Design Method of Submarine Vertical Plane Motion Controller Based on Closed-Loop Gain Shaping Algorithm

Abstract

This invention discloses a design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm. Based on this algorithm, the submarine depth controller and pitch controller designed according to the state-space equations of the submarine dynamics model can quickly and accurately achieve variable depth and constant depth maneuvers, optimizing the response time and steady-state error of the submarine's vertical plane motion controller, thus improving the submarine's underwater performance and safety. By constructing the depth and pitch nonlinear functions of the submarine's vertical plane motion controller, and combining these functions with the control laws of the bow and stern rudder actuators obtained from the vertical plane motion controller, the linear error feedback of the submarine's vertical plane motion controller is improved, enhancing the controller's robustness and energy-saving effect, reducing the impact of external interference, and decreasing the energy consumption of the servo motors.

Description

Technical Field

[0001] This invention relates to the field of submarine motion controller design technology, and in particular to a design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm. Background Technology

[0002] Submarine vertical plane motion control encompasses automatic rudder control for depth-changing maneuvers and depth-holding control. Depth-holding control refers to the ability of submarines and other underwater vehicles to maintain a desired depth or follow a depth trajectory under various disturbances and uncertainties, which is crucial for submarine safety, stealth, and mission execution. When performing various missions, submarines will navigate at a fixed depth as required by the mission, necessitating good depth-holding performance, which has consistently been a hot research topic in the field of submarine attitude control.

[0003] However, since depth-holding control is a multiple-input multiple-output system, traditional submarine attitude control aims to improve the overall performance of navigation and enhance the stability, accuracy, and speed of control. While it is effective during normal navigation, it is not robust in extreme situations and still requires the pilot to rely on their experience to operate the submarine. This cannot meet the precision control requirements for submarine movement. Furthermore, since submarines need to operate offshore for extended periods, energy conservation is also an important consideration for maintaining the combat effectiveness of conventionally powered submarines. Summary of the Invention

[0004] This invention provides a design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm to overcome the above-mentioned technical problems.

[0005] To achieve the above objectives, the technical solution of the present invention is as follows:

[0006] A design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm includes the following steps:

[0007] S1: Establish a mathematical model of the submarine that only considers the vertical plane motion of the submarine, and simplify the mathematical model of the submarine into a submarine dynamics model;

[0008] S2: Obtain the state-space equations of the submarine dynamics model based on the submarine dynamics model;

[0009] S3: Based on the closed-loop gain shaping algorithm, a submarine vertical plane motion controller is constructed according to the state space equation; the submarine vertical plane motion controller includes a submarine depth controller and a submarine pitch controller.

[0010] S4: Construct the depth nonlinear function and pitch nonlinear function of the submarine vertical plane motion controller, and obtain the control law of the submarine bow rudder actuator and the control law of the submarine stern rudder actuator based on the depth nonlinear function and pitch nonlinear function and the submarine vertical plane motion controller.

[0011] S5: Control of the submarine's vertical plane motion is achieved based on the control laws of the submarine's bow rudder actuator and the submarine's stern rudder actuator.

[0012] Furthermore, S1 specifically includes the following steps:

[0013] S11: Construct a mathematical model of the submarine that only considers the vertical plane motion of the submarine. The expression of the submarine mathematical model is as follows:

[0014]

[0015] In the formula: m represents the mass of the submarine; This represents the vertical acceleration along the Gz axis in the submarine's appendage coordinate system G-xyz; Represents the pitch acceleration; u represents the forward velocity along the Gx axis in the submarine appendage coordinate system G-xyz; I y δ represents the moment of inertia of the submarine about the Gy axis in the submarine appendage coordinate system G-xyz; b With δ s These represent the bow elevator angle and stern elevator angle of a submarine, respectively; θ represents the pitch angle; ζ represents the depth; X B With Z B These represent the distances from the center of buoyancy to the center of gravity along the Gx and Gz axes, respectively; B b ρ represents the buoyancy force acting on the submarine; W represents the static force on the submarine in the water. Z w , M w M q , All are hydrodynamic derivatives; Z wave With M wave These represent the wave disturbance force and torque, respectively; φ represents the submarine's roll angle.

[0016] S12: Assuming the submarine is sailing at a constant depth and the depth change is a weak maneuver, the submarine mathematical model is simplified to obtain the submarine dynamics model. The expression of the submarine dynamics model is as follows:

[0017]

[0018] Furthermore, the state-space equation expression for the submarine dynamics model obtained in S2 is as follows:

[0019]

[0020] In the formula:

[0021] In the formula: X represents the system state; δ represents the input of the submarine dynamics model; A, B, Q, and C all represent intermediate parameter variable matrices; Y represents the output of the submarine dynamics model.

[0022] Furthermore, S3 specifically includes the following steps:

[0023] S31: Based on the closed-loop gain shaping algorithm and ignoring the wave force and torque on the submarine, formula (3) is transformed into the submarine vertical plane transfer function matrix. The expression of the submarine vertical plane transfer function matrix is:

[0024] G = C(Is - A) -1 B (4)

[0025] In the formula: G represents the transfer function matrix of the submarine's vertical plane; I represents the identity matrix; s represents the Laplace operator;

[0026] S32: Construct the complementary sensitivity function matrix T, the expression of which is:

[0027]

[0028] In the formula: λ represents the coefficient matrix of the complementary sensitivity function matrix; T 11 With T 22 All represent the parameters of the complemented sensitivity matrix; s represents the Laplacian operator; λ 11 With λ 22 All represent parameters of the coefficient matrix λ;

[0029] S33: Obtain the submarine vertical plane motion controller K according to formulas (4) and (5). The expression for the submarine vertical plane motion controller K is as follows:

[0030]

[0031] In the formula: G represents the submarine's vertical plane transfer function matrix; T represents the complementary sensitivity function matrix; κ1, κ2, and κ3 all represent the model parameters of the submarine dynamics model; a 11 ,a 12 ,a 21 ,a 22 All represent the model matrix elements of the submarine dynamics model;

[0032] S34: Decouple the submarine vertical plane motion controller K to obtain the submarine depth controller and the submarine pitch controller; the expressions for the submarine depth controller and the submarine pitch controller are as follows:

[0033]

[0034] In the formula: K 11 Indicates submarine depth controller; K 22 This refers to the submarine's pitch control.

[0035] Furthermore, S4 specifically includes the following steps:

[0036] S41: Define the depth error of the submarine, the expression for which is:

[0037] e 11 =ζ-ζ d (8)

[0038] In the formula: e 11 ζ represents the submarine's depth error; ζ represents the submarine's current depth. d This represents the submarine's set depth.

[0039] S42: Using nonlinear feedback technology, a depth nonlinear function for the submarine's vertical plane motion controller is constructed based on the submarine's depth error. The expression for the depth nonlinear function is as follows:

[0040]

[0041] In the formula: a1 and b1 both represent the feedback design parameters of the deep nonlinear function; f1(e 11 ) represents a deep nonlinear function;

[0042] S43: Obtain the control law of the submarine bow rudder actuator based on the aforementioned depth nonlinear function and the submarine depth controller. The expression for the control law of the submarine bow rudder actuator is as follows:

[0043] δ b =f1(e 11 )K 11 (10)

[0044] Where: δ b This input represents the angle of the submarine's bow elevator;

[0045] S44: Define the pitch error of a submarine, the expression for which is:

[0046] e 22 =θ-θ d (11)

[0047] In the formula: e 22 θ represents the submarine's pitch error; θ represents the submarine's pitch at its current position; θ d The set pitch of the submarine;

[0048] S45: Based on the submarine's pitch error and the submarine's pitch controller, obtain the current control law δ′ of the submarine's pitch controller. s The control law δ of the current pitch controller s The expression for ′ is

[0049] δ s '=e 21 K 22 (12)

[0050] S46: Based on nonlinear modification technology, according to the control law δ′ of the current pitch controller s The pitch nonlinear function is obtained and used as the control law for the submarine stern rudder actuator. The expression for the control law of the submarine stern rudder actuator is as follows:

[0051]

[0052] In the formula: f2(e 21 K 22 ) represents the pitch nonlinear function; δ s represents the input of the submarine's stern elevator angle; a2 and b2 both represent the modification design parameters of the pitch nonlinear function.

[0053] Beneficial Effects: This invention provides a design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm. Based on this algorithm, the submarine depth controller and pitch controller designed according to the state-space equations of the submarine dynamics model can quickly and accurately achieve variable depth and constant depth maneuvers, optimizing the response time and steady-state error of the submarine's vertical plane motion controller, thus improving the submarine's underwater performance and safety. By constructing the depth and pitch nonlinear functions of the submarine's vertical plane motion controller, and combining these functions with the control laws of the bow and stern rudder actuators obtained from the vertical plane motion controller, the linear error feedback of the submarine's vertical plane motion controller is improved, enhancing the controller's robustness and energy-saving effect, reducing the impact of external interference, and decreasing the energy consumption of the servo motors. Attached Figure Description

[0054] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0055] Figure 1This is a flowchart of the design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm according to the present invention.

[0056] Figure 2 This is a schematic diagram of the submarine's inertial coordinate system and appendage coordinate system in this embodiment;

[0057] Figure 3 This is an overall structural diagram of the submarine vertical plane motion controller design method in this embodiment;

[0058] Figure 4 This is a schematic diagram of the submarine's "elevator mode" depth-deployment maneuver in this embodiment;

[0059] Figure 5 This is a diagram of the nonlinear feedback and nonlinear modification structure in this embodiment;

[0060] Figure 6 This is a simulation diagram of the first-order wave force and moment under sea state VI in this embodiment;

[0061] Figure 7 This is a graph showing the change in depth during submarine maneuvers in this embodiment;

[0062] Figure 8 This is a curve showing the pitch angle of the submarine's depth maneuver in this embodiment;

[0063] Figure 9 This is a graph showing the change in bow rudder angle during deep maneuvering of the submarine in this embodiment;

[0064] Figure 10 This is a graph showing the change in rudder angle during deep maneuvering of the submarine in this embodiment;

[0065] Figure 11 This is a graph showing the change in submarine depth under wave interference in this embodiment;

[0066] Figure 12 This is a curve showing the dynamic pitch angle of the submarine under wave interference in this embodiment;

[0067] Figure 13 This is a graph showing the change in the bow rudder angle of the submarine under wave interference in this embodiment.

[0068] Figure 14 This is a graph showing the change in the rudder angle of the submarine under wave interference in this embodiment. Detailed Implementation

[0069] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0070] This embodiment provides a design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm, such as... Figure 1 As shown, it includes the following steps:

[0071] S1: Establish a mathematical model of the submarine that only considers the vertical plane motion of the submarine, and simplify the submarine mathematical model into a submarine dynamics model, specifically including the following steps:

[0072] S11: This embodiment uses the International Towing Tank (ITTC) and the terminology bulletin of the Society of Naval Architects and Marine Engineers (SNAME) to establish the submarine coordinate system, such as... Figure 2 As shown, the submarine coordinate system includes two coordinate systems: the inertial coordinate system E-ξηζ and the appendage coordinate system G-xyz. Furthermore, the submarine motion model assumes that the horizontal plane motion has little impact on the vertical plane motion. The vertical plane motion is considered decoupled from the horizontal plane motion and axial motion. Therefore, when performing submarine depth control, only the vertical plane motion needs to be considered. Thus, a submarine mathematical model considering only the vertical plane motion is constructed. The expression for this submarine mathematical model is:

[0073]

[0074] In the formula: m represents the mass of the submarine; This represents the vertical acceleration along the Gz axis in the submarine's appendage coordinate system G-xyz; The acceleration represents the pitch angle; u represents the forward velocity along the Gx axis in the submarine's appendage coordinate system G-xyz (to simplify the calculation, the submarine can be assumed to be moving at a constant velocity, i.e., u is a constant value); I y δ represents the moment of inertia of the submarine about the Gy axis in the submarine appendage coordinate system G-xyz; b With δ s These represent the bow and stern elevator angles of the submarine, respectively, with downward-facing rudders defined as positive; θ represents the pitch angle, defined as positive when the bow is raised and negative when the bow is lowered; ζ represents depth; X B With Z BLet Gx and Gz represent the distances from the center of buoyancy to the center of gravity along the Gx and Gz axes, respectively; Bb is the buoyancy force acting on the submarine; W represents the static force on the submarine in the water. For simplified calculation, it can be assumed that the static force and buoyancy force on the submarine are in equilibrium, i.e., Bb = W, and that the point of application of the buoyancy force is on the same vertical line as the center of gravity of the submarine, i.e., X. B =0;; Z w Z q , M w M q , All are hydrodynamic derivatives; Z wave With M wave These represent the wave disturbance force and torque, respectively; φ represents the submarine's roll angle. To simplify the calculation, this invention assumes that the submarine has no roll during the depth-changing maneuver and depth-maintaining process, i.e., φ = 0.

[0075] S12: Based on actual submarine operation experience, assuming that depth changes during constant-depth navigation are weak maneuvers, the pitch angle θ is considered very small and negligible. The submarine mathematical model is simplified to obtain the submarine dynamics model, the expression of which is:

[0076]

[0077] S2: Obtain the state-space equations of the submarine dynamics model based on the submarine dynamics model; the expression of the state-space equations of the submarine dynamics model is:

[0078]

[0079] In the formula:

[0080] In the formula: X represents the system state; δ represents the input of the submarine dynamics model; A, B, Q, and C all represent intermediate parameter variable matrices; Y represents the output of the submarine dynamics model;

[0081] For example, when the speed is taken as 6 knots, or 3.086 m / s, substituting the hydrodynamic derivative at the corresponding speed yields the state-space equation for the submarine dynamics model.

[0082]

[0083] S3: As Figure 3As shown, a submarine vertical plane motion controller is constructed based on a closed-loop gain shaping algorithm and the state-space equations. This controller includes a depth controller and a pitch controller. There are two methods for maintaining submarine depth: one is to adjust buoyancy by pumping out ballast water to control depth, and the other is to control devices such as bow and stern elevators to maintain the submarine at the desired depth or height. The second method can be further divided into two types based on the steering method. A common method is the "sled mode," where the bow and stern rudders are steered in opposite directions to generate lift or descent, thus achieving a large depth change. This method is suitable for high-speed navigation or rapid surfacing or diving. Another common method is the "elevator mode," which uses the bow and stern rudders to generate constant lift, giving the submarine a certain vertical speed, thus achieving a small depth change or maintaining a constant depth. This method is suitable for low-speed navigation or precise depth control. This embodiment focuses on the design of a depth-holding controller for small-amplitude depth maneuvers in the low-speed range and under external interference conditions. Therefore, the bow and stern rudder adjustment method is selected, and a "lifter mode" is used for depth maneuvers, that is, maintaining a constant pitch angle to allow the submarine to submerge in parallel. The maneuvering process is as follows: Figure 4 As shown. During this process, to maintain the submarine's parallel descent, changes in the pitch angle will be considered a disturbance, and a controller will be designed to keep it in a pitch-free state. The submarine's vertical plane control structure is as follows. Figure 5 As shown, utilizing the bow rudder's dominant role in depth control, the depth returned from the depth control loop and the set depth are used as error inputs to the depth controller to obtain the bow rudder angle. The bow rudder angle is then input into the submarine's mathematical model via a servo system to obtain the output depth. Similarly, the stern rudder's precise control of pitch is utilized to obtain the output pitch angle. The specific steps include:

[0084] S31: Based on the closed-loop gain shaping algorithm and ignoring the wave force and torque on the submarine, formula (3) is transformed into the submarine vertical plane transfer function matrix. The expression of the submarine vertical plane transfer function matrix is:

[0085] G = C(Is - A) -1 B (4)

[0086] In the formula: G represents the transfer function matrix of the submarine's vertical plane; I represents the identity matrix; s represents the Laplace operator;

[0087] For example, when the speed is taken as 6 knots, or 3.086 m / s, the corresponding submarine vertical plane transfer function matrix is:

[0088]

[0089] S32: According to the closed-loop gain shaping algorithm, for the signal tracking problem of a dual-input dual-output system, the system transfer function matrix from input to output is actually the complemented sensitivity matrix.

[0090] GK0(I+GK0) -1 =T (5)

[0091] In the formula: G is the system model transfer function matrix, I is the identity matrix, T is the complemented sensitivity matrix, and K0 is the transfer function matrix of the dual-input dual-output system controller;

[0092] This leads to the formula for solving the controller matrix based on model G and the complement sensitivity matrix T.

[0093] K0 = G -1 (IT) -1 T (6)

[0094] Therefore, the elements of the complementary sensitivity matrix T can be assumed to be first-, second-, or third-order inertial elements with a maximum singular value of 1. In this embodiment, the submarine depth holding control is a typical dual-input dual-output system, which can be assumed to be a second-order inertial element with off-diagonal elements of 0 and diagonal elements with a maximum singular value of 1. The form of the T0 matrix shaped by the closed-loop gain is as follows:

[0095]

[0096] In the formula: T 11 ,T 22 All of these are parameters of the complemented sensitivity matrix.

[0097] Furthermore, since the two input / output channels for the submarine's depth control and pitch control are completely decoupled from each other, the coefficient matrix of the T matrix is ​​designed to be λ.

[0098]

[0099] Then, construct the complementary sensitivity function matrix T, the expression of which is:

[0100]

[0101] In the formula: λ represents the coefficient matrix of the complementary sensitivity function matrix; T 11 With T 22 All represent the parameters of the complemented sensitivity matrix; s represents the Laplacian operator; λ 11 With λ 22 All represent parameters of the coefficient matrix λ;

[0102] S33: Obtain the submarine vertical plane motion controller K according to formulas (4) and (9), and ignore the minimum terms in the formulas (i.e., the Laplace operator coefficients are less than 10) during the calculation process.-4 The item (which is considered to contribute little to the controller and whose removal has almost no impact on the control effect) is described in the expression for the submarine vertical plane motion controller K.

[0103]

[0104] In the formula: κ1, κ2, and κ3 all represent the model parameters of the submarine dynamics model; a 11 ,a 12 ,a 21 ,a 22 All of these represent the model matrix elements of the submarine dynamics model, which can all be calculated by equation (4); G represents the submarine vertical plane transfer function matrix; T represents the complementary sensitivity function matrix;

[0105] For example, when the speed is 6 knots, or 3.086 m / s, the expression for the submarine's vertical plane motion controller K is:

[0106]

[0107] in:

[0108] And here λT is the same as λT0 in formula (9);

[0109] S34: Decouple the submarine vertical plane motion controller K to obtain the submarine depth controller and the submarine pitch controller; the expressions for the submarine depth controller and the submarine pitch controller are as follows:

[0110]

[0111] In the formula: K 11 Indicates submarine depth controller; K 22 This refers to the submarine's pitch control.

[0112] S4: Construct the depth nonlinear function and pitch nonlinear function of the submarine vertical plane motion controller, and obtain the control law of the submarine bow rudder actuator and the control law of the submarine stern rudder actuator based on the depth nonlinear function and pitch nonlinear function and the submarine vertical plane motion controller.

[0113] like Figure 5 As shown, the specific steps include:

[0114] S41: Define the depth error of the submarine, the expression for which is:

[0115] e 11 =ζ-ζ d (12)

[0116] In the formula: e 11ζ represents the submarine's depth error; ζ represents the submarine's current depth. d This represents the submarine's set depth.

[0117] S42: Employing nonlinear feedback technology, to address the issue of large errors in the depth control channel, a nonlinear feedback term is used to improve the submarine's depth controller. A depth nonlinear function for the submarine's vertical plane motion controller is constructed based on the submarine's depth error. The expression for this depth nonlinear function is as follows:

[0118]

[0119] In the formula: a1 and b1 both represent the feedback design parameters of the deep nonlinear function; f1(e 11 ) represents a deep nonlinear function;

[0120] S43: Obtain the control law of the submarine bow rudder actuator based on the aforementioned depth nonlinear function and the submarine depth controller. The expression for the control law of the submarine bow rudder actuator is as follows:

[0121] δ b =f1(e 11 )K 11 (14)

[0122] Where: δ b This input represents the angle of the submarine's bow elevator;

[0123] S44: Define the pitch error of a submarine, the expression for which is:

[0124] e 22 =θ-θ d (15)

[0125] In the formula: e 22 θ represents the submarine's pitch error; θ represents the submarine's pitch at its current position; θ d The set pitch of the submarine;

[0126] S45: Based on the submarine's pitch error and the submarine's pitch controller, obtain the current control law δ of the submarine's pitch controller. s ′, the control law δ of the current pitch controller s The expression for ′ is

[0127] δ s '=e 21 K 22 (16)

[0128] S46: For cases where the pitch angle control channel error is small, a nonlinear modification technique is used to improve the submarine pitch controller, based on the current pitch controller's control law δ′. sThe pitch nonlinear function is obtained and used as the control law for the submarine stern rudder actuator. The expression for the control law of the submarine stern rudder actuator is as follows:

[0129]

[0130] In the formula: f2(e 21 K 22 ) represents the pitch nonlinear function; δ s represents the input of the submarine's stern elevator angle; a2 and b2 both represent the modification design parameters of the pitch nonlinear function.

[0131] This embodiment introduces nonlinear feedback and nonlinear modification theory without changing the original controller structure. It simply uses the nonlinear function of the error to replace the original error as the input and output of the controller, but it has a significant energy-saving effect.

[0132] S5: Control of the submarine's vertical plane motion is achieved based on the control laws of the submarine's bow rudder actuator and the submarine's stern rudder actuator.

[0133] Simulation Results: The depth and pitch control system of a submarine was simulated to verify the effectiveness of nonlinear feedback. To better reflect reality, a servo system was introduced into the simulation. According to specifications, the bow rudder angle was limited to 25° and the rudder speed to 3.5° / s; the stern rudder angle was limited to 20° and the rudder speed to 3° / s. When designing a submarine depth controller, the characteristics of the submarine's operation in shallow waters, including wave forces and torque interference, must be considered. These mainly include first-order wave forces and second-order wave forces (wave suction). First-order wave forces are regular, high-frequency periodic forces caused by short waves, oscillating at the same frequency as the wave waveform, but lagging behind the wave phase. Since a submarine is a system with high inertia, this high-frequency periodic force causes the submarine to rise and fall with long waves. Second-order average wave forces are irregular forces that exert an upward force on the submarine; they are the average component of a suction force. Their amplitude is smaller than that of the first-order force and remains constant over a long period, comparable to the rudder force. This force cannot be ignored because it is the cause of depth control problems and needs to be counteracted by the controller. This invention uses the simplified Pierson-Moskowitz spectrum from ITTC, the specific expression of which is:

[0134]

[0135] In the formula: S(ω) represents the spectral density (m 2 ·s); ω represents the wave frequency (rad / s); H s Indicates the meaningful wave height (m);

[0136] To better describe the effect of waves on a submarine, the Pierson-Moskowitz spectrum is approximated as the frequency ω at which white noise reaches its maximum at S(ω). M A filter with a center frequency is represented as:

[0137]

[0138] In the formula: v represents wave height; ι represents white noise; The gain K is used to adjust the output amplitude and variance of the filter to match S(ω).

[0139] The simplified wave force and moment can then be expressed as:

[0140]

[0141] In the formula: v represents the wave height obtained in equation (15); a, b, c, and d are all coefficients, the estimates of which are derived from the analysis of the Pierson-Moskowitz spectrum. The first-order wave force and moment vary under sea state VI as follows: Figure 6 As shown.

[0142] First, analyze the control effects during the submarine's depth-deep maneuver, such as Figure 7 The controller control system designed using the second-order closed-loop gain shaping algorithm responds rapidly to depth commands. At a submarine speed of 6 knots, the depth maneuver from 30m to the periscope depth of 10m takes 27.52s, with no overshoot and a steady-state error of 0. This demonstrates the excellent control performance of the closed-loop gain shaping design control system. In contrast, the controller designed using the Independent Channel Analysis and Design (ICAD) method has a depth adjustment time of 58.55s. Comparatively, the second-order closed-loop gain shaping algorithm improves upon the ICAD method by 53.00%. Furthermore, the submarine's pitch control effect is analyzed... Figure 8 As shown, the Independent Channel Analysis and Design (ICAD) method initially produces a stern trim of 2.36° with increasing depth, followed by a bow trim of 1.32°, and then gradually returns to a trim-free state. In contrast, the trim control under the closed-loop gain shaping algorithm produces a stern trim of 1.2° with increasing depth, followed by a negligible bow trim, before returning to a trim-free state. For submarine attitude control, it is clear that the smaller the number and magnitude of trim changes, the easier it is to control.

[0143] Secondly, a simulation of depth-holding control under disturbance was performed to analyze its effect on improving the nonlinear terms of the control system. The parameters for the nonlinear modification terms were a1 = 0.6 and b1 = 2.4, and the parameters for the nonlinear feedback terms were a2 = 0.8 and b2 = 3.2. The submarine's forward speed was 6 knots, maintaining the periscope depth (i.e., 10m), and it cruised at a constant depth with no pitch angle. Wave disturbance was assumed under sea state VI, with a significant wave height Hs = 3.962m. The simulation results are as follows: Figures 11 to 14 As shown.

[0144] This allows for a qualitative analysis of the improvement effect on the nonlinear term, from Figure 11 and Figure 12 It can be seen that the controller effect improved by the nonlinear term for the changes in depth and pitch angle is almost the same as that before the improvement, but from... Figure 13 and Figure 14 It can be seen that the bow and stern rudder angle amplitudes improved by the nonlinear term are significantly reduced compared to before the improvement.

[0145] To quantitatively characterize the improvement effect, a performance index function J for control energy (rudder angle δ) and a performance index function H for control effect (depth ζ and pitch angle θ) are introduced. The performance index functions are shown in equations (21) and (22), and the maximum rudder angle δ is used as the reference value. b-max and δ s-max Mean state change (i.e., mean depth ζ) mean and mean pitch angle θ mean The effect of nonlinear term improvement is evaluated using the control effect as an evaluation factor.

[0146]

[0147]

[0148] Table 1. Evaluation of the improvement effect of depth channel nonlinear modification (T0=1200)

[0149]

[0150] Table 2. Evaluation Table of Improvement Effect of Nonlinear Feedback in Tilting Angle Channel (T0=1200)

[0151]

[0152] Table 1 shows that, with the nonlinear modification applied to the depth channel, the average depth remains largely unchanged, but the control performance evaluation index decreases by 64.18%, the maximum bow rudder angle decreases by 97.25%, and the evaluation index decreases by 94.00%. Table 2 shows that, with the nonlinear feedback applied to the pitch angle channel, the average pitch angle decreases by 97.56%, the control performance evaluation index decreases by 35.08%, the maximum stern rudder angle decreases by 74.19%, and the evaluation index decreases by 77.16%. In summary, the nonlinear term improvement to the controller error channel significantly reduces steering amplitude and saves control energy while maintaining or even enhancing control performance.

[0153] Based on the simulation results of this implementation, a comparison is made. Figures 9 to 10 and Figures 13 to 14 After analyzing the bow and stern rudder angles, it was found that the bow rudder angle in the submarine depth-maintaining control simulation was very small, maintaining a value on the order of 10⁻¹, which is far different from the rudder angle during depth-changing maneuvers, and can even be ignored in actual steering. Therefore, when the submarine is sailing at a constant depth, the depth change is very small, and the bow rudder is dominant in depth control. When the submarine's depth changes slightly, the bow rudder hardly participates in depth adjustment, and the stern rudder is dominant in pitch angle control. At this time, the main consideration is the pitching phenomenon caused by wave forces and moments.

[0154] Based on the above experimental results, the controller designed using the submarine vertical plane motion controller design method based on the closed-loop gain shaping algorithm in this embodiment has the following three beneficial effects:

[0155] 1. This invention employs a closed-loop gain shaping algorithm to design a submarine vertical plane motion controller, enabling rapid and accurate depth-changing and depth-holding maneuvers, thereby improving the submarine's underwater performance and safety. This algorithm optimizes the controller's closed-loop gain, achieving optimal response time and steady-state error, while also considering the strong coupling and multi-input multi-output characteristics of the submarine model, effectively solving the challenge of submarine vertical plane motion control.

[0156] 2. This invention utilizes nonlinear feedback driven by the arctangent function and nonlinear modification to improve the original linear error feedback, enhancing the robustness and energy-saving effect of the controller, reducing the impact of external interference, and decreasing the energy consumption of the servo motor. This nonlinear feedback can adaptively adjust the feedback gain according to the magnitude of the error, giving the controller strong tracking capability at large errors and good stability at small errors, avoiding overshoot and oscillation. This nonlinear modification can adaptively adjust the modification coefficient according to the rudder angle, giving the controller strong control capability at large rudder angles and low energy consumption at small rudder angles, avoiding excessive use and wear of the servo motor.

[0157] 3. This invention, through simulation experiments, verifies the effectiveness and superiority of the closed-loop gain shaping algorithm and nonlinear feedback in the vertical plane motion control of submarines, providing a new approach and method for the research and application of submarine control systems. Simulation results show that during variable-depth maneuvers, the controller designed in this invention can reach the predetermined depth in a shorter time, with a response time 53.00% faster than the ICAD method, demonstrating the speed of this invention. During constant-depth navigation, the controller designed in this invention can maintain the control accuracy of depth and pitch angle even with increased wave interference, while reducing the rudder angle amplitude and control energy of the bow and stern rudders, demonstrating the robustness and energy-saving effect of this invention.

[0158] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features; and these modifications or substitutions 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 design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm, characterized in that, Includes the following steps: S1: Establish a mathematical model of the submarine that only considers the vertical plane motion of the submarine, and simplify the mathematical model of the submarine into a submarine dynamics model; S2: Obtain the state-space equations of the submarine dynamics model based on the submarine dynamics model; S3: Based on the closed-loop gain shaping algorithm, a submarine vertical plane motion controller is constructed according to the state space equation; the submarine vertical plane motion controller includes a submarine depth controller and a submarine pitch controller. S4: Construct the depth nonlinear function and pitch nonlinear function of the submarine vertical plane motion controller, and obtain the control law of the submarine bow rudder actuator and the control law of the submarine stern rudder actuator based on the depth nonlinear function and pitch nonlinear function and the submarine vertical plane motion controller. S5: Control of the submarine's vertical plane motion is achieved based on the control laws of the submarine's bow rudder actuator and the submarine's stern rudder actuator.

2. The design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm according to claim 1, characterized in that, S1 specifically includes the following steps: S11: Construct a mathematical model of the submarine that only considers the vertical plane motion of the submarine. The expression of the submarine mathematical model is as follows: （1） In the formula: Indicates the mass of the submarine; Indicates the coordinate system along the submarine appendages Down Vertical acceleration of the axis; Indicates the pitch angle acceleration; Indicates the coordinate system along the submarine appendages Down The forward speed of the shaft; Represents the submarine's coordinate system relative to its appendages. Down Moment of inertia of the shaft; and These represent the bow elevator angle and stern elevator angle of a submarine, respectively. Represents the pitch angle; Indicates depth; and These represent the distance between the center of buoyancy and the center of gravity. shaft and Distance on; This refers to the buoyancy force acting on the submarine. This indicates the static force of a submarine in the water; All are hydrodynamic derivatives; and These represent wave disturbance force and torque, respectively. Indicates the submarine's roll angle; S12: Assuming the submarine is sailing at a constant depth and the depth change is a weak maneuver, the submarine mathematical model is simplified to obtain the submarine dynamics model. The expression of the submarine dynamics model is as follows: (2)。 3. The design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm according to claim 2, characterized in that, The state-space equation expression for the submarine dynamics model obtained in S2 is: (3) In the formula: , , , , , , , ; In the formula: Indicates the system status; This represents the input to the submarine dynamics model; All represent intermediate parameter variable matrices; This represents the output of the submarine dynamics model.

4. The design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm according to claim 3, characterized in that, S3 specifically includes the following steps: S31: Based on the closed-loop gain shaping algorithm and ignoring the wave force and torque on the submarine, formula (3) is transformed into the submarine vertical plane transfer function matrix. The expression of the submarine vertical plane transfer function matrix is: (4) In the formula: Represents the transfer function matrix of the submarine's vertical plane; Represents the identity matrix; Represents the Laplace operator; S32: Constructing the complementary sensitivity function matrix T The complementary sensitivity function matrix T The expression is （5） In the formula: The coefficient matrix represents the complement sensitivity function matrix; and All represent the parameters of the complemented sensitivity matrix; Represents the Laplace operator; and All represent coefficient matrices Parameters; S33: Obtain the submarine vertical plane motion controller according to formula (4) and formula (5) The submarine vertical plane motion controller The expression is （6） In the formula: Represents the transfer function matrix of the submarine's vertical plane; Represented as a complemented sensitivity function matrix; All of these represent the model parameters of the submarine dynamics model; All represent the model matrix elements of the submarine dynamics model; S34: Vertical plane motion controller for the submarine Decoupling is performed to obtain the submarine depth controller and the submarine pitch controller; the expressions for the submarine depth controller and the submarine pitch controller are as follows: , （7） In the formula: Indicates the submarine depth controller; This refers to the submarine's pitch control.

5. The design method for a submarine vertical plane motion controller based on a closed-loop gain shaping algorithm according to claim 4, characterized in that, S4 specifically includes the following steps: S41: Define the depth error of the submarine, the expression for which is: （8） In the formula: This represents the depth error of the submarine. Represents the submarine's current depth; This represents the submarine's set depth. S42: Using nonlinear feedback technology, a depth nonlinear function for the submarine's vertical plane motion controller is constructed based on the submarine's depth error. The expression for the depth nonlinear function is as follows: （9） In the formula: and All of these represent feedback design parameters for deeply nonlinear functions; Represents a deep nonlinear function; S43: Obtain the control law of the submarine bow rudder actuator based on the aforementioned depth nonlinear function and the submarine depth controller. The expression for the control law of the submarine bow rudder actuator is as follows: （10） In the formula: This input represents the angle of the submarine's bow elevator; S44: Define the pitch error of a submarine, the expression for which is: （11） In the formula: Represents the pitch error of the submarine; The submarine's tilt indicates its current position; The set pitch of the submarine; S45: Based on the submarine's pitch error and the submarine's pitch controller, obtain the current control law of the submarine's pitch controller. The control law of the current pitch controller The expression is （12） S46: Based on nonlinear modification technology, according to the control law of the current pitch controller The pitch nonlinear function is obtained and used as the control law for the submarine stern rudder actuator. The expression for the control law of the submarine stern rudder actuator is as follows: （13） In the formula: This represents the pitch nonlinear function; This input represents the angle of the submarine's stern elevator. and Both represent the modified design parameters of the pitch nonlinear function.

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