Fully-distributed multi-underwater-robot optimization tracking control method

Abstract

The invention discloses a fully-distributed multi-underwater-robot optimization tracking control method. A multi-underwater-robot system with a high-order nonlinear dynamic model is constructed into a fuzzy linearization system by using a fuzzy theory; designing a fully-distributed fixed time integral sliding mode control protocol, and solving an equivalent control system moving on a sliding mode surface; constructing a state consistency error system based on a navigation-following mechanism, defining a global performance index function based on the error system, designing a fully-distributed state feedback nominal optimization control protocol, and ensuring that the performance index function is globally optimal; the state consistency of a multi-underwater robot system is analyzed by using the Lyapunov stability theory, and asymptotic stability of a state consistency tracking error system based on a navigation-following mechanism is ensured. According to the method provided by the invention, the state consistency of the multi-underwater robot system based on the navigation-following mechanism can be realized with smaller control input.

Description

Technical Field

[0001] The invention belongs to the technical field of robots, and in particular relates to an optimization tracking control method for fully distributed multiple underwater robots. Background Art

[0002] In the prior art, the document "Stability and robustness analysis of minmax solutionsfor differential graphical games, Automatica, 2020, 121: 109117" discloses a multi-robot collaborative tracking control method based on the minimax principle. This method studies the state consistency tracking control problem of a multi-robot system and designs a tracking control algorithm based on the minimax principle. The technical problem with the collaborative control algorithm described in the document is that the control protocol based on the minimax principle cannot achieve the desired control performance with the optimal control input. Summary of the invention

[0003] In order to overcome the shortcomings of the prior art, the present invention provides a fully distributed multi-underwater robot optimization tracking control method, which uses fuzzy theory to construct a multi-underwater robot system with a high-order nonlinear dynamic model as a fuzzy linearization system; designs a fully distributed fixed-time integral sliding mode control protocol, and simultaneously solves the equivalent control system moving on the sliding mode surface; constructs a state consistency error system based on the pilot-follower mechanism, and defines a global performance index function based on the error system, designs a fully distributed state feedback nominal optimization control protocol, and ensures that the performance index function is globally optimal; uses Lyapunov stability theory to analyze the state consistency of the multi-underwater robot system, and ensures that the state consistency tracking error system based on the pilot-follower mechanism is asymptotically stable. The method provided by the present invention can achieve the state consistency of the multi-underwater robot system based on the pilot-follower mechanism with a smaller control input.

[0004] The technical solution adopted by the present invention to solve the technical problem is as follows:

[0005] Step 1: Using fuzzy theory, the multi-underwater robot system with a high-order nonlinear dynamic model is constructed as a fuzzy linearized system;

[0006] Step 2: Design a fully distributed fixed-time integral sliding mode control protocol for the fuzzy linearization system constructed in step 1, and solve the equivalent control system moving on the sliding surface;

[0007] Step 3: For the equivalent control system obtained in step 2, a state consistency error system based on the pilot-follower mechanism is constructed, and a global performance index function is defined based on the error system. A fully distributed state feedback nominal optimization control protocol is designed to ensure the global optimality of the performance index function.

[0008] Step 4: Use Lyapunov stability theory to analyze the state consistency of the multi-underwater robot system to ensure that the state consistency tracking error system based on the leader-follower mechanism is asymptotically stable.

[0009] Furthermore, the step 1 is specifically as follows:

[0010] Step 1-1: Give the following six-degree-of-freedom multi-underwater robot dynamics model:

[0011]

[0012] In the formula, is the position and angle information of the underwater robot, They represent the x-axis coordinate, y-axis coordinate and heading angle of the underwater robot rotating around the z-axis in the earth coordinate system respectively; is the speed information of the underwater robot, They represent the x-axis velocity, y-axis velocity and heading angular velocity in the earth coordinate system respectively; u i represents the control input, w i represents the uncertain external interference input, M, C(q i ) and D(q i ) are the inertia matrix, Coriolis force and centripetal force matrix and fluid damping matrix of the system respectively; is the coordinate transformation matrix, expressed as follows:

[0013]

[0014] Step 1-2: Rewrite equation (1) as a high-order nonlinear system using state-space representation:

[0015]

[0016] In the formula, u i 、w i They represent the control input and uncertain external disturbance input of the i-th underwater robot respectively;

[0017]

[0018] Status information x i is a 6×1 dimensional vector. Based on the fuzzy system theory, the high-order nonlinear system described in equation (2) is constructed as the following fuzzy linearization system:

[0019] Fuzzy rule k: If x i1 is λ k1 , and x i2 is λ k2 , …, and x i6 is λ k6 ,So:

[0020]

[0021] In the formula, x i1 ,x i2 ,...,x i6 represents the fuzzy sampling point of the system, λ k1 ,...,λ k6 represents the fuzzy set of the system, r represents the total number of fuzzy rules;

[0022] The weighted function method is used to rewrite equation (3) into a fuzzy linear system:

[0023]

[0024] In the formula, represents the weight function, and the function φ k (x i )satisfy And φ k (x i )≥0.

[0025] Furthermore, the step 2 is specifically as follows:

[0026] Step 2-1: For the fuzzy linearization system (4) in step 1, design the integral sliding mode variables as follows:

[0027]

[0028] In the formula, x i (0) is the status information x i At the initial value at time t = 0, F is a constant matrix and satisfies that the matrix FB is a full rank matrix, u im is the fully distributed state feedback nominal optimization control protocol to be designed; k (·) represents the coefficient of the weight function, A k represents the weight matrix of the linearized state quantity of the system, and B represents the weight matrix of the control input quantity;

[0029] Deriving both sides of equation (5):

[0030]

[0031] Step 2-2: For the system given in equation (6), design a fully distributed fixed-time sliding mode control protocol as follows:

[0032]

[0033] In the formula, sig(s i )=[sig(s i1 ),...,sig(s ip )] T Represents variable s i The sign function of β1, β2, β3 and ρ represent the sliding mode gain parameters to be designed;

[0034] Step 2-3: Substituting equation (7) into equation (6) yields:

[0035]

[0036] Select sliding mode gain parameters β1>0, β3>0, ρ>0, and β2>α||FB||, α≥||w i || is the upper bound of the uncertain external interference input; the Lyapunov function related to formula (6) is selected as follows:

[0037]

[0038] The derivative of formula (9) is:

[0039]

[0040] Step 2-4: According to the fixed-time stability theory, the sliding mode control protocol designed in equation (7) can ensure that the integral sliding mode variable s i = 0 is realized in a fixed time; According to formula (6), we know that s i =0 is equivalent to u i -u im +w i =0, so the equivalent control system of the fuzzy linearized system (4) moving on the sliding surface can be expressed as:

[0041]

[0042] Furthermore, the step 3 is specifically as follows:

[0043] Step 3-1: First, for the equivalent control system (11) in step 2, construct the following state consistency tracking error system based on the pilot-follower mechanism:

[0044] δ i =x i -x0,i=1,…,N. (12)

[0045] In the formula, x0 represents the pilot underwater robot that only provides reference state trajectory information, and its dynamic model is x0 = A(x0)x0;

[0046] Step 3-2: Taking the derivative of both sides of equation (12), we get:

[0047]

[0048] According to the Kronecker product, the global form of equation (13) can be expressed as:

[0049]

[0050] In the formula, I N represents the N-order identity matrix;

[0051]

[0052] Step 3-3: For the error system (14), define the global performance index function as follows:

[0053]

[0054] In the formula, The weighting matrix representing the tracking error, represents the weighted matrix of the control input, Q>0 and R>0 are symmetric matrices, and P is a positive definite solution to the following algebraic Riccati equation:

[0055]

[0056] Step 3-4: Based on the error system (14) and the performance index function (15), a fully distributed state feedback nominal optimization control protocol is designed as follows:

[0057]

[0058] In the formula, a ij represents the communication relationship between the ith follower underwater robot and the jth follower underwater robot. When the ith follower can obtain the status information of the jth follower, a ij =1, otherwise a ij =0; represents the in-degree of the ith follower underwater robot; h i represents the communication relationship between the ith follower underwater robot and the leader underwater robot, h i =1 means that the i-th follower can directly obtain the status information of the leader, K = R -1 B T P represents the control gain matrix;

[0059] According to equations (14) and (15), the following Hamiltonian function is selected:

[0060]

[0061] In the formula, represents a co-variable,

[0062] According to the optimization theory, the fully distributed state feedback nominal optimization control protocol (17) can ensure the global optimality of the performance indicator function (15).

[0063] Furthermore, the step 4 is specifically as follows:

[0064] The Lyapunov function related to equation (14) is selected as follows:

[0065]

[0066] The derivative of formula (19) is:

[0067]

[0068] Substituting formula (17) into formula (20), we have:

[0069]

[0070] Since Q>0 holds, we have Established; According to Lyapunov stability theory, system (14) is asymptotically stable, that is Realize state consistency of multiple underwater robots based on leader-follower mechanism.

[0071] The beneficial effects of the present invention are as follows:

[0072] The fully distributed multi-underwater robot optimization tracking control method proposed in the present invention can achieve faster tracking error convergence with smaller control input. Compared with the nominal control protocol based on the maximum and minimum principle in the literature, the fully distributed state feedback nominal optimization control protocol proposed in the present invention has a faster tracking error convergence speed. In addition, under the action of the fully distributed state feedback nominal optimization control protocol proposed in the present invention, the maximum value of the global control input norm of the four follower underwater robots is 2.29×10 4 , while in the literature, under the nominal control protocol based on the maximum-minimum principle, the maximum value of the global control input norm of the four follower underwater robots is 1.52×10 5 . BRIEF DESCRIPTION OF THE DRAWINGS

[0073] Figure 1is a schematic diagram of a communication topology structure between one leader underwater robot (marked as 0) and four follower underwater robots (marked as 1 to 4) in an embodiment of the present invention;

[0074] Figure 2 In the embodiment of the present invention, the state components of the four follower underwater robots under the control protocol proposed in the present invention are The tracking error norm curve of

[0075] Figure 3 In the embodiment of the present invention, the state components of the four follower underwater robots under the control protocol proposed in the present invention are The tracking error norm curve of

[0076] Figure 4 In the embodiment of the present invention, the state components θ of the four follower underwater robots under the control protocol proposed in the present invention are i The tracking error norm curve of

[0077] Figure 5 In the embodiment of the present invention, the state components of the four follower underwater robots under the control protocol in the background technology document are The tracking error norm curve of

[0078] Figure 6 In the embodiment of the present invention, the state components of the four follower underwater robots under the control protocol in the background technology document are The tracking error norm curve of

[0079] Figure 7 is the state component θ of the four follower underwater robots under the control protocol in the background art document in the embodiment of the present invention. i The tracking error norm curve of

[0080] Figure 8 It is a global control input norm curve of four follower underwater robots under the control protocol proposed in the embodiment of the present invention and the control protocol used in the background technology document. DETAILED DESCRIPTION

[0081] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0082] In order to overcome the shortcomings of the tracking control algorithm based on the maximum and minimum principle in the literature in terms of network structure and the inability to optimize the control input, the present invention provides a fully distributed multi-underwater robot optimization tracking control method based on fuzzy system theory and sliding mode control theory. The multi-underwater robot system with a high-order nonlinear dynamic model is constructed as a fuzzy linearized system using fuzzy theory; in order to overcome the uncertain ocean current interference in the complex marine environment, a fully distributed fixed time integral sliding mode control protocol is designed for the constructed fuzzy linearized system, and the equivalent control system moving on the sliding mode surface is solved at the same time; for the equivalent control system on the sliding mode surface, a state consistency error system based on the pilot-follower mechanism is constructed, and a global performance index function is defined based on the error system, and a fully distributed state feedback nominal optimization control protocol is designed to ensure the global optimal performance index function; the state consistency of the multi-underwater robot system is analyzed using the Lyapunov stability theory to ensure that the state consistency tracking error system based on the pilot-follower mechanism is asymptotically stable. The method provided by the present invention can achieve state consistency of a multi-underwater robot system based on a leader-follower mechanism with smaller control input, and the solution of input parameters is independent of global information, which has the advantage of being fully distributed and is more conducive to the application of the controller in actual systems.

[0083] The technical solution adopted by the present invention to solve the technical problem is: a fully distributed multi-underwater robot optimization tracking control method, which is characterized by comprising the following steps:

[0084] Step 1: Use fuzzy theory to construct the multi-underwater robot system with high-order nonlinear dynamic model into a fuzzy linearized system. First, the following six-degree-of-freedom multi-underwater robot dynamic model is given:

[0085]

[0086] In the formula, is the position and angle information of the underwater robot, is the speed information of the underwater robot; u i represents the control input, w i represents the uncertain external interference input, M, C(q i ) and D(q i )q i are the inertia matrix, Coriolis force and centripetal force matrix, and fluid damping matrix of the system, The coordinate transformation matrix is ​​expressed as follows:

[0087]

[0088] Using state space representation, equation (1) can be rewritten as:

[0089]

[0090] In the formula,

[0091]

[0092] Considering the state information x i is a 6×1 dimensional vector. Based on the fuzzy system theory, the high-order nonlinear system described in equation (2) is constructed as the following fuzzy linearization system:

[0093] Fuzzy rule k: If x i1 is λ k1 And... and x i6 is λ k6 ,So

[0094]

[0095] In the formula, λ k1 ,...,λ k6 represents the fuzzy set of the system, and r represents the total number of fuzzy rules. For the convenience of analysis, the system described in equation (3) is rewritten using the weighted function method as follows:

[0096]

[0097] In the formula, represents the weight function, and the function φ k (x i )satisfy And φ k (x i )≥0.

[0098] Step 2: To overcome the uncertain ocean current interference in the complex ocean environment, a fully distributed fixed-time integral sliding mode control protocol is designed for the fuzzy linearized system (4) constructed in step 1, and the equivalent control system moving on the sliding mode surface is solved. First, for the fuzzy linearized system (4) in step 1, the integral sliding mode variable is designed as follows:

[0099]

[0100] In the formula, x i (0) is the status information x i At the initial value at time t = 0, F is a constant matrix and satisfies that the matrix FB is a full rank matrix, u im is the fully distributed state feedback nominal optimization control protocol to be designed. Taking the derivative of both sides of equation (5), we have

[0101]

[0102] For the system given in equation (6), the fully distributed fixed-time sliding mode control protocol is designed as:

[0103]

[0104] In the formula, sig(s i )=[sig(s i1 ),...,sig(s ip )] T Represents variable s i The sign function of β1, β2, β3 and ρ represent the sliding mode gain parameters to be designed. Substituting equation (7) into equation (6) yields

[0105]

[0106] Select sliding mode gain parameters β1>0, β3>0, ρ>0, and β2>α||FB||, α≥||w i || is the upper bound of the uncertain external interference input. The Lyapunov function related to equation (6) is selected as follows:

[0107]

[0108] The derivative of formula (9) is:

[0109]

[0110] According to the fixed time stability theory, the sliding mode control protocol designed in equation (7) can ensure that the integral sliding mode variable s i = 0 is achieved in a fixed time. According to formula (6), we know that s i =0 is equivalent to u i -u im +w i = 0, so the equivalent control system of the fuzzy linearized system (4) moving on the sliding surface can be expressed as

[0111]

[0112] Step 3: For the equivalent control system on the sliding surface obtained in step 2, a state consistency error system based on the pilot-follower mechanism is constructed, and the global performance index function is selected based on the error system, and a fully distributed state feedback nominal optimization control protocol is designed to ensure the global optimal performance index function. First, for the equivalent control system (11) in step 2, the following state consistency tracking error system based on the pilot-follower mechanism is constructed:

[0113] δ i =x i -x0,i=1,…,N. (33)

[0114] In the formula, x0 represents the pilot underwater robot that only provides reference state trajectory information, and its dynamic model is x0 = A(x0)x0. Taking the derivative of both sides of equation (12), we have

[0115]

[0116] According to the Kronecker product, the global form of equation (13) can be expressed as

[0117]

[0118] In the formula,

[0119]

[0120] For the error system (14), the global performance index function is defined as follows:

[0121]

[0122] In the formula, The weighting matrix representing the tracking error, represents the weighted matrix of the control input, Q>0 and R>0 are symmetric matrices, and P is a positive definite solution to the following algebraic Riccati equation:

[0123]

[0124] According to the error system (14) and the performance index function (15), the fully distributed state feedback nominal optimization control protocol is designed as follows:

[0125]

[0126] In the formula, a ij represents the communication relationship between the ith follower underwater robot and the jth follower underwater robot. When the ith follower can obtain the status information of the jth follower, a ij =1, otherwise a ij =0; represents the in-degree of the ith follower underwater robot; h i represents the communication relationship between the ith follower underwater robot and the leader underwater robot, h i =1 means that the i-th follower can directly obtain the status information of the leader, K = R -1 B T P represents the control gain matrix. According to equations (14) and (15), the following Hamiltonian function is selected:

[0127]

[0128] In the formula, represents a co-variable, According to optimization theory, it can be proved that the fully distributed state feedback nominal optimization control protocol (17) can ensure the global optimality of the performance indicator function (15).

[0129] Step 4: Use Lyapunov stability theory to analyze the state consistency of the multi-underwater robot system to ensure that the state consistency tracking error system based on the pilot-follower mechanism in step 3 is asymptotically stable. First, select the Lyapunov function related to equation (14) as follows:

[0130]

[0131] Taking the derivative of equation (19) we have

[0132]

[0133] Substituting equation (17) into equation (20), we have

[0134]

[0135] Since Q>0 holds, we have According to Lyapunov stability theory, system (14) is asymptotically stable, that is, Realize state consistency of multiple underwater robots based on leader-follower mechanism.

[0136] Example:

[0137] Assume that there is a multi-underwater robot cluster system with 1 leader and 4 followers. The communication relationship between the robots is as follows: Figure 1 As shown, the Laplace matrix corresponding to the communication topology between followers and the communication relationship matrix between followers and leaders are expressed as follows:

[0138]

[0139] The system parameter matrix of each underwater robot is selected as

[0140]

[0141] The fuzzy system is at the operating point q ix ={-0.6,0.6},q iy ={-0.3,0.3}, For A(x i ) are sampled and substituted into the original system (2) to obtain the coefficient matrix A corresponding to 16 sets of fuzzy rules: k , k=1,...,16. Let the external disturbance input acting on the four follower robots be

[0142] w1=[sin(0.1t),sin(0.1t),sin(0.1t)] T ,w2=[cos(0.1t),cos(0.1t),cos(0.1t)] T

[0143] w3=0.5[sin(0.1t)+1,sin(0.1t)+1,sin(0.1t)+1] T ,

[0144] w4=0.5[cos(0.1t)+1,cos(0.1t)+1,cos(0.1t)+1] T ,

[0145] Then the upper bound parameter of the uncertain external interference input can be taken as For the fully distributed fixed-time integral sliding mode control protocol designed for equation (7), let the matrix F = [0 3×3 ,I3], respectively select the sliding mode gain parameters as β1=β3=1, β2=α||FB||+0.1, and ρ=2. Further, let Q=10I6, R=I3, then the control gain matrix in the fully distributed state feedback nominal optimization control protocol (17) can be calculated as

[0146]

[0147] In addition, according to the control method adopted in the literature, while maintaining u is Under the same conditions, the following nominal control protocol based on the maximum and minimum principle is designed:

[0148]

[0149] Where P i and R i is a positive definite symmetric matrix and satisfies the following algebraic Riccati equation:

[0150]

[0151] Wherein, γ = 2 represents the coupling strength parameter.

[0152] The initial states of the leader and the four follower underwater robots are

[0153] x0(0)=[0,0,0,-0.6,-0.3,0.1] T ,

[0154]

[0155] Then, the state components of the four follower underwater robots under the control protocol proposed by the present invention and the control protocol described in the literature can be obtained respectively. θ i The tracking error norm curve is as follows Figures 2 to 7 As shown, the global control input norm Curve like Figure 8 As shown in the figure. It can be seen from the state component tracking error norm curve that compared with the nominal control protocol based on the maximum and minimum principle described in the literature (22), the fully distributed state feedback nominal optimization control protocol proposed in the present invention has a faster tracking error convergence speed. In addition, under the action of the fully distributed state feedback nominal optimization control protocol proposed in the present invention, the maximum value of the global control input norm of the four follower underwater robots is 2.29×10 4 , while in the literature, under the nominal control protocol based on the maximum-minimum principle, the maximum value of the global control input norm of the four follower underwater robots is 1.52×10 5 Therefore, the advantage of the fully distributed optimization tracking control method for multiple underwater robots proposed in the present invention is that it has a smaller control input and a faster convergence speed.

Claims

1. A fully distributed multi-underwater robot optimization tracking control method, characterized in that: The steps include: Step 1: Using fuzzy theory, the multi-underwater robot system with a high-order nonlinear dynamic model is constructed as a fuzzy linearized system; Step 2: Design a fully distributed fixed-time integral sliding mode control protocol for the fuzzy linearization system constructed in step 1, and solve the equivalent control system moving on the sliding surface; Step 3: For the equivalent control system obtained in step 2, a state consistency error system based on the pilot-follower mechanism is constructed, and a global performance index function is defined based on the error system. A fully distributed state feedback nominal optimization control protocol is designed to ensure the global optimality of the performance index function. Step 4: Use Lyapunov stability theory to analyze the state consistency of the multi-underwater robot system to ensure that the state consistency tracking error system based on the leader-follower mechanism is asymptotically stable.

2. The method for optimizing tracking and controlling a fully distributed multi-underwater robot according to claim 1, characterized in that: The step 1 is specifically as follows: Step 1-1: Give the following six-degree-of-freedom multi-underwater robot dynamics model: In the formula, is the position and angle information of the underwater robot, θ i They represent the x-axis coordinate, y-axis coordinate and heading angle of the underwater robot rotating around the z-axis in the earth coordinate system respectively; is the speed information of the underwater robot, q ix ,q iy , They represent the x-axis velocity, y-axis velocity and heading angular velocity in the earth coordinate system respectively; u i represents the control input, w i represents the uncertain external interference input, M, C(q i ) and D(q i ) are the inertia matrix, Coriolis force and centripetal force matrix and fluid damping matrix of the system respectively; is the coordinate transformation matrix, expressed as follows: Step 1-2: Rewrite equation (1) as a high-order nonlinear system using state-space representation: In the formula, u i 、w i They represent the control input and uncertain external disturbance input of the i-th underwater robot respectively; Status informationx i is a 6×1 dimensional vector. Based on the fuzzy system theory, the high-order nonlinear system described in equation (2) is constructed as the following fuzzy linearization system: Fuzzy rule k: If x i1 is λ k1 , and x i2 is λ k2 , …, and x i6 is λ k6 ,So: In the formula, x i1 ,x i2 ,...,x i6 represents the fuzzy sampling point of the system, λ k1 ,...,λ k6 represents the fuzzy set of the system, r represents the total number of fuzzy rules; The weighted function method is used to rewrite equation (3) into a fuzzy linear system: In the formula, represents the weight function, and the function φ k (x i )satisfy And φ k (x i )≥0.

3. A fully distributed multi-underwater robot optimization tracking control method according to claim 2, characterized in that: The step 2 is specifically as follows: Step 2-1: For the fuzzy linearization system (4) in step 1, design the integral sliding mode variables as follows: In the formula, x i (0) is the status information x i At the initial value at time t = 0, F is a constant matrix and satisfies that the matrix FB is a full rank matrix, u im is the fully distributed state feedback nominal optimization control protocol to be designed; k (·) represents the coefficient of the weight function, A k represents the weight matrix of the linearized state quantity of the system, and B represents the weight matrix of the control input quantity; Deriving both sides of equation (5): Step 2-2: For the system given in equation (6), design a fully distributed fixed-time sliding mode control protocol as follows: In the formula, sig(s i )=[sig(s i1 ),...,sig(s ip )] T Represents variable s i The sign function of β1, β2, β3 and ρ represent the sliding mode gain parameters to be designed; Step 2-3: Substituting equation (7) into equation (6) yields: Select sliding mode gain parameters β1>0, β3>0, ρ>0, and β2>α||FB||, α≥||w i || is the upper bound of the uncertain external interference input; the Lyapunov function related to formula (6) is selected as follows: The derivative of formula (9) is: Step 2-4: According to the fixed-time stability theory, the sliding mode control protocol designed in equation (7) can ensure that the integral sliding mode variable s i = 0 is realized in a fixed time; According to formula (6), we know that s i =0 is equivalent to u i -u im +w i =0, so the equivalent control system of the fuzzy linearized system (4) moving on the sliding surface can be expressed as:

4. The method for optimizing tracking and controlling a fully distributed multi-underwater robot according to claim 3 is characterized in that: The step 3 is specifically as follows: Step 3-1: First, for the equivalent control system (11) in step 2, construct the following state consistency tracking error system based on the pilot-follower mechanism: δ i =x i -x0,i=1,…,N. (12) In the formula, x0 represents the pilot underwater robot that only provides reference state trajectory information, and its dynamic model is x0 = A(x0)x0; Step 3-2: Taking the derivative of both sides of equation (12), we get: According to the Kronecker product, the global form of equation (13) can be expressed as: In the formula, I N represents the N-order identity matrix; Step 3-3: For the error system (14), define the global performance index function as follows: In the formula, The weighting matrix representing the tracking error, represents the weighted matrix of the control input, Q>0 and R>0 is a symmetric matrix, and P is a positive definite solution to the following algebraic Riccati equation: Step 3-4: Based on the error system (14) and the performance index function (15), a fully distributed state feedback nominal optimization control protocol is designed as follows: In the formula, a ij represents the communication relationship between the ith follower underwater robot and the jth follower underwater robot. When the ith follower can obtain the status information of the jth follower, a ij =1, otherwise a ij =0; represents the in-degree of the ith follower underwater robot; h i represents the communication relationship between the ith follower underwater robot and the leader underwater robot, h i =1 means that the i-th follower can directly obtain the status information of the leader, K = R -1 B T P represents the control gain matrix; According to equations (14) and (15), the following Hamiltonian function is selected: In the formula, represents a co-variable, According to the optimization theory, the fully distributed state feedback nominal optimization control protocol (17) can ensure the global optimality of the performance indicator function (15).

5. A fully distributed multi-underwater robot optimization tracking control method according to claim 4, characterized in that: The step 4 is specifically as follows: The Lyapunov function related to equation (14) is selected as follows: The derivative of formula (19) is: Substituting formula (17) into formula (20), we have: Since Q>0 holds, we have Established; According to Lyapunov stability theory, system (14) is asymptotically stable, that is Realize state consistency of multiple underwater robots based on leader-follower mechanism.

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