Pneumatic artificial muscle actuator fixed time iterative learning control method

Abstract

The application belongs to the technical field of pneumatic artificial muscle actuator control, and discloses a fixed-time iterative learning control method for a pneumatic artificial muscle actuator, which comprises the following steps: constructing a preset performance boundary decaying with time and implementing logarithmic error transformation, using a regularized double-power sliding mode variable to realize fixed-time robust regulation of single operation and adopting a radial basis function neural network to compensate for continuous unknown nonlinearity on line, so that stability and convergence of the pneumatic artificial muscle actuator on the time axis are realized; constructing a double-power iterative learning law according to the transformed error and the sliding mode variable, and updating the control input by using historical iterative error information, so that the trajectory tracking error can be quickly reduced within a given convergence upper limit.

Description

Technical Field

[0001] This invention belongs to the field of pneumatic artificial muscle actuator control technology, and specifically relates to a fixed-time iterative learning control method for a pneumatic artificial muscle actuator. Background Technology

[0002] Pneumatic artificial muscle (PAM) actuators possess advantages such as simple structure, light weight, good compliance, and high safety. They can generate contractile movements similar to biological muscles, thus showing promising application prospects in rehabilitation assistance and limb support. However, due to factors such as material nonlinearity, structural coupling, and the pneumatic drive process, pneumatic artificial muscle actuators exhibit significant hysteresis nonlinearity during operation. That is, during inflation and deflation, the output displacement and contractile force of the pneumatic artificial muscle actuator are not only related to the current input pressure but also affected by the loading history and the direction of motion.

[0003] For pneumatic artificial muscle actuators, tasks such as rehabilitation assistance and limb support typically exhibit repetitive characteristics. Iterative Learning Control (ILC) can utilize tracking error information from previous or historical iterations to correct the control input for the next run, thereby gradually improving the performance of repetitive trajectory tracking. Therefore, ILC provides an effective approach to eliminating hysteresis nonlinearity in pneumatic artificial muscle actuators.

[0004] Traditional iterative learning control methods typically focus on the asymptotic convergence of the error along the iteration axis, meaning that ideal tracking performance can only be achieved when the number of iterations approaches infinity. However, this is insufficient to meet the requirements of finite number of experiments and rapid convergence in practical engineering. To address this issue, finite-time stability theory provides an important approach to improving the convergence performance of iterative learning control. Its core lies in using a fractional power-law error convergence mechanism to ensure that the system error reaches the target neighborhood within a finite time. Although finite-time iterative learning control improves system learning efficiency, its convergence time or upper bound is usually still related to the initial error. Therefore, when the initial error is large or the initial state is difficult to predict accurately, the upper bound of the convergence time may increase, making it difficult to provide reliable performance guarantees for practical systems. Therefore, how to provide an iterative convergence guarantee independent of the initial error, thereby further improving the predictability of convergence in repetitive task control processes, has become a pressing technical challenge. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a fixed-time iterative learning control method for a pneumatic artificial muscle actuator. This method combines preset performance control, sliding mode control and iterative learning control to construct a composite controller, which can improve the tracking accuracy of multi-round repetitive trajectories.

[0006] The technical solution provided by this invention is as follows:

[0007] A fixed-time iterative learning control method for a pneumatic artificial muscle actuator includes:

[0008] A non-affine nonlinear system model characterizing the operation of a pneumatic artificial muscle actuator is established, and the control law of the pneumatic artificial muscle actuator is set as a function composed of a continuous time axis feedback control quantity and an iterative learning feedforward compensation term based on the system model.

[0009] And the control law is used to control the movement of the pneumatic artificial muscle actuator;

[0010] The continuous time axis feedback control quantity is determined by the following method:

[0011] The tracking error of the pneumatic artificial muscle actuator is obtained based on the expected displacement and the actual displacement, and the tracking error is converted into an unconstrained transformation error through error transformation.

[0012] Based on the transformation error, a non-singular fixed-time sliding mode variable containing regularized low-order power terms and high-order power terms is constructed, and the sliding mode is dynamically divided into computable dynamic terms, control input terms, nonlinear unknown terms, and robust terms.

[0013] The nonlinear unknown term is determined by online estimation based on the expected displacement, the transformation error, and the sliding mode variable;

[0014] A low-order fixed-time sliding mode term and a high-order fixed-time sliding mode term are constructed using fixed-time sliding mode variables; the computable dynamic term, the low-order fixed-time sliding mode term, the high-order fixed-time sliding mode term, the nonlinear unknown term, and the robust term are summed to obtain the continuous-time axis feedback control quantity.

[0015] Preferably, the control law of the pneumatic artificial muscle actuator is:

[0016] ;

[0017] In the formula, Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. The control voltage applied to the proportional valve at all times Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. Continuous time-axis feedback control quantity at any given moment Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. Iterative learning of feedforward compensation terms at each time step. To control voltage Equivalent input gain after entering sliding mode dynamics express The inverse operation.

[0018] Preferably, the fixed-time iterative learning control method for the pneumatic artificial muscle actuator further includes:

[0019] Based on the transformation error, the transformation error learning term is obtained:

[0020] ;

[0021] Based on the sliding mode variables, the learning terms for the sliding mode variables are obtained:

[0022] ;

[0023] The iterative learning feedforward compensation term is updated using the transformation error learning term and the sliding mode variable learning term;

[0024] in, ; ;

[0025] In the formula, Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. Iterative learning of feedforward compensation terms at each time step. For pneumatic artificial muscle actuators The process repeats for the [number]th time. Learning correction term for time-varying error. For pneumatic artificial muscle actuators The process repeats for the [number]th time. Learning correction term for sliding mode variable at time step. For pneumatic artificial muscle actuators The process repeats for the [number]th time. Total learning and correction amount at all times; All are positive learning gains; and Iterate over the lower-order and higher-order power exponents on the axis respectively; For pneumatic artificial muscle actuators The process repeats for the [number]th time. Error in time transformation, For pneumatic artificial muscle actuators The process repeats for the [number]th time. Sliding mode variables at time points; and These are the lower-order and higher-order power terms of the error transformation, respectively; and These are the lower-order and higher-order power terms of the sliding mode variable, respectively.

[0026] Preferably, the formula for calculating the transformation error is:

[0027] ;

[0028] in, ; ;

[0029] In the formula, Indicates the pneumatic artificial muscle actuator. Repeat the first run Tracking error at any time, For pneumatic artificial muscle actuators Repeat the first run Normalization error at time step; Indicates the first Preset performance function at any given time; The allowable error boundary at the initial moment of the system. This represents the allowable error boundary when the system is stable. For the error decay rate, This represents the symmetry performance constraint coefficient.

[0030] Preferably, the sliding mode variable is:

[0031] ;

[0032] in, It is a non-singular fixed-time sliding mode variable; To regularize lower-order power terms, It is a higher-order power term;

[0033] ;

[0034] In the formula, and All are designed for gain via sliding surface. Here is the regularization constant. and Let be the lower and higher order power exponents of the continuous time axis, respectively, and satisfy . ; for The first derivative.

[0035] Preferably, a radial basis function neural network is used to estimate the unknown nonlinear terms in the sliding mode dynamics online, as follows:

[0036] by Using the input vector, the approximation property of the radial basis function neural network is utilized to obtain the estimate of the unknown nonlinear term:

[0037]

[0038] in, For pneumatic artificial muscle actuators in The expected displacement trajectory at time t. For the desired displacement trajectory Regarding time The first derivative, For the desired displacement trajectory Regarding time The second derivative; For the weights of the radial basis function neural network, through Online updates, The learning rate matrix is ​​positive definite. yes The derivative of .

[0039] Preferably, the non-affine nonlinear system model of the pneumatic artificial muscle actuator is as follows:

[0040] ;

[0041] in, The iteration number for repeated runs. The continuous time within each repeated run, Duration of a single task run; and These represent the pneumatic artificial muscle actuators. The process repeats for the [number]th time. The displacement and velocity states at each moment. Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. The proportional valve drive voltage at any given time, Indicates the pneumatic artificial muscle actuator. Non-affine nonlinear function that runs repeatedly Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. External disturbances at any given moment. Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. The actual displacement output at any given time.

[0042] The beneficial effects of this invention are:

[0043] 1. This invention addresses the problem that single iterative learning control mainly corrects errors along the iteration axis and is insufficient in suppressing transient errors and external disturbances. It combines preset performance control, sliding mode control, and iterative learning control to construct a composite controller. This method can constrain the dynamic boundary of tracking error and enhance disturbance rejection capability during the current operation. At the same time, it can use historical error information to correct the control input during repeated operation, realizing the synergy of time-domain performance constraint, robust feedback adjustment, and iterative learning update, thereby improving the trajectory tracking performance of the pneumatic artificial muscle actuator under complex input signal and load conditions.

[0044] 2. This invention addresses the problem that traditional iterative learning control focuses primarily on asymptotic convergence and suffers from insufficient convergence performance of tracking errors within a finite number of iterations. It introduces the concept of fixed-time convergence into the iterative learning control framework of pneumatic artificial muscle actuators, constructing a nonlinear learning mechanism with fixed-time convergence characteristics. This method does not rely on an exact hysteresis model, but utilizes historical iteration error information to update the control input, enabling the trajectory tracking error to rapidly decrease within a given upper convergence bound. Furthermore, it establishes a convergence performance analysis framework independent of initial errors, thereby improving the trajectory tracking accuracy and adaptability to complex operating conditions of the pneumatic artificial muscle actuator. Attached Figure Description

[0045] Figure 1 This is a block diagram illustrating the fixed-time iterative learning control principle of the pneumatic artificial muscle actuator described in this invention.

[0046] Figure 2 This is a schematic diagram of the electrical and pneumatic connections of the pneumatic artificial muscle experimental platform in an embodiment of the present invention.

[0047] Figure 3 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a low-frequency composite sine wave under a 600g load in an embodiment of the present invention.

[0048] Figure 4 This is a graph showing the experimental results of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking a low-frequency composite sine wave under a 600g load in an embodiment of the present invention.

[0049] Figure 5 This is a graph showing the tracking error experimental results of a pneumatic artificial muscle actuator tracking a low-frequency composite sine wave under a 600g load in an embodiment of the present invention.

[0050] Figure 6 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a high-frequency composite sine wave under a 600g load in an embodiment of the present invention.

[0051] Figure 7The figure shows the experimental results of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking a high-frequency composite sine wave under a 600g load according to an embodiment of the present invention.

[0052] Figure 8 This is a graph showing the tracking error experimental results of a pneumatic artificial muscle actuator tracking a high-frequency composite sine wave under a 600g load in an embodiment of the present invention.

[0053] Figure 9 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a low-frequency composite triangular wave under a 600g load in an embodiment of the present invention.

[0054] Figure 10 The figure shows the experimental results of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking a low-frequency composite triangular wave under a 600g load in this embodiment of the invention.

[0055] Figure 11 This is a graph showing the tracking error experimental results of a pneumatic artificial muscle actuator tracking a low-frequency composite triangular wave under a 600g load in an embodiment of the present invention.

[0056] Figure 12 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a high-frequency composite triangular wave under a 600g load in an embodiment of the present invention.

[0057] Figure 13 The figure shows the experimental results of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking the high-frequency composite triangular wave under a 600g load in this embodiment of the invention.

[0058] Figure 14 This is a graph showing the experimental results of the tracking error of the pneumatic artificial muscle actuator tracking a high-frequency composite triangular wave under a 600g load in an embodiment of the present invention.

[0059] Figure 15 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a low-frequency composite sine wave under a 1200g load in an embodiment of the present invention.

[0060] Figure 16 This is an experimental result diagram of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking a low-frequency composite sine wave under a 1200g load in an embodiment of the present invention.

[0061] Figure 17 This is a graph showing the experimental results of the tracking error of the pneumatic artificial muscle actuator tracking a low-frequency composite sine wave under a 1200g load in an embodiment of the present invention.

[0062] Figure 18 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a high-frequency composite sine wave under a 1200g load in an embodiment of the present invention.

[0063] Figure 19 This is an experimental result diagram of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking a high-frequency composite sine wave under a load of 1200g in an embodiment of the present invention.

[0064] Figure 20 This is a graph showing the tracking error experimental results of a pneumatic artificial muscle actuator tracking a high-frequency composite sine wave under a 1200g load in an embodiment of the present invention.

[0065] Figure 21 This is an experimental result diagram of a pneumatic artificial muscle actuator tracking a low-frequency composite triangular wave under a 1200g load in an embodiment of the present invention.

[0066] Figure 22 This is an experimental result diagram of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking a low-frequency composite triangular wave under a load of 1200g in an embodiment of the present invention.

[0067] Figure 23 This is an experimental result of the tracking error of the pneumatic artificial muscle actuator tracking a low-frequency composite triangular wave under a load of 1200g in an embodiment of the present invention.

[0068] Figure 24 This is an experimental result diagram of the pneumatic artificial muscle actuator tracking a high-frequency composite triangular wave under a load of 1200g in an embodiment of the present invention.

[0069] Figure 25 The figure shows the experimental results of the maximum absolute error and root mean square error of the pneumatic artificial muscle actuator tracking the high-frequency composite triangular wave under a load of 1200g in this embodiment of the invention.

[0070] Figure 26 This is an experimental result of the tracking error of the pneumatic artificial muscle actuator tracking a high-frequency composite triangular wave under a load of 1200g in an embodiment of the present invention. Detailed Implementation

[0071] The present invention will now be described in further detail with reference to the accompanying drawings, so that those skilled in the art can implement it based on the description.

[0072] This invention provides a fixed-time iterative learning control method for a pneumatic artificial muscle actuator. Figure 1 A block diagram of the fixed-time iterative learning control principle of the PAM actuator is presented, illustrating the signal relationships between the preset performance error transformation, radial basis function neural network, fixed-time sliding mode control, double-power iterative learning update law, and iterative memory stage. The specific implementation process of this invention is as follows.

[0073] I. Establishing a proportional valve voltage-driven pneumatic artificial muscle actuator in the first... The nonlinear non-affine system model under repeated runs is as follows:

[0074] ;

[0075] in, The iteration number for repeated runs. The continuous time within each repeated run, Duration of a single task run; and These represent the pneumatic artificial muscle actuator in the first... The 1st iteration of the repeated process The displacement and velocity states at each moment. Indicates that the artificial muscle actuator is in the first The 1st iteration of the repeated process The system state at any given moment. This indicates that the pneumatic artificial muscle actuator is in the first... During the repeated run, the first The actual control voltage applied to the proportional valve at any given moment. Indicates the pneumatic artificial muscle actuator. During the repeated execution, at the... At any given moment, determined by the current system state and proportional valve control voltage The unknown non-affine nonlinear dynamics of the jointly determined pneumatic artificial muscle actuator Indicates the actuator in the first During the repeated run, the first The external disturbances that are constantly present. Indicates the actuator in the first During the repeated run, the first The actual displacement output at any given time.

[0076] The actual control voltage consists of the continuous time axis feedback control quantity and the iterative learning feedforward compensation quantity:

[0077] ;

[0078] in, This represents the pneumatic artificial muscle actuator constructed in the transformed sliding mode channel. The process repeats for the [number]th time. Continuous time-feedback adjustment amount. Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. The feedforward compensation amount is learned iteratively at every moment.

[0079] Within the predetermined working domain, assume an unknown non-affine function. Regarding the proportional valve voltage It is continuously differentiable, and the local direction of the voltage's effect on the acceleration of the pneumatic artificial muscle actuator remains unchanged. To avoid constructing a global inverse hysteresis model, a nominal operating point is selected within the operating domain, and the original non-affine input channel is locally decomposed into "state-time related terms + local equivalent input gain × control voltage + local representation error" according to the mean value theorem. This processing only performs a local equivalent representation of the input channel and does not linearize the overall state nonlinearity of the pneumatic artificial muscle actuator. Therefore, equation (1) can be equivalently expressed as:

[0080] ;

[0081] in, Indicates the first During each repeated run, within the selected local working domain, the current state of the pneumatic artificial muscle actuator is determined. and continuous time The state-time related nonlinear dynamic terms, which are jointly determined, have a different meaning from the original nonaffine function. ;at the same time Indicates the control voltage at the first Continuous time during each repeated run The local equivalent input gain after entering the local equivalent model; Indicates the first Successive moments in the repeated cycle The lumped uncertainty term generated during the local equivalent process is further decomposed in the subsequent sliding mode dynamics into a continuous state-related unknown nonlinear term that can be approximated by a radial basis function neural network, and a neural network approximation residual and non-repetitive perturbation suppressed by a robust term.

[0082] The local equivalent input gain satisfies the following in the operating domain: ,in and These are the known lower and upper bounds of the absolute value of the local equivalent input gain, respectively.

[0083] II. Controller Design

[0084] 2.1 Preset Performance Transformation

[0085] Tracking error is uniformly defined as the actual displacement minus the expected displacement:

[0086] ;

[0087] In the formula, Indicates the pneumatic artificial muscle actuator. The process repeats for the [number]th time. Tracking error at any time, Indicates the first pneumatic artificial muscle The process repeats for the [number]th time. The expected displacement at time t. For pneumatic artificial muscles The process repeats for the [number]th time. The actual displacement at any given moment.

[0088] Select the exponentially decreasing preset performance function:

[0089] ;

[0090] in, For continuous time during a single repeated run The preset performance boundary function corresponding to time . ; Represents the initial performance boundary parameters. This represents the performance boundary parameters at steady state. This represents the decay rate at the performance boundary.

[0091] Define normalization error:

[0092] ;

[0093] Select logarithmic error transformation:

[0094] , ;

[0095] in, For the first The repeated runs occur in continuous time. The normalized tracking error at each time step is used to characterize the current tracking error. Relative to the preset performance boundary at that moment The relative size; due to and Having the same dimensions, therefore These are dimensionless variables. The symmetry performance constraint coefficients are selected. And require the normalization error to satisfy Equivalent to Thus ensuring the first At any time during the repeated operation The tracking error is limited to within the preset performance boundaries; Indicates the pneumatic artificial muscle actuator. Iteration number Tracking error at any time, Indicates the first The preset performance function at each time step.

[0096] This can be achieved by reducing the transformation error. Bounded, thus ensuring that the original tracking error always remains within the bounded range. and Within the defined preset performance boundaries.

[0097] 2.2 Fixed-time sliding mode variables and sliding mode dynamics

[0098] Constructing non-singular fixed-time sliding mode variables:

[0099] ;

[0100] in, It is a non-singular fixed-time sliding mode variable; express The derivative; , All are designed for gain via sliding surface. Here is the regularization constant. and Let be the lower and higher order power exponents of the continuous time axis, respectively, and satisfy . Regularization of low-order power terms Maintaining continuity and boundedness within a small error region, higher-order power terms It provides strong control over large error regions. In the larger error phase, it... It plays a leading role and can reduce errors rapidly.

[0101] Based on the definition of error, the local equivalent system, and the preset performance error transformation, the sliding mode dynamics can be summarized as follows:

[0102] ;

[0103] in, It is a computable dynamic term consisting of the desired trajectory and its derivative, the preset performance function and its derivative, the transformation error, and related known terms; To control voltage Equivalent input gain after entering sliding mode dynamics; To define the total unknowns in sliding mode dynamics and clarify the roles of different compensation components, it is further decomposed into unknown nonlinearities related to the continuous state. Neural networks approximate residuals and non-repeating bounded perturbations ,Right now ;in Online approximation by radial basis function neural networks, let This is the residual term, which is suppressed by the robust term; that is, as long as a sufficiently large robust gain is chosen. , making ,in and The first The continuous time of each repeated cycle Neural network approximation residual During system operation The continuous time of each repeated cycle The assumption of non-repeating perturbations at a given location has a finite upper bound. This is a finite upper bound on the residual, thus suppressing the influence of neural network approximation of the residual and non-repetitive perturbations on the closed-loop system. This term will be used in the subsequent formula derivation. Therefore, the equivalent input gain is:

[0104] ;

[0105] in, This represents the gain parameter. Based on the local input gain condition and the preset performance transformation definition, The symbol is known and non-zero within the working domain, therefore exist.

[0106] 2.3 Radial Basis Function Neural Network Design

[0107] Radial basis function neural network The Gaussian radial basis functions are defined as follows:

[0108] ;

[0109] in, For the input vector, For the first The cycle repeats at time [time]. Preset performance transformation error, Transformation error Regarding continuous time The first derivative is used to characterize the rate of change of the transformation error. For the first Non-singular fixed-time sliding mode variable that runs repeatedly For pneumatic artificial muscle actuators at all times The expected displacement trajectory, For the desired displacement trajectory Regarding time The first derivative, i.e., the desired velocity, For the desired displacement trajectory Regarding time The second derivative of , i.e., the desired acceleration; For the first Each node center This represents the node width.

[0110] The basis function vector is:

[0111] ;

[0112] The online estimation values ​​of the unknowns related to continuous states by the neural network are:

[0113] ;

[0114] Adopting a continuous-time weight update law:

[0115] ;

[0116] in, Given a positive definite learning rate matrix, radial basis function neural networks are used to approximate unknown nonlinearities that are state-dependent and continuously varying within the working domain.

[0117] 2.4 Double-Power Iterative Learning Feedforward Compensation

[0118] Define the transformation error learning term respectively and sliding mode variable learning terms :

[0119] ;

[0120] ;

[0121] Therefore, the double-power learning correction is:

[0122] ;

[0123] in, All are related to continuous time Irrelevant positive learning gain; For transformation error, For sliding mode variables;

[0124] and Iterate over the lower-order and higher-order power exponents on the axis respectively, satisfying:

[0125] .

[0126] 2.5 Overall Control Law

[0127] No. The total control voltage actually applied to the proportional valve during each repeated operation is:

[0128] ;

[0129] in, This applies to all compensation terms within the parentheses, including the iterative learning feedforward term. Its function is to offset the input channel gain after transformation, and to make the feedback term and neural network compensation term (estimated value of unknown nonlinear term) equal. For robust terms; and for iterative learning terms, they act uniformly on sliding mode dynamics; , , , , and The definition is consistent with the one mentioned above; For fixed-time sliding mode gain, For robustness gain, and These are low-order fixed-time sliding mode terms and high-order fixed-time sliding mode terms, respectively. and These are the power exponents of the lower-order fixed-time sliding mode term and the higher-order fixed-time sliding mode term, respectively. It is a power exponent.

[0130] III. Convergence Proof

[0131] To avoid ambiguity in the proof, let's clarify the symbols first: for Ideal weights, Its estimated value, This refers to the weight estimation error; and These are the upper bounds for the neural network approximation residuals and non-repetitive perturbations, respectively. This represents a composite variable that includes sliding mode variables and weight errors. function, This represents the sliding mode energy function used for fixed-time arrival analysis; The positive constant is obtained by control gain and power transformation. , ; This represents the upper bound of the fixed time required for the sliding mode variable to reach the sliding surface. This represents the fixed upper bound of the time when the transformation error on the sliding mode surface enters the adjustable residual set. This represents the upper bound of the total fixed time for a single run.

[0132] Assumption 1: Expected trajectory and its first and second derivatives in The inner continuum is bounded, and the initial error of each round satisfies the preset initial feasibility condition for performance, namely:

[0133] , , ;

[0134] ;

[0135] Assumption 2: Radial basis function neural networks have ideal weights, minimum approximation error, and bounded non-repetitive perturbations.

[0136] , , ;

[0137] Assumption 3: In practice, initial state deviations, non-repetitive perturbations, neural network approximation residuals, and model changes between iterations are allowed to constitute a bounded remainder term.

[0138] Assumption 4: The closed-loop iterative mapping can be expanded by Fréchet first order within the working domain, the closed-loop learning gain operator satisfies the strong monotonicity condition, and the higher-order remainders and non-repeating terms satisfy the absorbing condition.

[0139] Theorem 1: Fixed-time stability along a continuous time axis. This assumes that Assumptions 1 and 2 hold and the control parameters satisfy... Furthermore, the robust gain can cover the neural network approximation error and non-repetitive perturbation conditions. The closed-loop system signal is bounded, and the sliding mode variable converges to zero or enters the adjustable residual set within a fixed time independent of the initial state. The original tracking error always satisfies the preset performance boundary.

[0140] Proof: Define a composite Lyapunov function:

[0141] ;

[0142] right Taking the derivative and substituting it into the closed-loop sliding mode dynamics and weight update law, the neural network cross terms cancel each other out, therefore...

[0143] ;

[0144] Depend on It can be known Monotonically decreasing. To obtain the fixed-time upper bound of the sliding mode variable, define...

[0145] ;

[0146] By differentiating and performing the operation, we can obtain

[0147] ;

[0148] in , , , According to the fixed-time stability lemma, the time when the sliding mode variable reaches 0 satisfies:

[0149] ;

[0150] when At that time, the transformation error satisfies:

[0151] ;

[0152] when Sometimes,

[0153] ;

[0154] Therefore, have

[0155] ;

[0156] Therefore, when Time satisfies:

[0157] ;

[0158] In summary, the upper bound of the total fixed time within each round of repetition is:

[0159] ;

[0160] This upper bound does not contain the initial sliding mode variable or the initial tracking error, and is therefore independent of the initial error of this round. Combined with the inverse PPC transform, it can be guaranteed that the original tracking error always satisfies the preset performance boundary. Theorem 1 is thus proved.

[0161] Theorem 2: Fixed-step convergence property of iterative axes. When Assumptions 1, 2, 3, and 4 hold, if the closed-loop iterative mapping corresponding to the double-power learning law satisfies the strong monotonicity condition and the absorbing condition of the remainder term, then under ideal repetition conditions, It converges to zero within a finite number of iterations independent of the initial iteration error; when there exists a bounded non-repeating remainder term, Enter the computable set of residuals within a finite number of iterations.

[0162] In iterative axis analysis, and These represent the trajectory energy of the transformation error and the sliding mode variable, respectively. These are weighting coefficients. The composite iterative error energy; Indicates by and The composite error vector; The first-order learning gain operator represents the mapping of closed-loop errors between two adjacent rounds. Indicates by , The learning direction is composed of double power error information. This represents the remainder term composed of higher-order expansion remainder, non-repetitive perturbation, approximation residual, and model changes between iterations. For a strong monotonic condition, it is a positive constant. The absorbability coefficient of the remainder term; , , , The positive constant is obtained from the norm relationship of finite time intervals and the learning gain; It is a unified upper bound for the energy difference of the bounded non-repeating remainder term.

[0163] Define composite error energy:

[0164] ;

[0165] in And thus, the composite iterative error energy is obtained:

[0166] ;

[0167] Proof: Mapping the closed-loop iteration between two adjacent rounds using Fréchet first-order expansion is as follows:

[0168] ;

[0169] in, For closed-loop composite learning gain operators, For the reason and The learning direction is composed of double power information. It includes higher-order expansion remainders, non-repetitive perturbations, neural network approximation residuals, and so on.

[0170] Assume there is , making

[0171] ;

[0172] Then the energy difference between two adjacent rounds is...

[0173] ;

[0174] If it exists The second-order learning term and the remainder term satisfy the following conditions:

[0175] ;

[0176] Then there is

[0177] ;

[0178] Within a limited time interval Above, according to norm relations, there exists a relationship with... irrelevant positive numbers , , making

[0179] ;

[0180] ;

[0181] make as well as

[0182] ;

[0183] but

[0184] ;

[0185] Under ideal replication conditions Next. When When the term is in a higher order, it dominates; otherwise, it is dominated by lower order terms; and the function is used. Given the monotonicity and mean value theorem, we can obtain the following from any initial energy... The required number of iterations shall not exceed Therefore, the maximum number of iterations under ideal repetition conditions satisfies:

[0186]

[0187] When there exists a bounded non-repeating remainder term At that time, select any Define the residual energy threshold:

[0188]

[0189] Define shift energy Then, outside the residual set, we have:

[0190]

[0191] Using the same partitioning estimation as the ideal conditions, the maximum number of iterations required to enter the residual set can be obtained as follows:

[0192]

[0193] The above and None of them contain initial composite error energy, and therefore are unrelated to the initial iteration error. As can be seen from the definition of composite error energy, and The trajectory energy decreases with iteration; combined with the boundedness of the continuous time axis and the inverse PPC transform, the original tracking error continues to satisfy the preset performance boundary. Theorem 2 is proved.

[0194] On a continuous time axis, Theorem 1 guarantees that the sliding mode variable in each round of repeated operation does not exceed [a certain value]. It reaches the sliding surface within a fixed time and ensures that the transformation error does not exceed The adjustable residual set is entered within a fixed time period, therefore the total practical fixed time for each round does not exceed [a certain value]. On the iteration axis, Theorem 2 guarantees that, under ideal repetition conditions, the composite error energy does not exceed [a certain value]. After the iterations converge to zero; when there is a bounded non-repeating remainder term, the composite error energy does not exceed After the next iteration, enter The residual set. This leads to a biaxial convergence conclusion that combines the fixed-time stability of the continuous-time axis with the finite-step learning convergence of the iterative axis.

[0195] Example Figure 2The electrical signal and pneumatic circuit connections of the experimental platform are presented. The platform mainly consists of a computer, data acquisition card, proportional valve, power supply, displacement sensor, force sensor and signal conditioning module, pneumatic artificial muscle actuator, and compressed air source. The computer sends control signals to the data acquisition card via MATLAB / Simulink. The data acquisition card converts the digital signals into analog voltage signals (D / A conversion) and outputs them to the proportional valve. The proportional valve adjusts the output air pressure according to the input voltage, which acts on the pneumatic artificial muscle actuator through the air tube, causing it to contract and displace. The displacement change of the pneumatic artificial muscle actuator is measured in real time by the displacement sensor. The measured signal is amplified and filtered by the signal conditioner before being transmitted to the data acquisition card. The data acquisition card then performs analog-to-digital conversion (A / D conversion) and returns the data to the computer for recording and feedback control. The air compressor provides stable compressed air for the experiment, and the pressure is kept constant and reliable after passing through the pressure regulating component. The entire system runs in MATLAB's real-time external mode. By compiling, downloading, and executing the control program, real-time driving of the pneumatic artificial muscle actuator and experimental data acquisition are achieved.

[0196] The ability of the pneumatic artificial muscle actuator to track different types of reference signals under different loads was tested, and the test results are as follows: Figures 3-26 As shown.

[0197] Figures 3 to 5 This demonstrates the tracking performance under a 600g load and a low-frequency composite sinusoidal trajectory. Figure 3 The actual trajectory and the reference trajectory remain basically consistent over multiple periods, and can follow the peak, trough and local changes of the composite sine trajectory well. Figure 4 The results show that as the number of iterations increases, the maximum absolute error and root mean square error generally show a downward trend and gradually stabilize, indicating that iterative learning feedforward compensation can reduce repetitive tracking errors round by round. Figure 5 This further indicates that the tracking error after convergence mainly fluctuates within a small range around zero, with short-term error peaks only appearing at some locations where the trajectory changes rapidly.

[0198] Figures 6 to 8 This demonstrates the tracking performance under a 600g load and a high-frequency composite sinusoidal trajectory. Compared to low-frequency conditions, high-frequency trajectories place higher demands on the system's dynamic response, but... Figure 6 The actual trajectory still follows the reference trajectory quite well. Figure 7 It can be seen that the maximum absolute error and the root mean square error decrease as a whole during the iteration process and gradually stabilize. Figure 8 The tracking error shown fluctuates around zero, indicating that the control method still has good learning and correction capabilities and tracking stability even after the trajectory frequency is increased.

[0199] Figures 9 to 11 The tracking performance under a 600g load and low-frequency composite triangular trajectory was demonstrated. The composite triangular trajectory exhibits significant slope changes and commutation characteristics. Figure 9 This indicates that the output of the PAM actuator can follow the rising, falling, and peak positions of the reference trajectory well. Figure 10 The two types of error indicators decreased significantly in the first few iterations, and then remained at a low level; Figure 11 This indicates that the final tracking error is mainly distributed near zero, with some transient fluctuations at trajectory reversal and local abrupt change locations, demonstrating that the present invention can effectively learn and compensate for non-smooth repeating trajectories.

[0200] Figures 12 to 14 This demonstrates the tracking performance under a 600g load and a high-frequency composite triangular trajectory. Figure 12 The reference trajectory is very close to the actual trajectory, and it can still maintain good trajectory tracking performance under conditions of multiple rapid reversals. Figure 13 The maximum absolute error and root mean square error shown generally decrease with increasing iteration number and remain stable in subsequent iterations; Figure 14 The error shown remains within a small range for most of the operating range, indicating that the present invention has good adaptability to high-frequency, rapid reversing repetitive trajectories.

[0201] Figures 15 to 17 This demonstrates the tracking performance under a 1200g load and a low-frequency composite sinusoidal trajectory. With increased load, Figure 15 The actual trajectory can still maintain continuous following of the reference trajectory; Figure 16 The maximum absolute error and root mean square error gradually decrease and tend to stabilize as the number of iterations increases; Figure 17 Although the error exhibits some periodicity and local fluctuations, it remains within a limited range overall, indicating that the composite control structure can maintain effective iterative learning and closed-loop tracking under load variation conditions.

[0202] Figures 18 to 20 This demonstrates the tracking performance under a 1200g load and a high-frequency composite sinusoidal trajectory. Figure 18 This indicates that under the combined effects of a large load and a high trajectory frequency, the PAM actuator can still follow the main variation patterns of the reference trajectory. Figure 19 Both types of error indicators generally decrease with the number of iterations and tend to stabilize in the later stages; Figure 20 The results show that the error after convergence mainly fluctuates around zero, indicating that the present invention still has a certain robust tracking capability when high dynamic requirements and load disturbances coexist.

[0203] Figures 21 to 23 This demonstrates the tracking performance under a 1200g load and a low-frequency composite triangular trajectory. Figure 21The actual trajectory can reproduce the piecewise linear changes and reversal characteristics of the composite triangular trajectory quite well; Figure 22 This indicates that the maximum absolute error and root mean square error decrease overall as the number of iterations increases; Figure 23 The tracking error shown is concentrated near zero in most operating ranges, indicating that under heavy load, iterative learning compensation can still effectively reduce the error of repeated trajectories.

[0204] Figures 24 to 26 The tracking performance under a 1200g load and high-frequency composite triangular trajectory was demonstrated. This operating condition includes a large load, high-frequency changes, and multiple commutations, which places higher demands on the hysteresis, pneumatic hysteresis, and dynamic compensation capabilities of the PAM actuator and controller. Figure 24 The actual trajectory and the reference trajectory generally maintain good consistency. Figure 25 The maximum absolute error and root mean square error generally decrease with the number of iterations and tend to stabilize. Figure 26 The final error mainly fluctuates around zero, indicating that the present invention can still maintain effective trajectory learning and error suppression capabilities under complex and repetitive trajectory conditions.

[0205] In conclusion, Figures 3 to 26 The control effect of this invention was verified using different loads, frequencies, and two types of repetitive trajectories: composite sine and composite triangular. In each group of experiments, the actual trajectory could follow the reference trajectory well. The maximum absolute error and root mean square error generally showed a decreasing trend with the number of iterations and gradually stabilized, with the final tracking error mainly remaining within a limited range near zero. The above results indicate that the controller can still achieve high-precision repetitive trajectory tracking even under conditions of PAM actuator hysteresis nonlinearity, aerodynamic hysteresis, load changes, and non-repetitive disturbances, and demonstrates good learning convergence and adaptability to operating conditions.

[0206] Although embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the specification and embodiments. They can be applied to various fields suitable for the present invention. For those skilled in the art, other modifications can be easily made. Therefore, without departing from the general concept defined by the claims and their equivalents, the present invention is not limited to the specific details and illustrations shown and described herein.

Claims

1. A fixed-time iterative learning control method for a pneumatic artificial muscle actuator, characterized in that, include: A non-affine nonlinear system model characterizing the operation of a pneumatic artificial muscle actuator is established, and the control law of the pneumatic artificial muscle actuator is set as a function composed of a continuous time axis feedback control quantity and an iterative learning feedforward compensation term based on the system model. And the control law is used to control the movement of the pneumatic artificial muscle actuator; The continuous time axis feedback control quantity is determined by the following method: The tracking error of the pneumatic artificial muscle actuator is obtained based on the expected displacement and the actual displacement, and the tracking error is converted into an unconstrained transformation error through error transformation. Based on the transformation error, a non-singular fixed-time sliding mode variable containing regularized low-order power terms and high-order power terms is constructed, and the sliding mode is dynamically divided into computable dynamic terms, control input terms, nonlinear unknown terms, and robust terms. The nonlinear unknown term is determined by online estimation based on the expected displacement, the transformation error, and the sliding mode variable; A low-order fixed-time sliding mode term and a high-order fixed-time sliding mode term are constructed using fixed-time sliding mode variables; the computable dynamic term, the low-order fixed-time sliding mode term, the high-order fixed-time sliding mode term, the nonlinear unknown term, and the robust term are summed to obtain the continuous-time axis feedback control quantity.

2. The fixed-time iterative learning control method for a pneumatic artificial muscle actuator according to claim 1, characterized in that, The control law for the pneumatic artificial muscle actuator is: ； In the formula, Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. The control voltage applied to the proportional valve at all times Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. Continuous time-axis feedback control quantity at any given moment Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. Iterative learning of feedforward compensation terms at each time step. To control voltage Equivalent input gain after entering sliding mode dynamics express The inverse operation.

3. The fixed-time iterative learning control method for a pneumatic artificial muscle actuator according to claim 2, characterized in that, Also includes: Based on the transformation error, the transformation error learning term is obtained: ； Based on the sliding mode variables, the learning terms for the sliding mode variables are obtained: ； The iterative learning feedforward compensation term is updated using the transformation error learning term and the sliding mode variable learning term; in, ; ; In the formula, Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. Iterative learning of feedforward compensation terms at each time step. For pneumatic artificial muscle actuators The process repeats for the [number]th time. Learning correction term for time-varying error. For pneumatic artificial muscle actuators The process repeats for the [number]th time. Learning correction term for sliding mode variable at time step. For pneumatic artificial muscle actuators The process repeats for the [number]th time. Total learning and correction amount at all times; All are positive learning gains; and Iterate over the lower-order and higher-order power exponents on the axis respectively; For pneumatic artificial muscle actuators The process repeats for the [number]th time. Error in time transformation, For pneumatic artificial muscle actuators The process repeats for the [number]th time. Sliding mode variables at time points; and These are the lower-order and higher-order power terms of the error transformation, respectively; and These are the lower-order and higher-order power terms of the sliding mode variable, respectively.

4. The fixed-time iterative learning control method for a pneumatic artificial muscle actuator according to claim 3, characterized in that, The formula for calculating the transformation error is as follows: ； in, ; ; In the formula, Indicates the first pneumatic artificial muscle actuator Repeat the first run Tracking error at any time, For pneumatic artificial muscle actuators Repeat the first run Normalization error at time step; Indicates the first Preset performance function at any given time; The allowable error boundary at the initial moment of the system. This represents the allowable error boundary when the system is stable. For the error decay rate, This represents the symmetry performance constraint coefficient.

5. The fixed-time iterative learning control method for a pneumatic artificial muscle actuator according to claim 4, characterized in that, The sliding mode variable is: ； in, It is a non-singular fixed-time sliding mode variable; To regularize lower-order power terms, It is a higher-order power term; ； In the formula, and All are designed for gain via sliding surface. Here is the regularization constant. and Let be the lower and higher order power exponents of the continuous time axis, respectively, and satisfy . ; for The first derivative.

6. The fixed-time iterative learning control method for a pneumatic artificial muscle actuator according to claim 4 or 5, characterized in that, The unknown nonlinear terms in sliding mode dynamics are estimated online using a radial basis function neural network, as follows: by Using the input vector, the approximation property of the radial basis function neural network is utilized to obtain the estimate of the unknown nonlinear term: ； in, For pneumatic artificial muscle actuators in The expected displacement trajectory at time t. For the desired displacement trajectory Regarding time The first derivative, For the desired displacement trajectory Regarding time The second derivative; For the weights of the radial basis function neural network, through Online updates, The learning rate matrix is ​​positive definite. yes The derivative of .

7. The fixed-time iterative learning control method for a pneumatic artificial muscle actuator according to claim 6, characterized in that, The non-affine nonlinear system model of the pneumatic artificial muscle actuator is as follows: ； in, The iteration number for repeated runs. The continuous time within each repeated run, Duration of a single task run; and These represent the pneumatic artificial muscle actuators. The process repeats for the [number]th time. The displacement and velocity states at each moment. Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. The proportional valve drive voltage at any given time, Indicates the first pneumatic artificial muscle actuator Non-affine nonlinear function that runs repeatedly Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. External disturbances at any given moment. Indicates the first pneumatic artificial muscle actuator The process repeats for the [number]th time. The actual displacement output at any given time.

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