A water surface ship control method based on optimization backstepping
By optimizing the backstepping control method and combining it with the Actor-Critic neural network, the problem of insufficient trajectory tracking accuracy of surface ships in deep-sea environments was solved, achieving high-precision trajectory tracking and improved stability, and simplifying complex nonlinear dynamics processing.
Patent Information
- Application Number
- CN202611023997.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2026-07-10
- Publication Date
- 2026-11-03
AI Technical Summary
Existing surface vessels struggle to accurately adapt to complex and ever-changing ocean disturbances in deep-sea environments, resulting in insufficient accuracy in trajectory tracking and path planning, which affects their autonomous navigation capabilities and operational efficiency.
An optimization-based backstepping control method is adopted, which combines the Actor-Critic neural network and the HJB equation to establish a surface ship motion model. Through hierarchical recursive optimization control, the treatment of high-order nonlinear terms is simplified, thereby improving system stability and trajectory tracking accuracy.
It achieves high-precision trajectory tracking of surface vessels in deep-sea environments, rapidly responds to time-varying signals, reduces transient lag and steady-state error, and improves the stability and control performance of the system.
Smart Images
Figure CN123029924A_ABST
Abstract
Description
Technical Field
[0001] This invention relates to a surface vessel control method based on optimized backstepping, belonging to the field of intelligent control. Background Technology
[0002] In recent years, with the rapid development of deep-sea resource development and the marine transportation industry, the requirements for the autonomous navigation capability and operational efficiency of surface vessels have been continuously increasing. Ship trajectory tracking and path planning technologies, as core technologies in the field of intelligent ship control, have become a research hotspot in this field. However, the deep-sea environment is highly complex and time-varying, and surface vessels are continuously affected by multi-source, unknown ocean disturbances such as wind, waves, and currents during navigation. Existing trajectory tracking and path planning schemes are difficult to accurately adapt to the complex and ever-changing deep-sea conditions, exhibiting significant shortcomings in tracking accuracy, planning rationality, and environmental robustness, severely restricting the practical engineering application of related technologies.
[0003] With the rapid development of intelligent control technology, hybrid control strategies that integrate neural networks and classical control have become a new research hotspot. How to effectively apply this hybrid control method to surface vessel systems is a pressing technical problem that needs to be solved in this field. Summary of the Invention
[0004] To address the problems existing in the prior art, this invention provides a control method for surface vessels based on optimized backstepping. This method can achieve tracking control of vessel motion, improving system stability and tracking accuracy.
[0005] To achieve the above objectives, the present invention provides a control method for surface vessels based on optimized backstepping, comprising the following steps:
[0006] S1. Establish the surface vessel motion model, the definition lemma of semi-global eventual uniform boundedness, and the assumptions;
[0007] S2. Combine the Actor-Critic neural network with the HJB equation to establish an evaluation mechanism for each level of subsystem;
[0008] S3. Perform system stability analysis;
[0009] S4. Conduct simulation experiments
[0010] S1.1 The ship motion model is as follows:
[0011]
[0012] in, These represent the ship's position and heading in the Earth's fixed coordinate system, respectively. These represent the pitch angular velocity, yaw angular velocity, and yaw rate of the ship in its fixed coordinate system. (Matrix) as follows:
[0013]
[0014] ,matrix The Jacobian transformation matrix is used for the coordinate transformation between the body fixed coordinate system and the Earth fixed coordinate system. The inertia matrix is defined as a matrix of positive constants. ; Here is the damping matrix. This is the restoring force vector of the ship under the combined effects of gravity and buoyancy. This is the control input for the system.
[0015] make Then the ship motion model (1) can be changed to the following form:
[0016]
[0017] in ,
[0018] , .
[0019] Finally, let The motion model of a surface vessel can then be represented by a nonlinear lower triangular model:
[0020]
[0021] S1.2 The semi-global eventually uniform bounded definition and lemma are as follows:
[0022] 1) For a nonlinear system, the system state vector is If for compact sets any initial state within Both have constants and with Initial state Related limited time , making when Time, always If true, then the solution to the system is semi-global eventually uniform and bounded.
[0023] 2) Let It is a continuous positive function, with initial value... Bounded. If the differential inequality This holds true (where a and c are both positive constants). Therefore, we have:
[0024]
[0025] S1.3 Assume the parameters of the actor neural network Bounded.
[0026] S2.1 with For the model's output If the desired output is obtained, then the error between the two is:
[0027]
[0028] Combining the motion model (3), the derivative of equation (5) with respect to time t yields:
[0029]
[0030] Will The virtual controller is considered as a first-order subsystem, and the value function of the first-order subsystem is... and cost function They are as follows:
[0031]
[0032]
[0033] The Hamiltonian equation related to equation (7) is as follows:
[0034]
[0035] The optimal controller for the first-order subsystem, i.e. The optimal value function of the first-order subsystem and optimal cost function They are as follows:
[0036]
[0037]
[0038] right Perform the following processing:
[0039]
[0040] Based on equations (6), (9), (10), (11), and (12), the Hamilton-Jacobi-Bellman equations (HJB equations) of the first-order subsystem of the surface ship motion model are as follows:
[0041]
[0042] Optimal Virtual Control Law It can be solved The following expression is obtained:
[0043]
[0044] in The expression is as follows:
[0045]
[0046] For ideal neural network weights, These are the basis functions of the neural network, and the input is the error between the model output and the desired trajectory. The expression for the basis functions is: , This represents the approximation error of the neural network.
[0047] Combining equations (12) and (14) with equation (15) respectively, we can obtain the following two expressions:
[0048]
[0049]
[0050] Due to ideal weights It is an unknown constant vector, therefore it cannot be applied directly. To obtain a usable control scheme, this invention uses an Actor-Critic neural network, and equations (16) and (17) take the following forms:
[0051]
[0052]
[0053] From equations (18) and (19), the following approximate HJB equation can be obtained:
[0054] Subtracting equation (20) from equation (13) yields the Bellman residual.
[0055]
[0056] The update rate of the Critic neural network aimed at reducing Bellman residuals is as follows:
[0057]
[0058] in The learning rate for the Critic neural network is... .
[0059] Minimizing the Bellman residual is equivalent to optimizing the output of the Actor network. Approaching the optimal control rate ,but Therefore, the update rate of the Actor neural network weights should be such that... After differentiation, it is .
[0060] Based on the objective of minimizing the Bellman residual, the update rate of the Actor neural network weights is as follows:
[0061]
[0062] Learning rate of Actor neural network Combining equations (22) and (23) Regarding time The derivative is as follows:
[0063]
[0064] S2.2 introduces errors into the second-order subsystem:
[0065]
[0066] Combining model (4), the derivative of equation (25) with respect to time t yields the following equation:
[0067]
[0068] The value function of the second-order subsystem is as follows:
[0069]
[0070] According to equations (27) and (28), the HJB equation is as follows:
[0071]
[0072] Perform the following processing:
[0073]
[0074] Combining equations (26), (28), and (29), we can obtain:
[0075]
[0076] Optimal control rate It can be solved The following expression is obtained:
[0077]
[0078] Approximated by neural networks :
[0079]
[0080] After using the AC neural network, we can obtain:
[0081]
[0082]
[0083] From equations (33) and (34), we can obtain the approximate HJB equation:
[0084] Subtracting equation (35) from equation (30) yields the Bellman residual.
[0085]
[0086] With the aim of reducing Bellman residuals, the AC neural network weight update rates are as follows:
[0087]
[0088] in
[0089] S3.1 Combining equation (6) with equation (25) yields the following equation:
[0090]
[0091] The Lyapunov function of the first-order subsystem is as follows:
[0092]
[0093] in , From equations (13), (14), and (15), we can obtain:
[0094]
[0095] in These are bounded terms. The simplification of the Bellman residual is shown below:
[0096]
[0097] Combining equations (22), (23), and (39), equation (40) relates to time. The derivative is as follows:
[0098]
[0099] According to Young's inequality, the following inequality can be obtained:
[0100]
[0101] Applying the above inequality to equation (43) yields:
[0102]
[0103] Combining equation (42) and equation (44), we get:
[0104]
[0105] According to Young's inequality, the following inequality can be obtained:
[0106]
[0107] Applying the above inequality to equation (45) yields:
[0108]
[0109] At this point, we can obtain:
[0110]
[0111]
[0112] Then inequality (46) can be transformed into:
[0113]
[0114] when When, inequality (48) can be expressed as:
[0115]
[0116] in It is a bounded term.
[0117] The inequality (49) is transformed into the following form:
[0118]
[0119]
[0120] The global Lyapunov function S3.2 is as follows:
[0121]
[0122] in ,
[0123] Equation (36) simplifies to:
[0124]
[0125] It is a bounded term.
[0126] Based on equations (26), (37), (38), and (51), the derivative of equation (50) with respect to time t is as follows:
[0127]
[0128] in .
[0129] It is a bounded term.
[0130] Inequality (52) can be transformed into the following form:
[0131]
[0132]
[0133] According to the definition of semiglobal eventually uniform boundedness and the lemma inequality (53), it takes the following form:
[0134]
[0135] but It is eventually uniformly bounded.
[0136] By comparing the overlap between the actual system output and the reference trajectory, the tracking performance of the control scheme can be intuitively evaluated. The two have a high degree of fit across the entire time domain, with no significant transient lag or overshoot. It can quickly track time-varying signals, with minimal deviation throughout the entire process and extremely low steady-state error. This proves that the optimized feedback control method of this invention has high time-varying tracking accuracy, excellent transient and steady-state performance, and that the scheme is effective.
[0137] The tracking errors of each subsystem converge rapidly and remain bounded, with no signs of divergence or instability. This indicates that the method of the present invention can effectively constrain tracking deviations, meets the semi-global consistent eventual bounded stability standard, and the closed-loop solution is stable with precise error adjustment.
[0138] Observing the evolution of the cost function of each subsystem at each order can characterize the closed-loop control quality and adaptive network characteristics. The cost value converges and is bounded throughout the entire process, which not only quantitatively confirms that the tracking error is controlled within the allowable range, but also corroborates the stable iteration of the AC network weights and the bounded control output, thus verifying the closed-loop reliability of the scheme.
[0139] Observing the evolution of the AC network weight norm in each subsystem can characterize the learning convergence and closed-loop stability of the adaptive network. The convergence of the Critic and Actor weights is bounded and drift-free, proving that the weight update law can avoid learning divergence, the value function estimation is accurate, and the closed-loop reliability of the scheme is verified.
[0140] This invention divides a surface vessel system into a position subsystem and a velocity subsystem based on optimized backstepping control. A backstepping recursive framework is responsible for progressively achieving system stabilization and tracking response. Trajectory tracking error and control cost are defined as the optimization objectives of each backstepping subsystem, simplifying the handling of high-order complex nonlinear terms. An AC neural network is introduced to approximate the value function online, overcoming the technical difficulty of analytically solving the HJB equations. An optimized backstepping composite control strategy based on neural network reinforcement learning is proposed. This invention, through hierarchical recursive optimized control, uses the optimal virtual control command output from the previous level as the reference input for the next level subsystem. This simplifies the handling of complex nonlinear dynamics of surface vessels while improving system stability, trajectory tracking accuracy, and optimal control performance. Attached Figure Description
[0141] Figure 1 It is a simulation diagram of the system output and the reference trajectory;
[0142] Figure 2 It is the tracking error between the three components of the system output and the reference output;
[0143] Figure 3 It is the tracking error of the three components of the ship's speed;
[0144] Figure 4 This is a simulation diagram illustrating the costs of the two subsystems;
[0145] Figure 5 This is a schematic diagram of the norms of the AC network parameters for each subsystem.
[0146] Figure 6 This is a schematic diagram of a surface ship model.
Claims
1. A surface vessel control method based on optimized backstepping, characterized in that, Includes the following steps: S1. Establish the surface vessel motion model, the definition lemma of semi-global eventual uniform boundedness, and the assumptions; S2. Combine the Actor-Critic neural network with the HJB equation to establish an evaluation mechanism for each level of subsystem; S3. Perform system stability analysis; S4. Conduct simulation experiments.
2. The surface vessel control method based on optimized backstepping according to claim 1, characterized in that, The specific process of S1 is as follows: S1.1 The ship motion model is as follows: in, These represent the ship's position and heading in a fixed Earth coordinate system, respectively. These represent the pitch angular velocity, yaw angular velocity, and yaw rate of the ship in its fixed coordinate system. (Matrix) as follows: ,matrix The Jacobian transformation matrix is used for the coordinate transformation between the body fixed coordinate system and the Earth fixed coordinate system. The inertia matrix is defined as a matrix of positive constants. ; Here is the damping matrix. This is the restoring force vector of the ship under the combined effects of gravity and buoyancy. For the system's control input, make Then the ship motion model (1) can be changed to the following form: in , ， ， Finally, let The motion model of a surface vessel can then be represented by a nonlinear lower triangular model: S1.2 The semi-global eventually uniform bounded definition and lemma are as follows: 1) For a nonlinear system, the system state vector is If for compact sets any initial state within Both have constants and with Initial state Related limited time , making when Time, always If true, then the solution to this system is semi-globally eventually uniform and bounded. 2) Let It is a continuous positive function, with initial value... Bounded. If the differential inequality This holds true (where a and c are both positive constants). Therefore, we have: S1.3 Assume the parameters of the actor neural network Bounded. S2.1 with For the model's output If the desired output is obtained, then the error between the two is: Combining the motion model (3), the derivative of equation (5) with respect to time t yields: Will The virtual controller is considered as a first-order subsystem, and the value function of the first-order subsystem is... and cost function They are as follows: The Hamiltonian equation related to equation (7) is as follows: The optimal controller is considered as the first-order subsystem. The optimal value function of the first-order subsystem and optimal cost function They are as follows: right Perform the following processing: Based on equations (6), (9), (10), (11), and (12), the Hamilton-Jacobi-Bellman equations (HJB equations) of the first-order subsystem of the surface ship motion model are as follows: Optimal Virtual Control Law It can be solved The following expression is obtained: in The expression is as follows: For ideal neural network weights, These are the basis functions of the neural network, and the input is the error between the model output and the desired trajectory. The expression for the basis functions is: , This represents the approximation error of the neural network. Combining equations (12) and (14) with equation (15) respectively, we can obtain the following two expressions: Due to ideal weights It is an unknown constant vector, therefore it cannot be applied directly. To obtain a usable control scheme, this invention uses an Actor-Critic neural network, and equations (16) and (17) take the following forms: From equations (18) and (19), the following approximate HJB equation can be obtained: Subtracting equation (20) from equation (13) yields the Bellman residual. The update rate of the Critic neural network aimed at reducing Bellman residuals is as follows: in The learning rate for the Critic neural network is... . Minimizing the Bellman residual is equivalent to optimizing the output of the Actor network. Approaching the optimal control rate ,but Therefore, the update rate of the Actor neural network weights should be such that... After differentiation, it is . Based on the objective of minimizing the Bellman residual, the update rate of the Actor neural network weights is as follows: Learning rate of Actor neural network Combining equations (22) and (23) Regarding time The derivative is as follows: S2.2 introduces errors into the second-order subsystem: Combining model (4), the derivative of equation (25) with respect to time t yields the following equation: The value function of the second-order subsystem is as follows: According to equations (27) and (28), the HJB equation is as follows: Perform the following processing: Combining equations (26), (28), and (29), we can obtain: Optimal control rate It can be solved The following expression is obtained: Approximated by neural networks : After using the AC neural network, we can obtain: From equations (33) and (34), we can obtain the approximate HJB equation: Subtracting equation (35) from equation (30) yields the Bellman residual. With the aim of reducing Bellman residuals, the AC neural network weight update rates are as follows: in S3.1 Combining equation (6) with equation (25) yields the following equation: The Lyapunov function of the first-order subsystem is as follows: in , From equations (13), (14), and (15), we can obtain: in These are bounded terms. The simplification of the Bellman residual is shown below: Combining equations (22), (23), and (39), equation (40) relates to time. The derivative is as follows: According to Young's inequality, the following inequality can be obtained: Applying the above inequality to equation (43) yields: Combining equation (42) and equation (44), we get: According to Young's inequality, the following inequality can be obtained: Applying the above inequality to equation (45) yields: At this point, we can obtain: Then inequality (46) can be transformed into: when When, inequality (48) can be expressed as: in It is a bounded term. The inequality (49) is transformed into the following form: The global Lyapunov function S3.2 is as follows: in , Equation (36) simplifies to: It is a bounded term. Based on equations (26), (37), (38), and (51), the derivative of equation (50) with respect to time t is as follows: in . It is a bounded term. Inequality (52) can be transformed into the following form: According to the definition of semiglobal eventually uniform boundedness and the lemma inequality (53), it takes the following form. but It is eventually uniformly bounded. By comparing the overlap between the actual system output and the reference trajectory, the tracking performance of the control scheme can be intuitively evaluated. The two have a high degree of fit across the entire time domain, with no significant transient lag or overshoot. It can quickly track time-varying signals, with minimal deviation throughout the entire process and extremely low steady-state error. This proves that the optimized feedback control method of this invention has high time-varying tracking accuracy, excellent transient and steady-state performance, and that the scheme is effective. The tracking errors of each subsystem converge rapidly and remain bounded, with no signs of divergence or instability. This indicates that the method of the present invention can effectively constrain tracking deviations, meets the semi-global consistent eventual bounded stability standard, and the closed-loop solution is stable with precise error adjustment. Observing the evolution of the cost function of each subsystem at each order can characterize the closed-loop control quality and adaptive network characteristics. The cost value converges and is bounded throughout the entire process, which not only quantitatively confirms that the tracking error is controlled within the allowable range, but also corroborates the stable iteration of the AC network weights and the bounded control output, thus verifying the closed-loop reliability of the scheme. Observing the evolution of the AC network weight norm in each subsystem can characterize the learning convergence and closed-loop stability of the adaptive network. The convergence of the Critic and Actor weights is bounded and drift-free, proving that the weight update law can avoid learning divergence, the value function estimation is accurate, and the closed-loop reliability of the scheme is verified.