A fixed-time stable control method for a single-link robot with input dead zone

Abstract

The present invention discloses a fixed-time stable control method for a single-link robot with an input dead zone, comprising the following steps: S1, modeling the input dead zone, establishing a single-link robot system model with input dead zone constraints, S2, designing an adaptive event triggering controller based on backstepping technology, extending radial basis function neural network adaptive control technology and fixed-time stable control theory, and S3, performing stability analysis. The present invention provides a control method, adopts a new control model and process, so that the control method has the advantages of realizing the convergence of system errors within a fixed time, reducing the utilization rate of communication resources, etc., and is used to solve the problem that the convergence time of the limited time control scheme in the prior art is affected by the initial state of the system, and the communication resources of the technical scheme for processing the input dead zone are limited, and a large amount of communication resources need to be occupied to process the input dead zone.

Description

Technical Field

[0001] The invention relates to the technical field of robot control, and in particular to a fixed-time stable control method for a single-link robot with an input dead zone. Background Art

[0002] As robots become the main force of intelligent manufacturing, industrial robots continue to penetrate from the field of automobile manufacturing to machinery, building materials, logistics, food and even aviation, aerospace, shipbuilding and other fields, driving the development of related industries. Therefore, the control performance and control accuracy requirements of robots are also constantly being improved. In many actual control systems, input dead zones are common in the actuators of robots, and nonlinear characteristics are inevitable. Input dead zone nonlinearity will reduce the stability of the system and seriously affect the performance of the system. While ensuring control accuracy, the input dead zone increases the communication burden, and the system needs to update a large number of control signals to compensate for the impact of the dead zone. However, the communication resources of actual network control systems are limited.

[0003] The existing technical solutions have the following disadvantages:

[0004] 1. Existing finite-time control methods require knowing the initial conditions of the system to measure the stabilization time. However, in many practical systems, the initial state of the system is unmeasurable, so the stabilization time cannot be calculated in advance.

[0005] 2. The existing technical solutions for processing input dead zones do not take into account that the communication resources of the system are limited, and a large amount of communication resources are required to process the input dead zones. Therefore, it is of great significance both in theory and practice to study how to reduce the occupation of system communication resources. Summary of the invention

[0006] 1. Technical issues to be resolved

[0007] In view of the shortcomings of the prior art, the present invention provides a fixed-time stable control method for a single-link robot with an input dead zone. A new control model and process are adopted to enable the control method to have the advantages of realizing the convergence of system errors within a fixed time and reducing the utilization rate of communication resources. It is used to solve the problem that the convergence time of the limited-time control scheme in the prior art is affected by the initial state of the system, and the technical scheme for processing the input dead zone does not take into account that the communication resources of the system are limited, and a large amount of communication resources need to be occupied to process the input dead zone.

[0008] (II) Technical solution

[0009] The present invention provides the following technical solutions:

[0010] A fixed time stable control method for a single-link robot with an input dead zone comprises the following steps:

[0011] S1. Model the input dead zone and establish a single-link robot system model with input dead zone constraints;

[0012] S2. Based on backstepping technology, the radial basis function neural network adaptive control technology and fixed time stable control theory are extended to design an adaptive event-triggered controller;

[0013] S3. Conduct stability analysis;

[0014] S4. Perform simulation.

[0015] Furthermore, in S1, the mathematical model of the single-link robot is given as follows:

[0016]

[0017] Where q(t) is the joint angle, is the joint angular velocity, J is the moment of inertia, B is the friction damping coefficient, m is the mass of the connecting rod, g is the acceleration due to gravity, l is the length of the connecting rod, and u(t) is the input torque.

[0018] Furthermore, in S1, the mathematical model of the given input dead zone is as follows:

[0019] D(u)=au+b (2)

[0020] Where, u represents the control output; a1, a2, b1 and b2 are all constants, and a1>0 and a2>0 represent unknown slopes of the curve on the negative axis and the positive axis, and b1<0 and b2>0 represent unknown turning points.

[0021] Furthermore, in S2, the design process of the adaptive event-triggered controller is as follows:

[0022] ① Design the first virtual control law α1: system state x1 and expected output signal y r After linear transformation, e1 is obtained, and then α1 is designed;

[0023] ② Design the second virtual control law α2: system state x2, the first-order derivative of the expected output signal The first virtual control law is transformed into e2 through linear transformation; the extended basis function neural network is used to estimate the uncertain parameter θ in the system, and the following is obtained: Then use e2 and Design α2;

[0024] ③Design Using α2 and e2 to construct Q and Furthermore, the design

[0025] ④ Design event trigger mechanism: A dynamic threshold event trigger mechanism is designed to reduce the update frequency of the input signal u, thereby alleviating the system communication pressure and achieving dead zone compensation;

[0026] ⑤The input signal u passes through the input dead zone and is then input into the robot system model to achieve closed-loop control of the system.

[0027] Furthermore, in S3, according to the fixed-time stable control theory, all error signals It converges to the following given compact set:

[0028]

[0029] The convergence time T is as follows:

[0030]

[0031] in, is a constant, and

[0032] Furthermore, in S3, according to the definition of V2, we can further obtain that the tracking error will converge to a compact set in a fixed time:

[0033]

[0034] From equation (54), we can see that the convergence time of the system has nothing to do with the initial state of the system. In addition, δ3 in equation (55) is λ1 and λ2 are adjustable, so a smaller tracking error can be obtained by adjusting the relevant design parameters.

[0035] Furthermore, in S3, for have

[0036]

[0037] represents the first-order derivative of the event-triggered control input ω(t). According to the expression (18) of the mathematical model of the event triggering mechanism, is continuous and bounded, and there exists a positive number because Therefore, the lower bound of the time interval t * ,and

[0038] Furthermore, in S4, the system parameters of the single-link robot are selected during simulation as follows: J=0.8, B=1, mgl=10, initial values ​​x1(0)=0.2, x2(0)=0, a1=a2=1, b1=-1 and b2=1.

[0039] Compared with the prior art, the fixed-time stable control method of a single-link robot with an input dead zone provided by the present invention has the following beneficial effects:

[0040] 1. The present invention adopts the mutual cooperation of new control models and processes, so that the control method can achieve the convergence of system errors within a fixed time under different system initial states, and the convergence time is independent of the system initial state, which solves the problem that the convergence time in the finite time control scheme is affected by the system initial state.

[0041] 2. The present invention can realize online dynamic compensation of input dead zone under the premise of ensuring control accuracy, while reducing the utilization rate of communication resources, solving the problem that the technical solution for processing input dead zone does not take into account that the communication resources of the system are limited and a large amount of communication resources are needed to process the input dead zone.

[0042] It is to be understood that the foregoing general description and the following detailed description are exemplary only and are not restrictive of the invention. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 An input dead zone model in a fixed-time stable control method for a single-link robot with an input dead zone according to an embodiment of the present invention;

[0044] Figure 2 A flowchart of controller design in a fixed-time stable control method for a single-link robot with an input dead zone according to an embodiment of the present invention;

[0045] Figure 3 It is a state trajectory of simulation results in a fixed-time stable control method for a single-link robot with an input dead zone according to an embodiment of the present invention;

[0046] Figure 4 It is the tracking error of the simulation result in the fixed time stable control method of the single-link robot with input dead zone according to the embodiment of the present invention;

[0047] Figure 5 A simulation result control signal in a fixed-time stable control method for a single-link robot with an input dead zone provided by an embodiment of the present invention;

[0048] Figure 6 It is the simulation result event triggering time interval in the fixed time stable control method of a single-link robot with input dead zone provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0049] The technical solutions in the embodiments of the present invention will be described clearly and completely below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0050] Examples of embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, and should not be construed as limiting the present invention.

[0051] like Figure 1-6 As shown, the fixed time stable control method of a single-link robot with a forehead band input dead zone provided by the present invention comprises the following steps:

[0052] S1. Model the input dead zone and establish a single-link robot system model with input dead zone constraints;

[0053] S2. Based on backstepping technology, the radial basis function neural network adaptive control technology and fixed time stable control theory are extended to design an adaptive event-triggered controller;

[0054] S3. Conduct stability analysis;

[0055] S4. Perform simulation.

[0056] Mathematical model analysis

[0057] Without loss of generality, the mathematical model of the single-link robot is given as follows:

[0058]

[0059] Where q(t) is the joint angle, is the joint angular velocity, J is the moment of inertia, B is the friction damping coefficient, m is the mass of the connecting rod, g is the acceleration due to gravity, l is the length of the connecting rod, and u(t) is the input torque.

[0060] Due to the structural characteristics of the mechanical structure itself, the input dead zone is unavoidable in the robot system and has a great impact on the system performance. The mathematical model of the given input dead zone is as follows:

[0061] D(u)=au+b (2)

[0062] Where, u represents the control output; a1, a2, b1 and b2 are all constants, and a1>0 and a2>0 represent unknown slopes of the curve on the negative and positive axes, and b1<0 and b2>0 represent unknown turning points. The model of the input dead zone is as follows: Figure 1 shown.

[0063] By performing coordinate transformation on (1), we can obtain:

[0064]

[0065] Among them, x1=q(t) is the first system state; is the second system state; represents the derivative of x1; represents the derivative of x2; y represents the system output; Represents the uncertain part of the system.

[0066] Design of adaptive event-triggered controller

[0067] In the process of designing an adaptive event-triggered controller, the lemmas and assumptions involved are as follows:

[0068] Lemma 1: Consider the following dynamical system:

[0069]

[0070] Where x represents the state vector and x∈R n , n is a positive integer; f(x) is a continuous function, and the initial value x(0) = f(0) = 0. If there exists a positive definite function V(x) such that the origin of system (4) is actually fixed-time stable, then V(x) satisfies:

[0071]

[0072] Among them, λ1, λ2, δ, p and q are all constants, and λ1>0, λ2>0, δ>0, 1<p<+∞, 0<q<1.

[0073] The convergence time T satisfies:

[0074]

[0075] in, is a constant, and The solution of the system can converge to the following compact set:

[0076]

[0077] Lemma 2: For The following inequality holds.

[0078]

[0079] Lemma 3: For The following inequality holds.

[0080]

[0081]

[0082] Lemma 4: For The following inequality holds.

[0083]

[0084] Lemma 5: For x∈R, y∈R, k1, k2 and m are all arbitrary positive numbers, the following inequality holds.

[0085]

[0086] Assumption 1: Expected output signal y r and its corresponding derivative are both bounded and continuous.

[0087] First, define the error system as follows:

[0088]

[0089] Among them, e1 and e2 are error variables, α1 is the first virtual control law, and y r is the expected output signal.

[0090] The Lyapunov function V1 is defined as follows:

[0091]

[0092] Taking the derivative of V1, we get:

[0093]

[0094] The first virtual control law α1 is designed as follows:

[0095]

[0096] Among them, c 11 、c 12 , p and q are all constants, and c 11 >0,c 12 >0,p>1,

[0097] Substituting formula (16) into formula (15), we get:

[0098]

[0099] Step 2: The event triggering mechanism is designed as follows:

[0100]

[0101] Where t represents the time; ω(t) represents the event trigger control input at time t; e(t) is the measurement error, and e(t) = ω(t) - u(t); δ, ο, ρ and k are both constants, and k∈Z + ,δ>0,ο>0, 0<ρ<1; represents the control law; u(t) represents the model input signal at time t; ω(t k ) indicates t k The event triggers the control input at time t k represents the time when the kth event is triggered, t k+1 Indicates the time when the k+1 event is triggered.

[0102] According to formula (18), we can get:

[0103] ω(t)=u(t)+ργ1(t)u(t)+ογ2(t) (19)

[0104] Among them, γ1(t) and γ2(t) represent time-varying parameters, and |γ1(t)|≤1, |γ2(t)|≤1, further we can get:

[0105]

[0106] Control Law The design is as follows:

[0107]

[0108] Where H = [1 / a, -b / a] T , H T represents the transpose of H, Q is a 1*2 matrix, and Q = [α2, 1] T , α2 represents the second virtual control law; in actual situations, the parameters of H are difficult to obtain. In order to deal with this problem, we design To estimate the value of H, define the estimation error and in and The estimated values ​​are and So we get The expression is:

[0109]

[0110] The Lyapunov function V2 is defined as follows:

[0111]

[0112] Among them, l is a positive number; represents the estimation error, and θ is a real number, is the estimated value of θ; K is a 2*2 positive definite matrix.

[0113] From equation (2), equation (3) and equation (13), we can get:

[0114]

[0115] in, represents the derivative of the second error variable e2; represents the derivative of the first virtual control law α1; Represents the second derivative of the desired output signal.

[0116] Taking the derivative of V2, we get:

[0117]

[0118] From formula (18) and formula (20), we can get:

[0119]

[0120] According to Lemma 3, we can get:

[0121]

[0122] From equation (17), equation (25) and equation (27), we can get:

[0123]

[0124] Where Ω(X) is an unknown combinatorial function, and

[0125] The radial basis function neural network (RBFNNs) is used to process the uncertain part of the system and approximate the model uncertainty term Ω(X). The general form of the RBFNNs function is as follows:

[0126] F=W T S(X) (29)

[0127] Where W is the weight vector, and W = [w1,w2,...,w m ] T ∈R m , m is a positive integer; W T represents the transpose of W; S(X) is the basis function vector, and S(X)=[s1(X),s2(X),...,s m (X)] T ∈R m , m is a positive integer; m is the number of nodes of BRFNNs, and m>1; represents the input vector, and X=[x1,x2,x3,x4,x5,x6] T , x1=q(t), x3=y r ,

[0128] s i The expression of (X)(i=1,2,...,m) is as follows:

[0129]

[0130] Among them, i is a label used to express the sequence number of the related elements; qi is the width of the Gaussian function, and q i ∈R;p i is the center of the Gaussian function, and

[0131] The expression of approximation processing Ω(X) is as follows:

[0132]

[0133] in, is the unknown ideal weight vector, and m is a positive integer; express is the transpose of ; τ(X) is the approximation error, and is an arbitrarily small constant, ||·|| represents the bi-norm, and

[0134] Furthermore, (31) can be transformed into:

[0135] Ω(X)=Φ T (X)Γ(X) (32)

[0136] in, Γ(X)=[S(X) T ,1] T ,S(X) T Represents the transpose of S(X).

[0137] Then, we can get:

[0138]

[0139] θ represents the unknown parameter and is defined as follows:

[0140]

[0141] Obviously, θ ≥ ||Φ(X)|| 2 .

[0142] The following inequality holds:

[0143]

[0144] Where θ is a real number and θ=||Φ||2 , ||·|| represents the bi-norm; μ is a constant, and μ>0.

[0145] The second virtual control law α2 is designed as follows:

[0146]

[0147] Among them, c 21 and c 22 are constants, c 21 >0;c 22 >0.

[0148] Combining (28)-(36) we get:

[0149]

[0150] in, represents the adaptive law, which is expressed as:

[0151]

[0152]

[0153] Among them, σ and η are both constants, and σ>0, η>0.

[0154] Substituting equation (38) and equation (39) into equation (37), we get:

[0155]

[0156] From Young's inequality we can get:

[0157]

[0158]

[0159] Substituting equation (41) and equation (42) into equation (40), we get:

[0160]

[0161] in,

[0162]

[0163] According to Lemma 5, by letting x = 1, k1=1-q,k2=q, You can get:

[0164]

[0165] in,

[0166] Similarly, we can get:

[0167]

[0168] Substituting equation (45) and equation (46) into equation (43), we get:

[0169]

[0170] Among them, δ2=δ1+2v.

[0171] According to Lemma 4, Equation (47) can be transformed into:

[0172]

[0173] in, λ max (K -1 ) represents K -1 The maximum eigenvalue of .

[0174] Then, we further get:

[0175]

[0176] in,

[0177] In addition, formula (49) can be transformed into:

[0178]

[0179] in,

[0180] According to Lemma 4, we can further get:

[0181]

[0182] in,

[0183] because and They are all bounded. is bounded. With δ2>0 as the boundary, we can get the following inequality:

[0184]

[0185] Among them, δ4=δ3+δ2>0.

[0186] At this point, the controller design is completed, and the design flow chart is as follows: Figure 2 shown.

[0187] Design process:

[0188] ① Design the first virtual control law α1: system state x1 and expected output signal y r After linear transformation, e1 is obtained, and then α1 is designed.

[0189] ② Design the second virtual control law α2: system state x2, the first-order derivative of the expected output signal The first virtual control law is transformed into e2 through linear transformation; the extended basis function neural network is used to estimate the uncertain parameter θ in the system, and the following is obtained: Then use e2 and Design α2.

[0190] ③Design Using α2 and e2 to construct Q and Furthermore, the design

[0191] ④ Design event trigger mechanism: A dynamic threshold event trigger mechanism is designed to reduce the update frequency of the input signal u, thereby alleviating the system communication pressure and achieving dead zone compensation.

[0192] ⑤The input signal u passes through the input dead zone and is then input into the robot system model to achieve closed-loop control of the system.

[0193] Stability analysis

[0194] According to the fixed-time stable control theory, all error signals It converges to the following given compact set:

[0195]

[0196] The convergence time T is as follows:

[0197]

[0198] in, is a constant, and

[0199] According to the definition of V2, we can further conclude that the tracking error will converge to a compact set in a fixed time:

[0200]

[0201] From equation (54), we can see that the convergence time of the system has nothing to do with the initial state of the system. In addition, δ3 in equation (55) is λ1 and λ2 are adjustable, so a smaller tracking error can be obtained by adjusting the relevant design parameters.

[0202] for have

[0203]

[0204] represents the first-order derivative of the event-triggered control input ω(t). According to the expression (18) of the mathematical model of the event triggering mechanism, is continuous and bounded, and there exists a positive number because Therefore, the lower bound of the time interval t * ,and Obviously, no Zeno phenomenon occurs.

[0205] simulation

[0206] The system parameters of the single-link robot are selected as follows: J=0.8, B=1, mgl=10, initial values ​​x1(0)=0.2, x2(0)=0, a1=a2=1, b1=-1 and b2=1.

[0207] Control parameters: c 11 =c 12 =25, c 21 =c 22 =15, μ=1, l=0.01, σ=0.1, K=[1,0;0,1], η=0.05, δ=0.01,ρ=0.1,ο=0.2,reference signal y r =sin(2t), simulation step size 0.01s.

[0208] The simulation results are as follows Figure 1-6

[0209]

[0210] Table 1 Event triggering times

[0211] analyze

[0212] from Figure 3 It can be seen from the figure that the system output can track the reference trajectory well when the system is subject to the input dead zone. Figure 4 It can be seen that in three different initial system states: 1) x1(0) = 0.20, x2(0) = 0; 2) x1(0) = 0.15, x2(0) = 0; 3) x1(0) = 0.10, x2(0) = 0; the absolute value of the tracking error |e1| is controlled at 0.05, and the convergence time of the three different initial system states is about 0.05s, indicating that the convergence time is independent of the initial state of the system; Figure 5 describes event-triggered control inputs; Figure 6It represents the event triggering time interval. It can be seen from the figure that the event triggering is non-periodic and the minimum interval time is 0.02s. Therefore, there is no Zeno phenomenon. It can be seen from Table 1 that within 10 seconds, the total number of triggers of event triggered control is only 266 times, the total number of triggers of time triggered control is 1000 times, and the simulation step is 0.01s (the event triggering rate is 26.6%), which saves 73.4% of communication resources compared with traditional time triggered control.

[0213] Although embodiments of the present invention have been shown and described, it will be appreciated by those skilled in the art that various changes, modifications, substitutions and variations may be made to the embodiments without departing from the principles and spirit of the present invention, and that the scope of the present invention is defined by the appended claims and their equivalents.

[0214] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0215] Obviously, the above embodiments are only some embodiments of the present invention, not all embodiments. Based on the embodiments and descriptions of the present invention, all other technical solutions obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

Claims

1. A fixed time stable control method for a single-link robot with an input dead zone, comprising the following steps: S1. Model the input dead zone and establish a single-link robot system model with input dead zone constraints. The input dead zone system model is given as follows: (2) in, Indicates control output; , , , , and are constants, and and represents the unknown slope of the curve on the negative and positive axes, and Indicates an unknown turning point; The single-link robot system model is given as follows: (1) in, is the joint angle, Joint angular acceleration, is the joint angular velocity, is the moment of inertia, is the friction damping coefficient, is the mass of the connecting rod, is the acceleration due to gravity, is the connecting rod length, is the input torque; S2. Based on backstepping technology, radial basis function neural network adaptive control technology and fixed time stable control theory are extended to design an adaptive event-triggered controller. The design process of adaptive event-triggered controller is as follows: ① Design the first virtual control law : System status and the expected output signal After linear transformation, we get , and then design ; ② Design the second virtual control law : System status , the first-order derivative of the expected output signal And the first virtual control law is obtained through linear transformation ; Using extended basis function neural network to analyze the uncertain parameters in the system Make an estimate and get ,in for ; and then use and design ; ③Design :use and Build and , further, the design is , control law The design is as follows: (21) in, , express The transpose of is a 1*2 matrix, and , represents the second virtual control law; in actual situations, The parameters are difficult to obtain. In order to deal with this problem, the design To estimate The value of , defines the estimation error ,and ,in and The estimated values ​​are and , so we get The expression is: (22)； ④ Design event trigger mechanism: A dynamic threshold event trigger mechanism is designed to reduce the input signal The update frequency can relieve the system communication pressure and realize the compensation of dead zone; ⑤Input signal After the input dead zone is applied, the robot system model is input to achieve closed-loop control of the system. S3, perform stability analysis, according to the fixed time stability control theory, all error signals It converges to the following given compact set: (53) in, is a custom Lyapunov function, ,in, is a positive number; represents the estimation error, and , is a real number, for An estimated value of is a 2*2 positive definite matrix; Convergence time as follows: (54) in, is a constant, and ; Convergence time satisfy: (6) in, is a constant, and , the solution of the system can converge to the following compact set: (7) Lemma 2: For , , the following inequality holds; according to Further, we can get the definition of tracking error will converge to a compact set in a fixed time: (55) From equation (54), we can see that the convergence time of the system has nothing to do with the initial state of the system. In addition, , , and is adjustable, so a smaller tracking error can be obtained by adjusting the relevant design parameters. ,have (56) Indicates event-triggered control input The first-order derivative of is expressed as follows according to the event-triggered mechanism system model (18): (18) in, Indicates the moment; express Event-triggered control input at the moment; is the measurement error, and ; , , , and are constants, and , , , , ; represents the control law; express The model input signal at time; express Event-triggered control input at the moment; Indicates The moment the event is triggered, express The time when the event is triggered; So we can get, is continuous and bounded, and there exists a positive number , ,because , so the lower bound of the time interval is ,and ; S4. Perform simulation.

2. A fixed time stable control method for a single-link robot with input dead zone according to claim 1, characterized in that: In S4, the single-link robot system parameters are selected during simulation: , , , initial value , , , and .

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