two or more of these concepts. For example, the paper [Cho, GR; Li, JH; Park, D.; Jung, JH Robust trajectory tracking of autonomous underwater vehicles using back-stepping control and time delay estimation. Ocean Eng. 2020, 201, 107-131] uses backstepping and time delay estimation (TDE) to design a robust controller and estimate environmental disturbances such as ocean currents. REMUS model simulations demonstrate that the system demonstrates good robust tracking performance under both ocean currents and external disturbances. The literature [Kim, E.; Fan, S.; Bose, N.; Nguyen, H. Current Estimation and Path Following for an Autonomous Underwater Vehicle (AUV) by Using a High-gain Observer Based on an AUV Dynamic Model. Int. J. Control Autom. Syst. 2020, 19, 478–490] uses a high-gain observer based on a dynamic model to estimate the unmeasurable state of the AUV, and obtains the ocean current velocity through a speed meter and an observer. The patent application document with publication number CN113110532A provides an adaptive terminal sliding mode trajectory tracking control method for a benthic AUV based on an auxiliary dynamic system. For the external interference and model uncertainty in the AUV trajectory tracking control, an adaptive method is used to approximate its boundaries. Patent application publication number CN105843224A proposes a horizontal path tracking control method for an underwater underwater vehicle (AUV) based on a neural dynamic model backstepping method. This method introduces neural dynamic model theory, which features smooth input and output. The virtual control variables generated during the backstepping design process are passed through the neural dynamic model, thus avoiding complex derivative calculations of the virtual control variables. Reference [Li Ruiqi. Hydrodynamic Modeling and Control Algorithm Design for Underwater Helicopters [D]. Zhejiang University, 2016] analyzes the open-loop transfer function of a simplified underwater underwater helicopter state-space model and uses a fuzzy PID controller to control the AUV under current disturbances.Reference [Li Guangyou, Wang Na, Yin Qinghua. Sliding mode control of underactuated AUV based on disturbance observer [J]. Mechanical Manufacturing & Automation, 2022, 51(02): 177-180] By constructing a nonlinear disturbance observer and designing a sliding mode controller based on the disturbance observer, it is ensured that the underactuated AUV still has good tracking control performance under the action of external disturbances.

[0005] While these solutions utilize different control strategies to address external disturbances such as ocean currents and thruster failures, they achieve good trajectory tracking results. However, they still lack comprehensive consideration of environmental factors, thruster saturation, and fault factors such as output loss and deviation. Furthermore, the AUH's unique operating mode, delicate operations, and large-scale deployment require control algorithms with high steady-state accuracy and dynamic performance. These also place higher demands on both experience-based fuzzy algorithms and data-intensive deep learning algorithms.

[0006] In addition, existing AUV trajectory tracking control schemes either focus on one of the factors, ocean current disturbances and thruster failures, or do not consider the problem of thruster saturation. Moreover, the fault tolerance mechanism, thruster saturation and anti-disturbance control design are usually separated from each other, resulting in a complex structure. Summary of the Invention

[0007] This paper proposes a trajectory tracking control method for underwater helicopters with preset performance, considering factors such as current disturbances, propeller saturation, losses, and deviations. Furthermore, a novel performance function is designed to replace the traditional one. This not only limits excessive overshoot in the initial phase, unifies initial errors of different signs under the same error constraint inequality, but also sets an expected convergence time, effectively guaranteeing the system's dynamic performance.

[0008] The specific technical solutions of the present invention are as follows:

[0009] A method for tracking control of an underwater helicopter with preset performance considering abnormal propeller output comprises:

[0010] 1) Construct the dynamic model of the AUH, use the thrust distribution matrix to represent the impact of the thruster failure, and generate the dynamic equations of the AUH represented by the state variables of the control system;

[0011] 2) establishing an improved performance function and determining the limit of trajectory tracking error through the improved performance function;

[0012] 3) constructing an error transform according to the limit of the trajectory tracking error;

[0013] 4) Introduce thruster output saturation, loss rate and deviation to design the thrust controller.

[0014] The kinetic model of AUH in step 1) is:

[0015]

[0016] Where: M is the mass inertia matrix, η=[η x ,η y ,η z ,η φ ,η θ ,η ψ ] T is the AUH six-degree-of-freedom position and attitude vector in the inertial coordinate system, v = [v u ,v v ,v w ,v p ,v q ,v r ] T is the AUH velocity and angular velocity vector in the motion coordinate system, J represents the transformation matrix connecting the inertial coordinate system and the motion coordinate system, the C(v) matrix represents the Coriolis force and centripetal force of the AUH, the D(v) matrix represents the hydrodynamic damping part of the AUH, and g η The matrix represents the forces and moments generated by the AUH gravity and buoyancy, and τ represents the control forces and moments output by the AUH thrusters.

[0017] The impact of the AUH's thruster failure is expressed in the form of a thrust distribution matrix, defined as ΔB; the actual control force and torque are expressed as:

[0018] τ+Δτ=(B0-KB)u=(B0+ΔB)u

[0019] Where: B0 represents the nominal value of the AUH thrust distribution matrix, B is the thrust distribution matrix of the AUH, u represents the control output of the AUH thruster, K is a diagonal matrix whose element k is ii ∈[0,1], represents the corresponding thruster failure degree, where 1 represents the highest failure degree and the thruster is completely ineffective;

[0020] The kinetic model of AUH can be rewritten as:

[0021]

[0022] Where: subscript 0 represents the nominal value of the corresponding variable; d represents the comprehensive disturbance of the system;

[0023]

[0024] Where: represents the impact of ocean current disturbance on AUH, where C Aη =C A (vr )J -1 , D η =D(v r )J -1 , η r is the displacement relative to water in the AUH inertial coordinate system.

[0025] In the step 1), let represents the state variables of the control system, then the dynamic equation of AUH can be expressed using state variables:

[0026]

[0027] In the step 2), an improved performance function is established under the framework of the corresponding constraint inequality;

[0028] The constrained inequality is of the following form:

[0029] P l (t)<e(t)<P r (t)

[0030]

[0031] Among them, P l (t) and P r (t) defines the lower and upper bounds of the error respectively, sgn(g) represents the sign function, 0≤δ1≤1, 0≤δ2≤1.

[0032] The improved performance function is as follows:

[0033]

[0034] Where a0=a1t 3 +a2t 2 +a3t+a4cos(a5t); a1~a5 are the parameters to be designed, ρ tf =ρ ∞ , preset parameter t f The performance function is defined to reach ρ ∞ Deadline;

[0035] Construct error transformation:

[0036]

[0037] Define an error transformation function at (-∞,+∞):

[0038]

[0039] When the ε(t) transformation error is bounded, the constraint inequalities will be satisfied simultaneously.

[0040] The control signal of the AUH trajectory tracking system must be within a bounded range, limiting the control input to the following saturation function:

[0041]

[0042] Where u s is the actual effective output after actuator saturation limitation, u max and -u min They represent the upper and lower bounds of the output signal respectively.

[0043] Design input compensation term adaptive law

[0044]

[0045] Where k4 is the design parameter, which is a positive constant, and u Δ =u s -u.

[0046] Preferably, considering the propeller anomaly including loss and deviation, the actual control output is:

[0047] u θ =f × u+θ

[0048] where f × , θ are unknown functions, the former represents the output loss rate, and the latter represents the deviation.

[0049] Furthermore, in step 4), the extended disturbance term d is used as the new state variable x3 to obtain a new state equation:

[0050]

[0051] make get:

[0052]

[0053] In the formula E=[I n 0 0]

[0054] Design Observer

[0055]

[0056] Where β1, β2, β3 are design parameters;

[0057] Its compact form is:

[0058]

[0059] In the formula

[0060] Compared with the traditional preset performance method, this paper uses a new error constraint framework to unify the initial errors of different signs under the same error constraint inequality; at the same time, a new performance function is developed, which can not only limit the excessive overshoot in the initial stage, but also can be used to preset the parameter t f Artificially setting the deadline for the convergence of the performance function ensures good dynamic performance of the system, which is in good agreement with the AUH control requirements.

[0061] The proposed preset performance control method is attractive for solving such stringent performance challenges, effectively ensuring stable trajectory tracking in the presence of external disturbances, thruster failures, saturation, output loss, and deviations. Furthermore, the combination with ESO allows for intuitive estimation of unknown AUV states and combined disturbances without the need for extensive data sets. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 Schematic diagram of the improved performance function designed for an embodiment of the present invention;

[0063] Figure 2 AUH trajectory tracking error e1 curve (translational degree of freedom) under sudden thruster failure of the present invention;

[0064] Figure 3 is the AUH trajectory tracking error e1 curve (rotational degree of freedom) under sudden thruster failure of the present invention;

[0065] Figure 4 AUH trajectory tracking error e2 curve (translational degree of freedom) under sudden thruster failure of the present invention;

[0066] Figure 5 This is the AUH trajectory tracking error e2 curve (rotational degree of freedom) under sudden thruster failure of the present invention. DETAILED DESCRIPTION

[0067] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the present invention is not limited to the specific embodiments disclosed below.

[0068] Key technologies involved in the technical solution of this application:

[0069] Inertial coordinate system: The origin can be selected at a certain point on the sea surface, and the positive directions of the three axes point to the north, east and the center of the earth respectively.

[0070] Motion coordinate system: The origin is taken at the center of gravity of AUH, and the positive directions of the three axes point to the forward direction, the right swing direction and the sinking direction respectively.

[0071] The AUH kinematic equations can refer to Fossen's outline six-degree-of-freedom nonlinear dynamic model

[0072]

[0073] Where: M is the mass inertia matrix, η=[η x ,η y ,η z ,η φ ,η θ ,η ψ ] T represents the six-degree-of-freedom position and attitude of AUH in the inertial coordinate system, v = [v u ,v v ,v w ,v p ,v q ,v r ] T represents the velocity and angular velocity of AUH in the motion coordinate system, J is the transformation matrix between the inertial coordinate system and the motion coordinate system, C(v) is the Coriolis force and centripetal force matrix of AUH, D(v) is the hydrodynamic damping matrix of AUH, g η are the force and torque vectors generated by the AUH gravity and buoyancy, and τ is the control force and torque generated by the AUH propulsion system.

[0074] Preset performance control: This is a method that converts the original "constrained" system into an equivalent "unconstrained" system by introducing a performance function and error transformation, and proves the stability of the "unconstrained" system so that the convergence speed, overshoot, and tracking error achieve preset performance.

[0075] Second-order differentiator: A differentiator that takes an unknown differential variable as input, constructs a second-order differential relationship, and outputs the approximate differential of the variable.

[0076] Extended State Observer (ESO): An observer that expands the uncertainties in the control system, such as external disturbances, into new observation states and compensates for the unknown states by designing a suitable gain matrix.

[0077] In this embodiment, the parameters are defined as follows:

[0078] η=[η x ,η y ,η z ,η φ ,η θ ,η ψ ] Tis the six-degree-of-freedom position and attitude value of AUH in the inertial coordinate system; J represents the transformation matrix between the inertial coordinate system and the motion coordinate system; v = [v u ,v v ,v w ,v p ,v q ,v r ] T represents the velocity and angular velocity of AUH in the motion coordinate system; M is the mass inertia matrix of AUH; C(v) is the Coriolis force and centripetal force matrix of AUH; D(v) is the hydrodynamic damping matrix of AUH; g η are the force and torque vectors generated by the AUH gravity and buoyancy; τ is the control force and torque generated by the AUH propulsion system; B is the thrust distribution matrix of the AUH; B0 is the nominal value of the AUH thrust distribution matrix; u is the control output of the AUH thruster.

[0079] The core of the present invention is to design a controller to enable the AUH to track the desired trajectory η while taking into account environmental disturbances such as ocean currents, propeller saturation, losses and deviations. d , and the tracking error e=η-η d Meet the pre-set steady-state response and dynamic performance. The specific steps are as follows:

[0080] Step 1: Dynamic model transformation of AUH.

[0081] The impact of AUH thruster failure can be expressed in the form of a thrust distribution matrix, defined as ΔB. Therefore, the actual control force and torque can be rewritten as τ + Δτ:

[0082] τ+Δτ=(B0-KB)u=(B0+ΔB)u (3)

[0083] Where: B0 represents the nominal value of the AUH thrust distribution matrix, B is the thrust distribution matrix of the AUH, u represents the control output of the AUH thruster, K is a diagonal matrix whose element k is ii ∈[0,1], represents the corresponding thruster failure degree, where 1 represents the highest failure degree and the thruster is completely ineffective. Therefore, equation (2) can be rewritten as:

[0084]

[0085] Where: subscript 0 represents the nominal value of the corresponding variable; d represents the total uncertainty of the system, which is expressed as follows:

[0086]

[0087] Where: Indicates the impact of ocean current disturbance on AUH.

[0088] make represents the system state, the dynamic equation of AUH can be expressed as:

[0089]

[0090] Step 2: Performance function and error transformation.

[0091] The present invention proposes a new performance function, which is as follows:

[0092]

[0093] Where a0=a1t 3 +a2t 2 +a3t+a4cos(a5t). a1, a2, a3, a4, a5 are the parameters to be designed, ρ(0)=ρ0, ρ(t f )=ρ tf =ρ ∞ , preset parameter ρ ∞ Given the upper bound of the allowable steady-state error, the parameter t f The performance function is defined to reach ρ tf The tracking error is subject to the inequality of the performance function as follows:

[0094]

[0095] e(t)=xx d Defines the deviation between the actual motion trajectory and the expected value, x d =η d The desired motion trajectory is defined. If the initial error satisfies the constraint inequality, the AUH's six-degree-of-freedom motion trajectory vector η is strictly constrained within the performance function boundary. Therefore, by designing an appropriate performance function ρ(t), the desired system error response can be obtained.

[0096] The present invention adopts the following error transformation:

[0097]

[0098] in

[0099] An error transformation function is defined at (-∞,+∞):

[0100]

[0101] When the transformation error ε(t) is bounded, the constraint inequality will be satisfied at the same time. Therefore, the constraint (8) on the tracking error can be achieved by designing a controller to make ε(t) bounded.

[0102] Define tracking error:

[0103] e1=x1-x 1d (12)

[0104]

[0105] According to the relationship between tracking error and transformation error:

[0106]

[0107] in

[0108] Design virtual control rate:

[0109]

[0110] Where k1 is the design parameter;

[0111] In order to obtain the time derivative of α, the following second-order differentiator is designed:

[0112]

[0113] Where χ is the intermediate variable, ε a >0 is the disturbance parameter, υ∈(0,1), and:

[0114]

[0115] where ε b >0 is a constant.

[0116] make:

[0117]

[0118] Substitute (15) and (19) into (14):

[0119]

[0120] in

[0121] Step 3: Introduce thruster output saturation, loss and deviation.

[0122] 1. Output saturation:

[0123] In practice, due to actuator limitations, the control signal of the AUH trajectory tracking system must be within a bounded range. Therefore, the control input is limited to the following saturation function:

[0124]

[0125] Where u sis the actual effective output after actuator saturation limitation, u max and -u min They represent the upper and lower bounds of the output signal respectively.

[0126] Design input compensation term adaptive law:

[0127]

[0128] Where k4 is the design parameter, which is a positive constant, and u Δ =u s -u

[0129] The present invention introduces the input compensation ξ into the controller to offset the effect of output saturation.

[0130] 2. Output loss rate and deviation:

[0131] Actuator control inputs often exhibit a certain degree of loss and deviation due to factors such as device aging, friction between actuators, battery life, and environmental factors. This section designs an adaptive law to estimate the multiplicative loss rate and demonstrates that the loss rate estimation error converges over time. In the subsequent controller design section, the loss rate estimate will be used to adjust the control inputs, demonstrating that the controller can still achieve the desired system performance even in the presence of unknown drift errors.

[0132] u θ =f × u+θ (23)

[0133] where f × , θ is an unknown function, the former represents the output loss rate, and the latter represents the deviation;

[0134] The following loss rate estimation adaptive law is introduced:

[0135]

[0136] Where m is the observer variable, is the intermediate state quantity, L × =-K × sgn(T * )M η0 , T * =diag{u0},K × is a positive definite diagonal matrix.

[0137] Step 4: Design of extended state observer.

[0138] Due to the influence of complex ocean environment, the first-order derivatives of position and attitude vectors It is difficult to measure directly, and the controller design includes unknown disturbances. Therefore, the present invention designs an extended state observer for the state variable x2 and the disturbance d to estimate the unknown quantity.

[0139] First, expand the disturbance term d to the new state variable x3, and obtain the new state equation:

[0140]

[0141] make Writing the system in compact form, we get:

[0142]

[0143] In the formula E=[I n 0 0];

[0144] Design observer:

[0145]

[0146] Where β1, β2, β3 are design parameters;

[0147] Its compact form is:

[0148]

[0149] In the formula

[0150] By selecting a suitable gain matrix L, the observation error matrix can be made Converge to a fixed bounded set, satisfying the stable observation of velocity state and disturbance.

[0151] Step 5: Preset performance controller design.

[0152] The following adaptive law of perturbation boundary estimation is introduced:

[0153]

[0154] The trajectory tracking controller designed by the present invention is as follows:

[0155]

[0156] Where k2 and k3 are positive design parameters. Considering the output loss rate, the The adaptive law (24) adjusts the control input:

[0157]

[0158] For the AUH trajectory tracking system (6) with external disturbances, considering the thruster output saturation, loss and deviation, if the controllers (31) and (32) are used with the help of the extended state observer (28), and the appropriate gain matrix and design parameters are selected, the tracking error constraint can be achieved as shown in Equation (8).

[0159] The theoretical basis involved in the technical solution of the present invention:

[0160] Kinetic model of AUH:

[0161] The AUH kinematic equations can refer to Fossen's outline six-degree-of-freedom nonlinear dynamic model

[0162]

[0163] Where: M is the mass inertia matrix, η=[η x ,η y ,η z ,η φ ,η θ ,η ψ ] T represents the six-degree-of-freedom position and attitude of AUH in the inertial coordinate system, v = [v u ,v v ,v w ,v p ,v q ,v r ] T represents the velocity and angular velocity of AUH in the motion coordinate system, J is the transformation matrix between the inertial coordinate system and the motion coordinate system, C(v) is the Coriolis force and centripetal force matrix of AUH, D(v) is the hydrodynamic damping matrix of AUH, g η are the force and torque vectors generated by the AUH gravity and buoyancy, and τ is the control force and torque generated by the AUH propulsion system.

[0164] The impact of AUH thruster failure can be expressed in the form of a thrust distribution matrix, defined as ΔB. Therefore, the actual control force and torque can be rewritten as τ + Δτ:

[0165] τ+Δτ=(B0-KB)u=(B0+ΔB)u (35)

[0166] Where: B0 represents the nominal value of the AUH thrust distribution matrix, B is the thrust distribution matrix of the AUH, u represents the control output of the AUH thruster, K is a diagonal matrix whose element k is ii ∈[0,1], represents the corresponding thruster failure degree, where 1 represents the highest failure degree and the thruster is completely ineffective. Therefore, equation (34) can be rewritten as:

[0167]

[0168] Where: subscript 0 represents the nominal value of the corresponding variable; d represents the total uncertainty of the system, which is expressed as follows:

[0169]

[0170] Where: Indicates the impact of ocean current disturbance on AUH.

[0171] make represents the state variables of the control system, then the dynamic equation of AUH can be expressed in terms of state variables:

[0172]

[0173] The core of the present invention is to design a controller to enable the AUH to track the desired trajectory η while taking into account environmental disturbances such as ocean currents, propeller saturation, losses and deviations. d , and the tracking error e=η-η d To achieve this goal, the present invention proposes the following assumptions based on AUH and application scenarios:

[0174] Assumption 1: The external disturbance has an upper bound, that is, ||d||≤d M , d M is an unknown positive constant.

[0175] Performance function and error transformation construction:

[0176] Definition: If a smooth function ρ(t):R + →R satisfies monotonically decreasing and If the condition is met, then this function can be defined as a performance function.

[0177] A common performance function is as follows:

[0178]

[0179] Where: ρ0, ρ ∞ and is a pre-given positive constant. e(t)=xx d Defines the deviation between the actual motion trajectory and the expected value, x d =η d The desired motion trajectory is defined. If the initial error satisfies the constraint inequality, the AUH's six-degree-of-freedom motion trajectory vector η is strictly limited within the performance function boundary. Parameters limits the minimum convergence rate of the tracking error, and ρ ∞Given the upper bound of the allowable steady-state error, the desired system error response can be obtained by designing an appropriate performance function ρ(t).

[0180] An improved new performance function is proposed under the framework of a special constrained inequality. The constrained inequality is as follows:

[0181] P l (t)<e(t)<P r (t) (40)

[0182]

[0183] Among them, P l (t) and P r (t) defines the lower and upper bounds of the error respectively, sgn(g) represents the sign function, 0≤δ1≤1, 0≤δ2≤1.

[0184] The new performance functions designed are as follows:

[0185]

[0186] Where a0=a1t 3 +a2t 2 +a3t+a4cos(a5t). ρ tf =ρ ∞ , preset parameter t f The performance function is defined to reach ρ ∞ The initial and terminal conditions of this novel performance function are the same as those of the traditional performance function, i.e., ρ(0)=ρ0, ρ(t f )=ρ tf Among them, the first and second order derivatives of ρ(t) with respect to time are both continuous functions, that is, The parameters a1, a2, a3, a4, and a5 can be calculated using the above conditions.

[0187]

[0188] When t f = 30s, the graph of the performance function is as follows Figure 1 shown.

[0189] Construct error transformation:

[0190]

[0191] It can be seen

[0192] Define an error transformation function at (-∞,+∞):

[0193]

[0194] When the ε(t) transformation error is bounded, the constraint inequalities will be satisfied simultaneously.

[0195] prove:

[0196] From formula (10), we can get:

[0197]

[0198] So we have:

[0199]

[0200] If the transformation error is bounded, then |ε(t)|≤ε M ;

[0201] according to have:

[0202]

[0203] Right now:

[0204]

[0205] After rearranging, we can get formula (40).

[0206] Therefore, as long as the transformation error is bounded, the tracking error will be constrained within the boundaries specified by the performance function.

[0207] This combination of the new performance function and error transformation not only makes the controller design process unnecessary to call different constraint inequalities due to the sign of the initial error, but also avoids excessive overshoot of the tracking error in the initial stage, and can be achieved by presetting the parameter t f Artificially setting the deadline for the convergence of the performance function ensures good dynamic performance of the system, which is in good agreement with the AUH control requirements.

[0208] Introducing thruster output saturation, loss rate and deviation:

[0209] 1. Output saturation:

[0210] In practice, due to actuator limitations, the control signal of the AUH trajectory tracking system must be within a bounded range. Therefore, the control input is limited to the following saturation function:

[0211]

[0212] Where u s is the actual effective output after actuator saturation limitation, u max and -u min They represent the upper and lower bounds of the output signal respectively.

[0213] Design input compensation term adaptive law:

[0214]

[0215] Where k4 is the design parameter, which is a positive constant, and u Δ =u s -u;

[0216] Assumption 2: Saturation deviation u Δ Bounded, satisfied is an unknown constant.

[0217] The boundedness of the input compensation term ξ will be proved in the controller design section.

[0218] 2. Output loss rate and deviation:

[0219] Actuator control inputs often exhibit a certain degree of loss and deviation due to factors such as device aging, friction between actuators, battery life, and environmental factors. This section designs an adaptive law to estimate the multiplicative loss rate and demonstrates that the loss rate estimation error converges over time. In the subsequent controller design section, the loss rate estimate will be used to adjust the control inputs, demonstrating that the controller can still achieve the desired system performance even in the presence of unknown drift errors.

[0220] u θ =f × u+θ (52)

[0221] where f × , θ is an unknown function, the former represents the output loss rate, and the latter represents the deviation.

[0222] Assumption 3: is an unknown constant.

[0223] The following loss rate estimation adaptive law is introduced:

[0224]

[0225] Where m is the observer variable, is the intermediate state quantity, L × =-K × sgn(T * )M η0 , T * =diag{u0},K × is a positive definite diagonal matrix.

[0226] prove:

[0227] Define the loss rate estimation error:

[0228]

[0229] Notice is a negative definite matrix, and It is bounded within a specific observation period, so the above differential equation satisfies the input-state stability and the input loss rate satisfies the robust observation condition.

[0230] Extended State Observer Design:

[0231] In designing the trajectory tracking control strategy of AUH, the position and attitude vector η in the fixed coordinate system and its first-order derivative However, due to the complex ocean environment, It is difficult to measure directly, and the controller design includes unknown disturbances. Therefore, we introduce the extended state observer technique to estimate the speed state variable and unknown disturbances.

[0232] Assumption 4: The rate of change of the disturbance is bounded and satisfies is an unknown positive constant.

[0233] First, expand the disturbance term d to the new state variable x3 to obtain the new state equation;

[0234]

[0235] make Writing the system in compact form, we get:

[0236]

[0237] In the formula E=[I n 0 0];

[0238] Design observer:

[0239]

[0240] Where β1, β2, β3 are design parameters;

[0241] Its compact form is:

[0242]

[0243] In the formula

[0244] By selecting a suitable gain matrix L, the observation error matrix x% can be converged to a fixed bounded set, satisfying the stable observation of velocity state and disturbance.

[0245] prove:

[0246] Define observation error:

[0247]

[0248] Taking its derivative we get:

[0249]

[0250] Take the Lyapunov function:

[0251]

[0252] Taking the derivative with respect to time, we get:

[0253]

[0254] According to Assumption 4 and combined with Young's inequality, we have:

[0255]

[0256] Where σ1 is a positive constant.

[0257] Substitute (64):

[0258]

[0259] It can be seen that as long as a suitable gain matrix L is selected so that A-LE is a Hurwitz matrix, the observation error matrix can be made Converges to the following set:

[0260]

[0261] The proof of the observer will also be combined with the subsequent system stability proof in the controller design part.

[0262] Preset performance controller design:

[0263] Lemma 1: For any a, b ≥ 0 and ε * >0, for p>1,q>1, The following inequality holds;

[0264]

[0265] Lemma 2: For any q and h∈R + ,have:

[0266]

[0267] Where κ = 0.2758.

[0268] The tracking error is defined according to model (38):

[0269] e1=x1-x1d (70)

[0270]

[0271] According to the relationship between tracking error and transformation error:

[0272]

[0273] in Design virtual control rate:

[0274]

[0275] Where k1 is the design parameter.

[0276] In order to obtain the time derivative of α, the following second-order differentiator is designed:

[0277]

[0278] Where χ is the intermediate variable, ε a >0 is the disturbance parameter, υ∈(0,1), and:

[0279]

[0280] where ε b >0 is a constant.

[0281] Take the following Lyapunov function:

[0282]

[0283] make:

[0284]

[0285] Substitute (73) and (79) into (72):

[0286]

[0287] in Substitute (80) into (78):

[0288]

[0289] If e2 is bounded, the upper bound is a constant, that is, Then we have:

[0290]

[0291] Among them, σ2 and σ3 are positive constants.

[0292] Substitute (82) into (81):

[0293]

[0294] make Then the estimation error x% and the transformation error ε1 will be limited to the following sets respectively:

[0295]

[0296] in Given the control input:

[0297]

[0298] Where k2 and k3 are positive design parameters. Considering the output loss rate, the The adaptive law (89) adjusts the control input:

[0299]

[0300] The adaptive law for perturbation boundary estimation is given as:

[0301]

[0302] The perturbation boundary estimation error is:

[0303]

[0304] Define the following Lyapunov function:

[0305]

[0306] Find its time derivative:

[0307]

[0308] Substitute (88) into (87):

[0309]

[0310] Substitute (91) and (95) into (94):

[0311]

[0312] According to Lemma 2, we can get:

[0313]

[0314] According to Lemma 1, we can get:

[0315]

[0316] Substitute (98) into (97):

[0317]

[0318] Note:

[0319]

[0320] Among them, σ4 and σ5 are positive constants.

[0321] Combining hypothesis 3, (62) and (67), we can conclude that Pick is an unknown positive constant. It can be found Also note Considering the above analysis and substituting (99)-(102) into (96), we can further obtain:

[0322]

[0323] make You can get:

[0324]

[0325] In the formula

[0326] So we have:

[0327]

[0328] but ε2 and and ξ will be restricted to the following sets respectively:

[0329]

[0330] Next, we prove that the controller can keep the signals bounded when the output signals are biased. The input v is given by the input loss and the actual input after considering the drift error becomes

[0331] Similar to formula (95), we have:

[0332]

[0333] Substitute (91) and (108) into (94):

[0334]

[0335] because:

[0336]

[0337] Where σ6 is a positive constant.

[0338] Notice Substitute (99)-(102) and (110) into (109):

[0339]

[0340] make have:

[0341]

[0342] in:

[0343]

[0344] The following convergence form is met:

[0345]

[0346] Then the observer estimation error Transformation error ε2 and perturbation boundary estimation error and the input compensation ξ will be limited to the following sets respectively:

[0347]

[0348] In summary, this section proves that when the output signal has deviations, the controllers (88) and (90) provided can satisfy the boundedness of each signal, so that the tracking error is limited to the inequality constrained by the performance function.

[0349] In this paper, a fully driven AUH is introduced into the simulation, consisting of four vertical thrusters and four horizontal thrusters, each of which operates independently to provide bidirectional thrust. The initial position and attitude vectors of the AUH are set to η(0) = [1.5; 1.5; 1.5; 1.5; 1.5], and the initial and angular velocity vectors of the AUH are set to v(0) = [0; 0; 0; 0; 0; 0]. Table 1 provides the dimensionless hydrodynamic coefficients of the AUH.

[0350] Table 1 Dimensionless hydrodynamic coefficients of AUH

[0351]

[0352]

[0353] Taking the common spiral dive as the expected trajectory of AUH, its specific expression is as follows:

[0354] η d=[2sin(0.1t);2cos(0.1t)+2;-0.5144t;0;0;0] (115)

[0355] In order to prove that the designed scheme can resist external disturbances, the ocean current disturbance is added to the simulation, as shown in Equation (116). Here, it is assumed that the ocean current is parallel to the positive direction of the x-axis of the inertial coordinate system.

[0356]

[0357] In addition, to prove that the controller proposed in this invention can handle thruster failures, we introduce a thruster sudden failure mode to verify the effectiveness of the algorithm, which is expressed as follows:

[0358]

[0359] Under the conditions of ocean current disturbance, thruster output saturation, output loss and deviation, the preset performance controller and extended state observer proposed in this invention are applied to obtain the AUH trajectory tracking error curves e1 and e2 through simulation.

[0360] from Figure 2-Figure 5 It can be seen that the preset performance control scheme designed by the present invention based on the new performance function and the extended state observer can effectively compensate for the effects caused by external disturbances, thruster output saturation, output loss and deviation, so that the AUH can track the desired trajectory (115). In addition, the scheme can unify the initial tracking errors of different signs under the same constraint conditions, so that the tracking error is limited within the performance function boundary and converges to the specified steady-state accuracy within the preset time with good dynamic performance.

[0361] The above description is only an example of a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for tracking control of underwater helicopter preset performance considering propeller abnormality, characterized in that: include: 1) Construct the dynamic model of the AUH, use the thrust distribution matrix to represent the impact of the thruster failure, and generate the dynamic equations of the AUH represented by the state variables of the control system; The kinetic model of AUH in step 1) is: Where: M is the mass inertia matrix, η=[η x ,η y ,η z ,η φ ,η θ ,η ψ ] T is the AUH six-degree-of-freedom position and attitude vector in the inertial coordinate system, v = [v u ,v v ,v w ,v p ,v q ,v r ] T is the AUH velocity and angular velocity vector in the motion coordinate system, J represents the transformation matrix connecting the inertial coordinate system and the motion coordinate system, the C(v) matrix represents the Coriolis force and centripetal force of the AUH, the D(v) matrix represents the hydrodynamic damping part of the AUH, and g η The matrix represents the forces and moments generated by the AUH gravity and buoyancy, and τ represents the control forces and moments output by the AUH thrusters; The impact of the AUH's thruster failure is expressed in the form of a thrust distribution matrix, defined as ΔB; the actual control force and torque are expressed as: τ+Δτ=(B0-KB)u=(B0+ΔB)u Where: B0 represents the nominal value of the AUH thrust distribution matrix, B is the thrust distribution matrix of the AUH, u represents the control output of the AUH thruster, K is a diagonal matrix whose element k is ii ∈[0,1], represents the corresponding thruster failure degree, where 1 represents the highest failure degree and the thruster is completely ineffective; The kinetic model of AUH can be rewritten as: Where: subscript 0 represents the nominal value of the corresponding variable; d represents the comprehensive disturbance of the system; Where: represents the impact of ocean current disturbance on AUH, where C Aη =C A (v r )J -1 , D η =D(v r )J -1 , C A Equivalent to the C(v) matrix, representing the Coriolis force and centripetal force of AUH, v r is the water velocity of AUH, D(v r ) is equivalent to the D(v) matrix, representing the hydrodynamic damping part of the AUH, η r is the displacement relative to water in the AUH inertial coordinate system; In the step 1), let represents the state variables of the control system, then the dynamic equation of AUH can be expressed using state variables: Where: x1 is the AUH position information η, x2 is the AUH speed information y is the system output, Equivalent to is the nominal inertial mass matrix, C RBη0 is the rigid body part of the AUH Coriolis force and centripetal force nominal matrix, C Aη0 is the part of the additional mass in the nominal matrix of AUH Coriolis force and centripetal force, D η0 is the nominal hydrodynamic damping matrix, g η0 Express the forces and moments due to the nominal AUH gravity and buoyancy; 2) establishing an improved performance function and determining the limit of trajectory tracking error through the improved performance function; In the framework of corresponding constrained inequalities, an improved performance function is established; The constrained inequality is of the following form: P l (t)<e(t)<P r (t) Among them, P l (t) and P r (t) defines the lower and upper bounds of the error, sgn(g) represents the sign function, 0≤δ1≤1, 0≤δ2≤1, t is the system running time, ρ ∞ is the final value of the performance function, e(0) is the initial error; The improved performance function is as follows: Where a0=a1t 3 +a2t 2 +a3t+a4cos(a5t); a1~a5 are the parameters to be designed, ρ0 is the initial value of the performance function, ρ tf =ρ ∞ , preset parameter t f The performance function is defined to reach ρ ∞ Deadline; 3) constructing an error transform according to the limit of the trajectory tracking error; Construct error transformation: An error transformation function is defined at (-∞,+∞): When the ε(t) transformation error is bounded, the constraint inequalities will be satisfied simultaneously; 4) Introducing thruster output saturation, loss rate, and deviation to design a propulsion controller; The control signal of the AUH trajectory tracking system must be within a bounded range, which is limited to the following saturation function: Where u s is the actual effective output after actuator saturation limitation, u max and -u min Represent the upper and lower bounds of the output signal respectively; Design input compensation term adaptive law: Where k4 is the design parameter, which is a positive constant, ξ is the compensation signal without gain multiplication, and u Δ =u s -u; Taking into account thruster anomalies including losses and deviations, the actual control output is: in θ =f × u+θ where f × , θ are unknown functions, the former represents the output loss rate, and the latter represents the deviation; In step 4), the comprehensive disturbance d is used as the new state variable x3 to obtain a new state equation: make get: E=[I n 00];I n is the identity matrix, h is the first-order derivative of the integrated perturbation; Design observer: Where β1, β2, β3 are design parameters; Its compact form is: In the formula

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