Time-varying formation cost preserving tracking control method for mobile robot system

Abstract

The application discloses a time-varying formation cost preserving tracking control method of a mobile robot system. The method designs an interference observer to estimate the real value of unknown interference of the mobile robot system, and gives a corresponding anti-interference controller. In addition, the whole process control cost of formation tracking is not infinite, and the cost upper limit of formation tracking control is estimated through a cost function. Finally, with the help of Lyapunov stability theorem, several sufficient criteria are established to ensure consistent ultimate boundedness, and time-varying tracking control is realized.

Description

Technical Field

[0001] This invention relates to a cost-preservation tracking control method, specifically to a time-varying formation cost-preservation tracking control method for a mobile robot system. Background Technology

[0002] Many dynamic models used in mobile robot systems may experience communication breakdowns between followers and leaders under unknown external disturbances, affecting formation and tracking performance. Therefore, discussions on time-varying formation tracking control for mobile robot systems are crucial for ensuring accurate observation of unknown external disturbances. Furthermore, the control cost of the entire formation tracking process is not infinite; estimating the upper bound of the cost and designing a cost-effective control scheme is essential for practical engineering applications. Based on this idea, researchers have conducted extensive research on mobile robot systems under unknown disturbances and achieved a series of results. Summary of the Invention

[0003] The purpose of this invention is to propose a time-varying formation cost-preserving tracking control method for mobile robot systems, which can effectively improve the performance of mobile robot control systems.

[0004] The specific technical solution of the present invention is as follows: A time-varying formation cost-preserving tracking control method for a mobile robot system, comprising the following steps:

[0005] Using the output function of a mobile robot system, design an interference observer to estimate the true value of unknown interference in the system, and provide a corresponding anti-interference compensator.

[0006] The dynamic coupling weights in a time-varying formation distributed controller are estimated using an adaptive control method, where the time-varying formation is given in advance.

[0007] Using Lyapunov's stability theorem, the uniform final bounded condition and upper bound of the cost function for mobile robot systems are derived, thereby enabling time-varying formation tracking control.

[0008] Under external disturbances, the mobile robot system model for time-varying formation tracking control is as follows:

[0009]

[0010] In the formula, x i (t) represents the system state of the i-th node, u i (t) represents the control input of the i-th node, y i (t) represents the system state at the i-th node, and A, C, and D represent known parameter matrices, which are expressed as follows:

[0011]

[0012] Where I2 represents the second-order identity matrix. Denotes the Kronecker product, d i (t) represents the unknown external input of the i-th node, which can be represented by one of the following external systems.

[0013]

[0014] Where η i (t) represents the state of the external system, and M and N represent known parameter matrices, specifically as follows:

[0015]

[0016] To construct an observer based on output perturbation, the first derivative of the output function is calculated. achievable

[0017]

[0018] Where C * It is a pseudo-inverse matrix that satisfies C * =(C * TC) -1 C T ;

[0019] For unknown interference d i (t), design the following interference observer:

[0020]

[0021] in For d i The estimated value of (t), For η i The estimated value of (t), I i (t) represents the integral term in the observer, and L1 and L2 are the unknown observer gains;

[0022] Define an intermediate variable The interference observer can be further written as

[0023]

[0024] Define the interference estimation error as The estimation error system can be written as

[0025]

[0026] To achieve time-varying formation tracking control, the corresponding leader system can be represented as:

[0027]

[0028] In the formula, x0(t) represents the system state of the 0th node, u0(t) represents the control input of the 0th node, and y0(t) represents the system state of the 0th node.

[0029] Define e i (t)=x i (t)-h i (t)-x0(t) is the formation error, e y (t)=y i If (t)-y0(t) is the system output error, then the error system can be expressed as:

[0030]

[0031] Where h i (t) represents the formation.

[0032] Design a distributed controller u i (t) to implement time-varying formation tracking control and compensate for unknown disturbances.

[0033]

[0034] Where K is the feedback gain, v i (t) represents the compensation signal function, ω ij and ω i0 This represents the unknown coupling weights, and they can be estimated in the following ways.

[0035] in For unknown coupling weights ω ij The estimated value, For unknown coupling weights ω i0 The estimated value, ξ ij and ξ i0 All represent positive constants, and P is an unknown positive definite gain matrix;

[0036] Substituting the distributed controller into the error system yields...

[0037]

[0038] Define the cost function J C for

[0039] J C (t)=J Ce (t)+J Cu (t)

[0040] in

[0041]

[0042] In the above equation, W1 and W2 are two positive definite matrices, and ζ i (t)=Ke i (t)

[0043] The formation tracking and control proof process is as follows:

[0044] C001: Select a Lyapunov function of the following form:

[0045]

[0046] C002: Where Q1 and Q2 are arbitrary positive definite matrices;

[0047] C003: Calculate the first derivative of V(t).

[0048]

[0049] C004: When Sometimes,

[0050]

[0051] C005: Among them Given a matrix, which satisfies

[0052] C006: Definition Therefore:

[0053]

[0054] C007: Assumption It is a non-singular matrix, from which we can obtain

[0055]

[0056] C008: With the help of condition ω ij (0)=ω ji (0) and ξ ij =ξ ji >0, we can obtain ω ij (t)=ω ji (t) and

[0057]

[0058] C009: Combining C003, C007, and C008 to arrive at...

[0059]

[0060] C010: Let K = DT Q1, P = Q1DD T Q1, then C009 can be rewritten as

[0061]

[0062] C011: Among them

[0063]

[0064] C012: Based on Lyapunov stability theory, when... Sometimes, Therefore, the mobile robot system achieves consistency and is eventually bounded.

[0065] C013: Based on cost function J C The definition of the cost-preservation function can be rewritten as follows:

[0066]

[0067] C014: Then for any time T≥0, we have

[0068]

[0069] C015: Condition-based At that time, there was J T If ≤V(0)-V(T), then J * The upper bound is represented as

[0070] Attached Figure Description

[0071] Figure 1 The topology diagram for seven mobile robots, including one leader and six followers;

[0072] Figure 2 Dynamic trajectory diagram for time-varying formation tracking control of mobile robots;

[0073] Figure 3 Dynamic trajectory diagram of adaptive coupling weights;

[0074] Figure 4 and 5 A dynamic trajectory diagram of the mobile robot's position along the x and y axes;

[0075] Figure 6 and 7 A dynamic trajectory diagram of the mobile robot's speed along the x and y axes;

[0076] Figure 8 and 9A dynamic trajectory diagram of the control input for a mobile robot on the x and y axes; Detailed Implementation

[0077] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that these embodiments are for illustrative purposes only and are not intended to limit the scope of the invention. After reading the present invention, any modifications of the present invention in various equivalent forms by those skilled in the art will fall within the scope defined by the appended claims.

[0078] A time-varying formation cost-preserving tracking control method for a mobile robot system includes the following steps:

[0079] Step 1: Set the various system parameters;

[0080] Step 2: Set up an observer for unknown disturbances to estimate their true values;

[0081] Step 3: Set up an adaptive control method to estimate the dynamic coupling weights in the time-varying formation distributed controller;

[0082] Step 4: Set up the estimation error system;

[0083] Step 5: Set up the tracking error system;

[0084] An embodiment of the present invention is described below:

[0085] Consider a mobile robot system, whose corresponding dynamic models are as follows:

[0086]

[0087]

[0088] The topology diagram of the seven mobile robots is as follows: Figure 1 As shown, the dynamic trajectory diagram of the mobile robot performing formation tracking control is as follows: Figure 2 As shown, the dynamic trajectory diagram of the unknown coupling weights is as follows: Figure 3 As shown, the dynamic trajectory diagram of the mobile robot's position on the x-axis and y-axis is as follows: Figure 4 and 5 As shown, the velocity trajectory diagram of the mobile robot along the x-axis and y-axis is as follows. Figure 6 and 7 As shown, the dynamic trajectory diagram of the mobile robot's control input on the x-axis and y-axis is as follows: Figure 8 and 9 As shown.

Claims

1. A time-varying formation cost-preserving tracking control method for a mobile robot system, characterized in that, Includes the following steps: Design an interference observer using the output function of a mobile robot system to estimate the true value of unknown interference in the system, and provide a corresponding anti-interference compensator, specifically: Define an intermediate variable Represented as The interference observer can be designed as in For the first Unknown external input at each node The estimated value, For the first External system status of each node The estimated value, For the first The integral term in each node observer and For unknown observer gain, Indicates the first Control input for each node Indicates the first The system output of each node , , , , Given a parameter matrix, It is a pseudo-inverse matrix that satisfies , Indicates transpose; the anti-interference compensator is given by the following formula. in It is feedback gain. Indicates the first The system status of each node Indicates the first The system status of each node Indicates the first Control input for each node For formation, Represents the compensation signal function. and Indicates unknown coupling weights. and Represents the weighted connection matrix; The dynamic coupling weights in a time-varying formation distributed controller are estimated using an adaptive control method, where the time-varying formation is given in advance, specifically as follows: in Unknown coupling weights The estimated value, Unknown coupling weights The estimated value, and All represent positive numbers. It is an unknown positive definite gain matrix; Using Lyapunov's stability theorem, a uniform final bounded condition and an upper bound on the cost function of a mobile robot system are given, thereby realizing time-varying formation tracking control, specifically: The consistent eventual bounded condition of a mobile robot system is in , , , Unknown coupling weights The final estimate, Weighted connection matrix In matrix form, express Unit matrix of order, Represents the Kronecker product. and Let be any positive definite matrix; upper bound of the cost function for in For formation error The value at time 0, For interference estimation error The value at time 0, for In matrix form, for In matrix form.

2. The time-varying formation cost-preserving tracking control method for a mobile robot system according to claim 1, characterized in that... Under interference from external systems, an interference observer is designed using the output function of the mobile robot system to estimate the true value of the unknown interference in the system, and a corresponding anti-interference compensator is given. The specific steps are as follows: Under external disturbances, the mobile robot system model for time-varying formation tracking control is as follows: In the formula, Indicates the first The system status of each node Indicates the first Control input for each node Indicates the first The system status of each node and Given a parameter matrix, its specific representation is as follows: ， ， in Describing a second-order identity matrix, Represents the Kronecker product. Indicates the first The unknown external inputs of a node can be represented by one of the following external systems. In the formula Indicates the state of the external system. and This represents a known parameter matrix, specifically as follows: ， ； To construct an observer based on output perturbation, the first derivative of the output function is calculated. achievable ， in It is a pseudo-inverse matrix that satisfies ; For unknown interference Design the following interference observer: in for The estimated value, for The estimated value, For the integral term in the observer, and For unknown observer gain; Define the interference estimation error as The estimation error system can be written as 。 3. The time-varying formation cost-preserving tracking control method for a mobile robot system according to claim 1, characterized in that... The dynamic coupling weights in a time-varying formation distributed controller are estimated using an adaptive control method, where the time-varying formation is given in advance. The specific steps are as follows: To achieve time-varying formation tracking control, the corresponding leader system can be represented as: In the formula, Indicates the first The system status of each node Indicates the first Control input for each node Indicates the first The system status of each node; definition For formation error, Let the system output error be the error, then the error system can be represented as: in For formation; Substituting the anti-interference compensator into the error system yields... Define cost function for in In the above equation and They are two positive definite matrices. .

4. The time-varying formation cost-preserving tracking control method for the mobile robot system according to claim 1, using Lyapunov's stability theorem, provides a uniform final bounded condition and an upper bound on the cost function for the mobile robot system, thereby achieving time-varying formation tracking control. The specific proof process is as follows: B001: Select a Lyapunov function of the following form: In the formula Let be any positive definite matrix; B002: Calculation first derivative : B003: When Sometimes, in For a given matrix to satisfy , B004: Definition Then we have: ， B005: Assumption It is a non-singular matrix, from which we can obtain ， B006: With the help of conditions and It can be concluded that as well as B007: Combining B003-B006, we can conclude that... B008: Order , Then B007 can be rewritten as in , , ， ， ； B009: Based on Lyapunov stability theory, when... Sometimes, Therefore, the mobile robot system achieves consistency and is eventually bounded. B0010: Based on cost function By definition, the cost-preservation function can be rewritten as B0011: So for any time... ,have B0012: Based on conditions Sometimes, ,So The upper bound is represented as 。

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