Second-order terminal sliding mode control method based on time-delay estimation fuzzy observer backstepping

Abstract

The application discloses a second-order terminal sliding mode control method based on time delay estimation fuzzy observer backstepping method, and aims at unknown disturbance and trajectory tracking problems of a mechanical arm system in many practical problems, estimates kinematics and dynamics parameters of the system through time delay estimation, estimates total disturbance existing in the system through a novel fuzzy observer, combines the second-order sliding mode control to increase instantaneous response and reduce steady-state error, suppresses chattering phenomenon while guaranteeing finite time convergence of tracking error, and finally proves stability of the system by adopting a Lyapunov method, and according to comparison of different methods, fast and accurate tracking of a desired trajectory in the method is shown.

Description

Technical Field

[0001] This invention belongs to the field of high-order robotic arm trajectory tracking technology, and more specifically relates to a second-order terminal sliding mode control method based on the backstepping method of a time delay estimation fuzzy observer. Background Technology

[0002] With the development of control theory and mechanical technology, time delay estimation methods have been around for 30 years and have been proven by many scholars to be a simple and robust control method. They are widely used in exoskeleton systems, underwater robots, etc. However, the robotic arm system is a complex nonlinear model. At the same time, the model parameters cannot be accurately measured, which causes a mismatch between modeling and simulation. Moreover, in actual control, the model will be affected by unknown external disturbances. Estimating and compensating for unknown dynamic parameters of the robotic arm and external disturbances, and then combining them with an observer for disturbance error tracking, has become a hot research direction.

[0003] The inversion method has significant advantages in implementing robust or adaptive control for uncertain nonlinear systems, especially when disturbances are uncertain or matching conditions are not met. However, the inversion method itself lacks a good solution to the term expansion caused by the virtual control derivative and the problems arising from it. This shortcoming is particularly prominent in high-order systems. To address the term expansion problem, a dynamic sliding surface control method is adopted, using a first-order integral filter to calculate the virtual control derivative and eliminate the expansion of the differential terms. For flexible robotic arm systems with high precision requirements, non-matching disturbances, response time, and tracking error are all significant factors. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a second-order terminal sliding mode control method based on the backstepping method of the time delay estimation fuzzy observer.

[0005] This invention provides a second-order terminal sliding mode control method based on the backstepping method of a fuzzy observer using time delay estimation, and applies it to a practical robotic arm. The specific design scheme is as follows:

[0006] Step 1: Estimate the kinematic and dynamic parameters of the system by time delay estimation.

[0007] Step 2: Based on the model in Step 1, design a novel fuzzy observer to predict disturbances.

[0008] Step 3: Based on the novel fuzzy observer in Step 2, set an adaptive law to ensure the stability of the system.

[0009] Step 4: Set up a novel non-singular second-order terminal sliding surface and combine it with the disturbance estimated by the observer in the previous step. Use the inversion control method to construct the torque, and use Lyapunov to prove the stability of the system.

[0010] Further preferred, in step 1, the kinematic equations are established:

[0011]

[0012] in: Let M(q) be the vector of angular displacement, angular velocity, and angular acceleration of the robotic arm joint; M(q) is the positive definite inertia matrix. The correlation matrix between centrifugal force and Coriolis force; G(q) is the friction force matrix; G(q) is the gravity vector; τ is the joint control torque input vector. d For uncertainties related to external disturbances to the robotic arm, but Due to precision issues, G(q) is decomposed into M0(q). G0(q) and the uncertain part ΔM(q), ΔG(q).

[0013] In actual operation, it is often difficult to obtain an accurate model. In the system, N2 is generally unknown. Time delay estimation can effectively estimate and define the unknown. To simplify the subsequent controller design, a positive definite matrix is ​​set. And rewrite the above formula as:

[0014]

[0015] Where the definition The time delay estimation adopts the following structure as an estimate of N²:

[0016]

[0017]

[0018] in It is the external disturbance term in kinematics. Since the external disturbance is unknown, The value is an estimate of the external disturbance, which may change rapidly in a short period of time, potentially leading to a large time delay estimation error. The kinematic equations of the system at time tT can be written as:

[0019]

[0020]

[0021] Further preferred, step 2 involves the following steps, transforming the system into the following form:

[0022] Consider the following nth-order nonlinear system

[0023]

[0024] A novel observer is designed based on the above system.

[0025]

[0026] Where the definition

[0027]

[0028] in For z i The estimated value, the observer error is defined as κ1 and κ2 are observer parameters and are positive constants.

[0029] The novel observer reuses pole placement techniques to design gain parameters, as follows:

[0030] (s+w0) 2 =s n +κ1s n-1 +κ2

[0031] w0 is the observer bandwidth. Compared to the traditional linearly extended observer gain, which changes faster and faster with the increase of error, w0 also changes accordingly as the speed changes, enabling the system to better and more accurately estimate the disturbance error.

[0032]

[0033]

[0034] Observer error of fuzzy systems Design a fuzzy system using its derivative as input and bandwidth as output:

[0035]

[0036] THEN y(z)is B l1l2

[0037] Where l1 = 1, 2, 3...N1, l2 = 1, 2, 3...N2, and the total number of rules M is N1*N2, a product inference engine, a single-value fuzzy logic generator, and a center-average unfuzzy logic generator are used to construct a fuzzy system based on M rules, and the output is:

[0038]

[0039] Simultaneously introducing the weight coefficient vector θ and the fuzzy basis vector

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046] definition For the output of the fuzzy system:

[0047] The weighting coefficients are set as follows:

[0048]

[0049] Where β is a positive constant. The above formula can be rewritten as:

[0050]

[0051]

[0052] Further preferably, in steps three and four, the definition is... First, the sliding surface of the integration fast terminal is designed as follows:

[0053]

[0054]

[0055]

[0056] Where k1, k2, k3, and k4 are all positive real numbers, 1 < a < 2, and 1 < b < 2. Define p and q as positive real numbers. The second derivative is as follows:

[0057]

[0058] Wherein the definition is:

[0059]

[0060] The above formula can be rewritten as:

[0061]

[0062] in

[0063] The control input of the system is designed using the backstepping method:

[0064]

[0065] Where A1 is a virtual control variable, taking the derivative of f1(t) yields: f1(t) = f2(t) + A1. The torque set is as follows:

[0066]

[0067] The present invention, by adopting the above technical solution, achieves the following beneficial effects:

[0068] (1) This invention combines time delay estimation with the dynamic model of the robotic arm, which can better simulate the operating state of the robotic arm in actual work. Based on this model, the controller can be designed to effectively improve the control accuracy.

[0069] (2) Combining the second-order sliding mode with the integral term can achieve error convergence in a very short time, while having a smaller steady-state error compared to other control methods, increasing transient response and reducing steady-state error.

[0070] (3) A novel fuzzy reduced-order observer is used to estimate lumped disturbances and joint velocities. This avoids the problem of difficulty in measuring joint velocity information, and at the same time, the feedforward compensation of disturbances by the state observer effectively avoids chattering.

[0071] (4) The torque of the control input is designed based on the inversion method, which can achieve global asymptotic stability based on Lyapunov and solve the singularity problem in fast terminal sliding mode. Attached Figure Description

[0072] Figure 1 This is a structural block diagram of the controller in this invention;

[0073] Figure 2 This is a schematic diagram of the torque model of a two-link rigid robotic arm in an embodiment of the present invention;

[0074] Figure 3 and Figure 4 This is a simulation diagram of the torques at two joints of the robotic arm;

[0075] Figure 5-8 A simulation diagram showing the expected and error estimates of the position and velocity of two joints of a robotic arm compared with different algorithms;

[0076] Figure 9-12 To compare different algorithms, a simulation comparison diagram of the position and velocity errors of the two joints of the robotic arm is shown.

[0077] Figure 13-14 A schematic diagram showing the torque simulation of two joints of a robotic arm compared with different algorithms;

[0078] Figure 15-16 A schematic diagram showing the comparison of external disturbance measurements using different observers; Detailed Implementation

[0079] The invention is further illustrated below with specific embodiments. To better illustrate the invention, the proposed controller is verified using MATLAB numerical simulation, and the results are as follows. Figures 1 to 14 As shown. The specific steps are as follows:

[0080] Step 1: Establish a dynamic model of the robotic arm that combines time delay estimation. The model is as follows:

[0081]

[0082] in: Let M(q) be the vector of angular displacement, angular velocity, and angular acceleration of the robotic arm joint; M(q) is the positive definite inertia matrix. The correlation matrix between centrifugal force and Coriolis force; G(q) is the friction force matrix; G(q) is the gravity vector; τ is the joint control torque input vector. d For uncertainties related to external disturbances to the robotic arm, but Due to precision issues, G(q) is decomposed into M0(q). G0(q) and the uncertain part ΔM(q), ΔG(q).

[0083] In actual operation, it is often difficult to obtain an accurate model. In the system, N2 is generally unknown. Time delay estimation can effectively estimate and define the unknown. To simplify the subsequent controller design, a positive definite matrix is ​​set. And rewrite the above formula as:

[0084]

[0085] Where the definition The time delay estimation adopts the following structure as an estimate of N²:

[0086]

[0087]

[0088] in It is the external disturbance term in kinematics. Since the external disturbance is unknown, The value is an estimate of the external disturbance, which may change rapidly in a short period of time, potentially leading to a large time delay estimation error. The kinematic equations of the system at time tT can be written as:

[0089]

[0090]

[0091] The specific steps of step 2 are as follows, transforming the system into the following form:

[0092] Consider the following nth-order nonlinear system

[0093]

[0094] A novel observer is designed based on the above system.

[0095]

[0096] Where the definition

[0097]

[0098] in For z i The estimated value, the observer error is defined as κ1 and κ2 are observer parameters and are positive constants.

[0099] The novel observer reuses pole placement techniques to design gain parameters, as follows:

[0100] (s+w0) 2 =s n +κ1s n-1 +κ2

[0101] w0 is the observer bandwidth. Compared to the traditional linearly extended observer gain, which changes faster and faster with the increase of error, w0 also changes accordingly as the speed changes, enabling the system to better and more accurately estimate the disturbance error.

[0102]

[0103]

[0104] Observer error of fuzzy systems Design a fuzzy system using its derivative as input and bandwidth as output:

[0105]

[0106] THEN y(z) is B l1l2

[0107] Where l1 = 1, 2, 3...N1, l2 = 1, 2, 3...N2, and the total number of rules M is N1*N2, a product inference engine, a single-value fuzzy logic generator, and a center-average unfuzzy logic generator are used to construct a fuzzy system based on M rules, and the output is:

[0108]

[0109] Simultaneously introducing the weight coefficient vector θ and the fuzzy basis vector

[0110]

[0111]

[0112]

[0113]

[0114]

[0115]

[0116] definition For the output of the fuzzy system:

[0117] The weighting coefficients are set as follows:

[0118]

[0119] Where β is a positive constant. The above formula can be rewritten as:

[0120]

[0121]

[0122] In steps three and four, define The sliding surface of the integration fast terminal is designed as follows:

[0123]

[0124]

[0125]

[0126] Where k1, k2, k3, and k4 are all positive real numbers, 1 < a < 2, and 1 < b < 2. Define p and q as positive real numbers. The second derivative is as follows:

[0127]

[0128] Wherein the definition is:

[0129]

[0130] The above formula can be rewritten as:

[0131]

[0132] in

[0133] The control input of the system is designed using the backstepping method:

[0134]

[0135] Where A1 is a dummy control variable, taking the derivative of f1(t) gives: f1(t) = f2(t) + A1

[0136] The set torque is as follows:

[0137]

[0138] This invention utilizes Simulink in the MATLAB 2018a environment to simulate and verify the non-singular fast terminal sliding mode controller (NSSM) based on the backstepping method of the fuzzy reduced-order observer designed in this invention on a two-joint robotic arm. It is compared with other control algorithms (Adaptive Inversion NFTSM Controller ABNFTMSC, Adaptive Integral Sliding Mode Controller (AISMC), and Adaptive Non-singular Second-Order Fast Terminal Sliding Mode Controller (SONFTSM)).

[0139] (1) The simulation parameters are as follows

[0140] The parameters of the robotic arm's dynamic model are as follows:

[0141]

[0142]

[0143]

[0144] m1=0.5kg, m2=1.5kg, r1=1m, r2=2m;

[0145] g = 9.8 m / s 2 J1 = 5 kg·m 2 J2 = 5 kg·m 2 ;

[0146]

[0147] Let the initial position and initial angular velocity of the robotic arm joints be respectively The expected trajectory to be tracked is as follows:

[0148]

[0149] The interference applied to the system is as follows:

[0150]

[0151]

[0152]

[0153] The controller parameters are designed as follows:

[0154]

[0155] Figure 3-4 The simulation diagram of the torque of the two joints of the robotic arm shows that the signals of the two robotic arms in this invention are continuous and without singularity, and there is little chattering.

[0156] Figure 5-8 The figure shows a simulation diagram of the position, velocity, expected value and error estimation of the two joints of the robotic arm. As can be seen from the figure, the steady-state error of the two robotic arms in this invention is very small, which reflects the advantage of high tracking accuracy of this invention.

[0157] Figure 9-12 To compare different algorithms, a simulation comparison diagram of the position and speed errors of the two joints of the robotic arm is shown. As can be seen from the figure, the control input of the two joints in this invention remains continuous, without any chattering, and the accuracy is higher.

[0158] Figure 13-14 To compare the torque diagrams of different control methods, the torque of this invention is more concentrated, stable, and exhibits less chattering.

[0159] Figure 15-16 The diagrams show different observers measuring external disturbances, demonstrating that the observer of this invention can track them faster and more accurately.

[0160] In summary, the control scheme designed in this invention only requires joint position information to enable the robotic arm to achieve high-precision tracking of the desired trajectory in a short time. It also exhibits strong robustness against disturbances and has global asymptotic stability.

[0161] The specific implementation examples described above are merely for the purpose of helping those skilled in the art to understand the present invention. However, the present invention is not limited to the cases described above. Any variations are obvious as long as they fall within the spirit and scope of the invention as defined and determined by the appended claims.

Claims

1. A second-order terminal sliding mode control method based on the backstepping method of a fuzzy observer using time delay estimation, characterized in that, Includes the following steps: Step 1: Estimate the kinematic and dynamic parameters of the system through time delay estimation; Step 2: Based on the model from Step 1, design a novel fuzzy observer to predict disturbances; Step 3: Based on the novel fuzzy observer in Step 2, set an adaptive law to ensure the stability of the system; Step 4: Set up a novel non-singular second-order terminal sliding surface and combine it with the disturbance estimated by the observer in the previous step. Use the inversion control method to construct the torque, and use Lyapunov to prove the stability of the system. Step four includes: defining The sliding surface of the integration fast terminal is designed as follows: ， ， in All are positive real numbers. , ,definition , is a positive real number. , , The second derivative is as follows: ， Wherein the definition is: , ， The sliding surface of the fast integration terminal is rewritten as follows: ， in ; The inverse of a positive definite matrix; This represents the joint control torque input vector; This represents the observer bandwidth for online adjustment of the fuzzy system; , and Represents state variables , and The estimated value; The control input of the system is designed using the backstepping method: ， in As a virtual control variable, for Differentiation yields: , The set torque is as follows: 。 2. The second-order terminal sliding mode control method based on the time delay estimation fuzzy observer backstepping method according to claim 1, characterized in that, Step one includes: Establish the kinematic equations: ， in: , represents the angular displacement, angular velocity, and angular acceleration vector of the robotic arm joint; It is a positive definite inertia matrix; The correlation matrix between centrifugal force and Coriolis force; Here is the friction force matrix; It is the gravity vector. For the joint control torque input vector, For external disturbances, but , , Due to accuracy issues, it is decomposed into , , and uncertain parts , , ; In actual operation, accurate models are often difficult to obtain, and in the system... Generally, the unknowns are unknown. Time delay estimation can effectively estimate and define the unknowns. To simplify the subsequent controller design, a positive definite matrix is ​​set. The kinematic equations are then rewritten as follows: ， Where the definition , for The estimated value, the time delay estimate adopts the following structure: ， ， in It is the external disturbance term in kinematics. Since the external disturbance is unknown, The estimated value of the external disturbance may change drastically in a short period of time, potentially leading to a large time delay estimation error in the system. The kinematic equations can be written as follows: ， 。 3. The second-order terminal sliding mode control method based on the backstepping method of the fuzzy observer based on time delay estimation as described in claim 1, characterized in that, The novel fuzzy observer design in step two is as follows: The system is transformed into the following form: Consider the following nth-order nonlinear system ， A novel observer is designed based on the aforementioned nth-order nonlinear system. ， Where the definition , ， in for The estimated value, the observer error is defined as , , These are observer parameters and are positive constants. The novel observer utilizes pole placement techniques to design gain parameters, as follows: ， The observer bandwidth, compared to the traditional linearly extended observer gain, changes faster and faster with increasing error. Simultaneously with this speed change... The corresponding changes enable the system to better and more accurately estimate disturbance errors; ， ， Observer error of fuzzy systems Design a fuzzy system using its derivative as input and bandwidth as output: ， in The total number of rules M is Using a product inference engine, a single-value fuzzy logic generator, and a center-average unfuzzy logic generator, a fuzzy system is constructed based on M rules, and the output is: ， Simultaneously, a weight coefficient vector is introduced. and fuzzy basis vectors , ， ， definition For the output of the fuzzy system: , The weighting coefficients are set as follows: ， in For positive integers, The above formula can be rewritten as: ， 。

Images