Time-varying formation insurance cost tracking control method of mobile robot system

Abstract

The invention discloses a time-varying formation insurance cost tracking control method of a mobile robot system. According to the method, an interference observer is designed to estimate an unknown interference true value of a mobile robot system, and a corresponding anti-interference controller is provided. In addition, the control cost of the whole process of formation tracking is not endless, and the upper bound of the cost of formation tracking control is estimated through a cost function. And finally, by means of the Lyapunov stability theorem, several sufficient criteria are established and are respectively ensured to be consistent and finally bounded, so that time-varying tracking control is realized.

Description

Technical Field

[0001] The invention relates to a guaranteed cost tracking control method, and in particular to a guaranteed cost tracking control method for a time-varying formation of a mobile robot system. Background Art

[0002] Many dynamic models for mobile robot systems may cause the communication information between followers and leaders to be disconnected when subject to unknown external disturbances, thus affecting the formation and tracking performance of the formation. Therefore, the discussion on the time-varying formation tracking control of mobile robot systems is crucial to ensure the accurate observation of unknown external disturbances of the system. In addition, the control cost of the entire process of formation tracking is not infinite. How to estimate the upper bound of the cost and design a cost-guaranteed control scheme is close to practical engineering applications. Based on this idea, researchers have conducted a lot of research on mobile robot systems under unknown disturbances and achieved a series of results. Summary of the invention

[0003] The purpose of the present invention is to propose a time-varying formation guaranteed cost tracking control method for a mobile robot system, which can effectively improve the performance of the mobile robot control system;

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

[0005] Using the output function of the mobile robot system, a disturbance observer is designed to estimate the true value of the unknown disturbance of the system, and the corresponding anti-interference compensator is given.

[0006] The adaptive control method is used to estimate the dynamic coupling weights in the time-varying formation distributed controller, where the time-varying formation is given in advance.

[0007] With the help of Lyapunov stability theorem, the uniform final bounded conditions and upper bounds of the cost function of the mobile robot system are obtained, thereby realizing time-varying formation tracking control.

[0008] Under external disturbance, 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 of the i-th node, A, C and D represent the known parameter matrices, which are expressed as Where I2 represents the second-order unit matrix, represents the Kronecker product, d i (t) represents the unknown external input of the i-th node, which can be represented by an external system as follows:

[0011]

[0012] where η i (t) represents the state of the external system, M and N represent the known parameter matrices, which are specifically expressed as:

[0013] To construct an output disturbance observer, we calculate the first-order derivative of the output function Available

[0014]

[0015] Among them C * is a pseudo-inverse matrix that satisfies C * =(C * TC) -1 C T ;

[0016] For unknown interference d i (t), design the following disturbance observer:

[0017]

[0018] in is d i The estimated value of (t), is η i The estimated value of (t), I i (t) is the integral term in the observer, L1 and L2 are the unknown observer gains;

[0019] Define an intermediate variable The disturbance observer can be further written as

[0020]

[0021] The interference estimation error is defined as The estimated error system can be written as

[0022]

[0023] To achieve time-varying formation tracking control, the corresponding leader system can be expressed as

[0024]

[0025] 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.

[0026] Definition i (t) = x i (t)-h i (t)-x0(t) is the formation error, e y (t) = y i (t)-y0(t) is the system output error, so the error system can be expressed as

[0027]

[0028] where h i (t) is the formation

[0029] Design a distributed controller i (t) To achieve time-varying formation tracking control and compensate for unknown disturbances

[0030]

[0031] Where K is the feedback gain, v i (t) represents the compensation signal function, ω ij and ω i0 denote the unknown coupling weights, and they can be estimated by

[0032] in is the unknown coupling weight ω ij The estimated value of is the unknown coupling weight ω i0 The estimated value of ξ ij and i0 All represent positive constants, and P is the unknown positive definite gain matrix;

[0033] Substituting the distributed controller into the error system, we can get

[0034]

[0035] Define the cost function J C for

[0036] J C (t) = J Ce (t)+J Cu (t)

[0037] in

[0038]

[0039] In the above equation, W1 and W2 are two positive definite matrices, ζ i (t) = Ke i (t)

[0040] The formation tracking control proof process is as follows:

[0041] C001: Select a Lyapunov function of the following form:

[0042]

[0043] C002: where Q1, Q2 are any positive definite matrices;

[0044] C003: Calculate the first derivative of V(t)

[0045]

[0046] C004: When Sometimes, there are

[0047]

[0048] C005: is a given matrix that satisfies

[0049] C006: Definition Then we have:

[0050]

[0051] C007: Assumption is a non-singular matrix, we can get

[0052]

[0053] C008: With the help of condition ω ij (0) = ω ji (0) and ξ ij =ξ ji >0, we can get ω ij (t) = ω ji (t) and

[0054]

[0055] C009: Combine C003, C007 and C008 to obtain

[0056]

[0057] C010: Let K = D T Q1,P=Q1DD TQ1, then C009 can be rewritten as

[0058]

[0059] C011:

[0060] C012: Based on Lyapunov stability theory, when Sometimes, there are Then the mobile robot system achieves consistent ultimate boundedness;

[0061] C013: Based on cost function J C The definition of guaranteed cost function can be rewritten as

[0062]

[0063] C014: Then for any time T ≥ 0, we have

[0064]

[0065] C015: Condition-based When there is J T ≤V(0)-V(T), then J * The upper bound of

[0066] BRIEF DESCRIPTION OF THE DRAWINGS

[0067] Figure 1 The topological structure diagram of seven mobile robots includes a leader and six followers;

[0068] Figure 2 Dynamic trajectory diagram for time-varying formation tracking control of mobile robots;

[0069] Figure 3 Dynamic trajectory diagram of adaptive coupling weights;

[0070] Figure 4 and 5 It is the dynamic trajectory diagram of the mobile robot position on the x-axis and y-axis;

[0071] Figure 6 and 7 It is the dynamic trajectory diagram of the mobile robot speed on the x-axis and y-axis;

[0072] Figure 8 and 9 Dynamic trajectory diagrams on the x-axis and y-axis are input for mobile robot control; DETAILED DESCRIPTION

[0073] The present invention is further explained below in conjunction with specific embodiments. It should be understood that these embodiments are only used to illustrate the present invention and are not used to limit the scope of the present invention. After reading the present invention, various equivalent forms of modifications to the present invention by those skilled in the art all fall within the scope defined by the claims attached to this application.

[0074] A guaranteed cost tracking control method for a time-varying formation of a mobile robot system comprises the following steps:

[0075] Step 1: Set various system parameters;

[0076] Step 2: Set up an observer for unknown disturbance to estimate its true value;

[0077] Step 3: Setting an adaptive control method to estimate the dynamic coupling weights in the time-varying formation distributed controller;

[0078] Step 4: Set up the estimation error system;

[0079] Step 5: Set up the tracking error system;

[0080] An embodiment of the present invention is described below:

[0081] Consider a mobile robot system, and its corresponding dynamic models are:

[0082]

[0083]

[0084] The topological structure of the seven mobile robots is shown in the figure below: Figure 1 As shown in the figure, the dynamic trajectory of the mobile robot performing formation tracking control is shown in Figure 2 As shown, the dynamic trajectory of the unknown coupling weight is shown in Figure 3 As shown, the dynamic trajectory of the mobile robot on the x-axis and y-axis is shown in Figure 4 and 5 As shown, the velocity state trajectory of the mobile robot on the x-axis and y-axis is shown in Figure 6 and 7 As shown in the figure, the dynamic trajectory diagram of the control input of the mobile robot on the x-axis and y-axis is as follows Figure 8 and 9 shown.

Claims

1. A guaranteed cost tracking control method for a time-varying formation of a mobile robot system, characterized in that: The following steps are involved: A disturbance observer is designed using the output function of the mobile robot system to estimate the true value of the unknown disturbance of the system, and the corresponding anti-disturbance compensator is given; Adaptive control method is used to estimate the dynamic coupling weights in the distributed controller of time-varying formation, where the time-varying formation is given in advance. With the help of Lyapunov stability theorem, the uniform final bounded conditions and the upper bound of the cost function of the mobile robot system are derived, and then the time-varying formation tracking control is realized.

2. The guaranteed cost tracking control method for a time-varying formation of a mobile robot system according to claim 1 is characterized in that Under the interference caused by the external system, use the output function of the mobile robot system to design a disturbance observer to estimate the true value of the unknown disturbance of the system and give the corresponding anti-interference compensator. The specific steps are as follows: Under external disturbances, the mobile robot system model for time-varying formation tracking control is as follows: y i (t)=Cx i (t), 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 of the i-th node, A, C and D are known parameter matrices, which are specifically expressed as Where I2 represents the second-order unit matrix, represents the Kronecker product, d i (t) represents the unknown external input of the i-th node, which can be represented by an external system as follows: d i (t)=Mη i (t), Where η i (t) represents the state of the external system, M and N represent the known parameter matrices, which are specifically expressed as To construct an output disturbance observer, we calculate the first-order derivative of the output function Available Among them C * is a pseudo-inverse matrix that satisfies C * =(C * TC) -1 C T ; For unknown interference d i (t), design the following disturbance observer: in is d i The estimated value of (t), is η i The estimated value of (t), I i (t) is the integral term in the observer, L1 and L2 are the unknown observer gains; Define an intermediate variable The disturbance observer can be further written as The interference estimation error is defined as The estimated error system can be written as 3. The time-varying formation guaranteed cost tracking control method for a mobile robot system according to claim 2 is characterized in that The adaptive control method is used to estimate the dynamic coupling weights in the time-varying formation distributed controller, 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 expressed as y0(t)=Cx0(t), 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. Definition i (t) = x i (t)-h i (t)-x0(t) is the formation error, e y (t) = y i (t)-y0(t) is the system output error, so the error system can be expressed as it is y (t)=What(t), where h i (t) is the formation of the team, Design a distributed controller i (t) To achieve time-varying formation tracking control and compensate for unknown disturbances Where K is the feedback gain, v i (t) represents the compensation signal function, ω ij and ω i0 denote the unknown coupling weights, and they can be estimated by in is the unknown coupling weight ω ij The estimated value of is the unknown coupling weight ω i0 The estimated value of ξ ij and i0 All represent positive constants, and P is the unknown positive definite gain matrix; Substituting the distributed controller into the error system, we can get Define the cost function J C for J C (t)=J Ce (t)+J Cu (t) in In the above equation, W1 and W2 are two positive definite matrices, ζ i (t) = Ke i (t).

4. According to the time-varying formation guaranteed cost tracking control method of the mobile robot system of claim 3, with the help of Lyapunov stability theorem, the consistent final bounded conditions and the upper bound of the cost function of the mobile robot system are obtained, and then the time-varying formation tracking control is realized. The proof process is as follows: B001: Select the following form of Lyapunov function: B002: where Q1, Q2 are any positive definite matrices; B003: Calculate the first derivative of V(t) B004: When Sometimes, there is in is a given matrix that satisfies B004: Definition Then we have: B005: Assumption is a non-singular matrix, we can get B006: With the help of condition ω ij (0) = ω ji (0) and ξ ij =ξ ji >0, we can get ω ij (t) = ω ji (t) and B007: Combining B003, B005 and B006, we can conclude B008： B009: Let K = D T Q1,P=Q1DD T Q1, then B008 can be rewritten as B0010： in B0011: Based on Lyapunov stability theory, when Sometimes, there is Then the mobile robot system achieves consistent ultimate boundedness; B0012: Based on cost function J C The definition of guaranteed cost function can be rewritten as B0013: Then for any time T ≥ 0, we have B0014： B0015: Condition-based When there is J T ≤V(0)-V(T), then J * The upper bound of

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