Time-varying state constraint terminal sliding mode tracking control method based on adaptive network

Abstract

The application discloses a time-varying state constraint terminal sliding mode tracking control method based on an adaptive network, and relates to the technical field of adaptive control.The method comprises the following steps: step 1, constructing a constraint potential energy error for each actual state of an engineering controlled object; step 2, constructing a fixed rational exponential composite sliding mode variable; all the sliding mode variables of all the actual states of the engineering controlled object form an integral sliding mode surface in a vector layer; step 3, using a single hidden layer radial basis network to perform online approximation on the actual dynamics of the engineering controlled object, introducing the network output as a dynamic compensation item into a control input, so as to ensure that the integral sliding mode surface is driven to zero in a limited time and real-time compensation of the actual state of the engineering controlled object is simultaneously completed. The application realizes limited-time high-precision tracking control of a controlled engineering controlled object with unknown nonlinear and double-sided time-varying boundaries.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of adaptive control, in particular to a time-varying state constraint terminal sliding mode tracking control method based on an adaptive network. BACKGROUND

[0002] In modern control engineering, the stable control and high-precision tracking of complex nonlinear engineering controlled objects are always the hot issues of research and application. Especially in the fields of aerospace, high-speed rail transportation, medical robots and high-precision manufacturing, the controlled objects often have significant uncertainty, external disturbance and strong coupling characteristics. In addition, the engineering controlled objects are usually subject to various safety constraint limits in the running process, such as physical boundaries of temperature, voltage, torque or displacement, and these constraints may also change dynamically over time in some applications. Therefore, designing a control method that can simultaneously achieve high-precision tracking, strong robust control, finite-time convergence and strict satisfaction of dynamic state constraints has become an important issue in the field of intelligent control.

[0003] Traditional control methods such as PID control, although simple in structure and easy to implement, have significant shortcomings in dealing with nonlinear dynamics, uncertainty modeling and time-varying state constraints. For engineering controlled objects with unknown nonlinear structure, PID control cannot provide a compensation mechanism for the nature of the engineering controlled object, making it difficult to ensure that the engineering controlled object can still operate stably under disturbance or modeling error. In addition, due to the fixed control gain of PID, the response of the engineering controlled object cannot adapt to multi-scale dynamics, and the convergence speed and steady-state error cannot be considered. For multivariable coupled engineering controlled objects, their performance will be severely degraded due to mutual interference between channels.

[0004] To make up for the limitations of classical control, in recent years, sliding mode control methods have gradually become a research hotspot. Sliding mode control uses sliding surface design and switching logic to achieve robust control by using the sliding motion of the engineering controlled object on the sliding surface, and has good fault tolerance for the uncertainty of the engineering controlled object model. However, the equal-order sliding surface used by traditional sliding mode control often leads to too slow engineering controlled object reaching process on the sliding surface, especially in the small error stage, the approaching speed decreases significantly, affecting the response performance of the controlled engineering object. To improve the convergence speed, terminal sliding mode control is proposed, which uses a rational power nonlinear sliding surface design to achieve the convergence of the engineering controlled object state to the reference trajectory in a finite time. However, this method may cause control discontinuity when the state approaches zero, and the handling of state constraints relies on saturation functions or forced truncation, which can easily cause chattering phenomenon. SUMMARY

[0005] To address the aforementioned technical challenges, a time-varying state-constrained sliding mode tracking control method based on an adaptive network is provided. This method achieves high-precision finite-time tracking control of controlled engineering objects with unknown nonlinearities and bilateral time-varying boundaries. The method naturally embeds state constraint information into the control law, avoiding the discontinuities caused by traditional projection or saturation functions, while ensuring rapid convergence of all states within the constraint range. After convergence on the sliding surface, it automatically switches to a low-frequency learning mode and freezes the exponential parameters, thereby reducing computational resource consumption and suppressing steady-state chattering. This method exhibits strong robustness, adaptability, and engineering applicability.

[0006] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0007] A time-varying state-constrained terminal sliding mode tracking control method based on adaptive networks, the method comprising:

[0008] Step 1: Construct constraint potential energy error for each actual state of the engineering controlled object with bilateral time-varying state constraints, and simultaneously map the deviation of the actual state relative to the reference trajectory to the logarithmic potential energy interval between its upper and lower boundaries that change with time.

[0009] Step 2: Construct composite sliding mode variables with fixed rational exponents; the sliding mode variables of all actual states of the controlled object in the project form an overall sliding surface at the vector level. When the overall sliding surface approaches zero, all constraint potential energy errors are eliminated at the same time, thereby ensuring that the actual state of the controlled object in the project tracks the reference trajectory within a finite time and does not exceed any time-varying upper or lower boundary.

[0010] Step 3: Use a single hidden layer radial basis network to approximate the actual dynamics of the controlled object in the project online, and introduce the network output as a dynamic compensation term into the control input to ensure that the overall sliding surface is driven to zero within a limited time and the real-time compensation of the actual state of the controlled object is completed synchronously.

[0011] Furthermore, the method also includes: Step 4: By selecting a Lyapunov function that includes a quadratic term of the composite sliding mode variable and a quadratic term of the weight estimation error of the single hidden layer radial basis network, a gradient-type weight update law based on the product of the composite sliding mode variable and the basis function of the single hidden layer radial basis network is given; Under the action of a fixed diagonal learning rate, the time derivative of the Lyapunov function is always non-positive, and decays to zero simultaneously with the composite sliding mode variable and the weight estimation error, thereby ensuring that the controlled object of the project achieves sliding mode arrival, bounded convergence of the weights of the single hidden layer radial basis network, and satisfies the two-sided time-varying state constraints for all times within a finite time.

[0012] Furthermore, step 1 specifically includes:

[0013] Step 1.1: At the beginning of each control cycle, read the upper bound and the lower bound corresponding to the current actual state in real time, both of which are updated synchronously with time to fully reflect the instantaneous position of the two-sided time-varying state constraint;

[0014] Step 1.2: Subtract the actual state from the reference trajectory to obtain the first deviation; at the same time, subtract the upper bound from the reference trajectory and the reference trajectory from the lower bound to obtain the second deviation and the third deviation, respectively;

[0015] Step 1.3: Perform a natural logarithm operation on the ratio of the second deviation to the first deviation to obtain the upper side potential energy term; perform a natural logarithm operation on the ratio of the first deviation to the third deviation to obtain the lower side potential energy term;

[0016] Step 1.4: Algebraically sum the upper side potential energy term and the lower side potential energy term with opposite signs to obtain the constraint potential energy error; when the actual state approaches the upper bound or the lower bound infinitely, the amplitude of the constraint potential energy error correspondingly tends to infinity in a finite time.

[0017] Further, the composite sliding mode variable in step 2 is the sum of a linear normalization part, a rational exponent part, and a time integral part; the linear normalization part is stored in the first buffer area of the register stack as a reference quantity for subsequent superposition operations to ensure that the composite sliding mode variable has a linear response characteristic when the error amplitude is small; the rational exponent part is used to enhance the convergence rate of the error interval and ensure the finite time convergence of the engineering controlled object; the time integral part is used to compensate in real time for the influence of slow boundary movement on the expansion of the error interval, ensuring the symmetric convergence behavior of the overall sliding mode surface in a dynamic constraint environment.

[0018] Further, in step 2, at the beginning of the current control cycle, a set of fixed and rational number related exponents is uniformly selected for all actual states; the selection of the exponent set follows the following principles: the numerator and the denominator are both positive integers to ensure that the exponent value is positive and can accurately describe the finite time convergence characteristic; different actual states share the same exponent set to eliminate the imbalance of relative convergence speed caused by the difference in exponents; the exponent value remains the same as the previous control cycle to avoid control discontinuity caused by frequent adjustment of exponents. Through this uniform selection process, the subsequent composite sliding mode variable is ensured to evolve cooperatively under the same order homogeneous structure, providing consistent convergence power for the overall sliding mode surface.

[0019] Further, in step 2, after confirming the index, the current constraint potential energy error of each actual state is calculated in real time, and the absolute amplitude of the error is mapped with the index to obtain a first sliding mode primitive; at the same time, the sign information of the constraint potential energy error is extracted independently to form a second sliding mode primitive. The two types of sliding mode primitives cooperate with each other: the first sliding mode primitive amplifies the error far from zero through power, ensuring rapid attenuation; the second sliding mode primitive records the error direction, ensuring that the switching does not produce directional ambiguity when the sliding mode tends to zero; thus, each actual state corresponds to a pair of original sliding mode primitives; a continuous, monotonic and saturated buffer mapping interval is added to the amplitude part of the original sliding mode primitive; when the constraint potential energy error is in the preset range, the buffer mapping does not affect the output of the original sliding mode primitive; when the constraint potential energy error tends to infinity, the buffer mapping is gradually smoothed and truncated within a limited amplitude, realizing soft clipping of the control gain to limit the control amplitude and avoid high-frequency chattering near the overall sliding mode surface.

[0020] Further, in step 2, after obtaining the buffered two types of sliding mode primitives, the two types of sliding mode primitives are spliced in the form of weighted algebraic sum with a fixed rational index as the weight to form a composite sliding mode variable; when splicing, odd-even cancellation design is used: the amplitude primitive and the sign primitive are added in the same direction in terms of sign, but are staggered in terms of index order, so as to balance the high-gain convergence and low-gain steady-state performance; to synchronously process multi-dimensional actual states, all composite sliding mode variables are arranged in order of states to form an equal-dimension vector; then, the overall sliding mode surface is defined in the form of an equal-dimension vector as a column vector; the zero vector of the overall sliding mode surface represents that all constraint potential energy errors are simultaneously completely offset; when the overall sliding mode surface is far from the zero vector, its components depict the convergence distance of the corresponding state; when the overall sliding mode surface approaches the zero vector, it indicates that all actual states have approached the reference trajectory and are kept within the safety interval of the double-sided time-varying constraint.

[0021] Further, in step 2, when the Euclidean norm of the overall sliding mode surface falls below the preset sliding mode threshold and remains stable for a number of consecutive sampling periods, it is determined that the overall sliding mode surface has effectively approached the zero vector; at this time, the following processes are triggered: freezing the index update logic of the composite sliding mode variable to keep the current index set from changing, so as to avoid reactivating the high-gain area; switching the network learning rate from high-frequency online update to low-frequency gradual calibration to reduce the consumption of computing resources.

[0022] Further, in step 3, a set of mutually non-overlapping single-hidden-layer radial basis network basis function centers is constructed for each actual state, and an initial width is uniformly set for all single-hidden-layer radial basis network basis functions; the network weight is initialized as a zero vector to avoid unknown bias of network output on control input in the initial stage; at the same time, in order to prevent outliers from affecting the convergence speed, all single-hidden-layer radial basis network basis function centers are selected by clustering according to the historical difference between the actual state and the reference trajectory within a limited window, so as to ensure that the initial network structure covers the main dynamic mode.

[0023] Further, in step 3, the component error signal of the overall sliding mode surface on the composite sliding mode variable is used as the driving force to calculate the product of the single-hidden-layer radial basis network basis function and the weight increment, and the weight increment is weighted to fix the network learning rate; the weight increment is added to the original weight vector, and the projection operator is used to limit it in the pre-defined bounded region, so as to ensure that the network weight is always bounded.

[0024] Compared with the prior art, the method has the beneficial effects that: by constructing a constraint potential error with a logarithmic barrier characteristic, the method realizes natural embedded control of the bilateral time-varying state constraint, avoids the non-continuous control problem caused by the traditional projection method, and improves the response sensitivity when the state approaches the boundary. Secondly, by constructing a composite sliding mode variable containing a linear normalization term, a rational exponential term and a time integral term, the method takes into account the steady-state accuracy in the small error region and the convergence speed in the large error region, and realizes the control goal of approaching the reference trajectory in a limited time. Thirdly, the single-hidden-layer radial basis function network is introduced as an online modeler to approximate the unknown nonlinear dynamics of the engineering controlled object, and the network weight is updated by the sliding mode error signal driving, so that the controller has dynamic compensation ability and self-learning characteristics during operation. In addition, after the overall sliding mode surface converges, the network learning rate is automatically switched and the exponential parameter is frozen, so as to effectively reduce the consumption of computing resources and the risk of control chattering, and adapt to the natural transition of the engineering controlled object from the dynamic stage to the steady state. At the same time, the network weight update is constrained by the projection operator, which ensures that the network parameters are always bounded, and further improves the closed-loop stability of the engineering controlled object. In summary, the present application provides a solution for engineering controlled objects in terms of handling time-varying state constraints, improving the response speed and anti-interference ability of the controlled engineering object, and optimizing the network training efficiency, and is suitable for various high-requirement, high-precision and strong-constraint complex nonlinear engineering controlled object control scenarios. BRIEF DESCRIPTION OF DRAWINGS

[0025] Figure 1 The method flowchart of the adaptive network-based time-varying state constraint terminal sliding mode tracking control method proposed in the present application is shown in the figure.

[0026] Figure 2Fig. 1 is a schematic diagram of a constraint potential error variation process in an embodiment of the present application;

[0027] Figure 3 Fig. 2 is a schematic diagram of a composite sliding mode variable convergence characteristic in an embodiment of the present application. DETAILED DESCRIPTION

[0028] The following description is provided so as to enable any person skilled in the art to make or use the application. The preferred embodiments described in the following description are only examples of the application and are not intended to limit the scope, applicability or configuration of the application in any way. Various changes to the preferred embodiments described herein can be made by those skilled in the art without departing from the scope of the application.

[0029] Reference Signs List Figure 1 As shown, the adaptive network-based time-varying state constraint terminal sliding mode tracking control method comprises:

[0030] Step 1: constructing a constraint potential error for each actual state of the engineering controlled object with double-sided time-varying state constraints, and mapping the deviation of the actual state from the reference trajectory to the logarithmic potential interval between the upper and lower boundaries thereof varying with time;

[0031] Step 2: constructing a composite sliding mode variable with a fixed rational index; the sliding mode variables of all actual states of the engineering controlled object form an overall sliding surface in the vector layer, and when the overall sliding surface tends to zero, all constraint potential errors are simultaneously eliminated, thereby ensuring that the actual state of the engineering controlled object tracks the reference trajectory in a finite time and does not exceed any upper and lower boundaries varying with time;

[0032] Step 3: using a single-hidden-layer radial basis network to online approximate the actual dynamics of the engineering controlled object, and introducing the network output as a dynamic compensation term into the control input, thereby ensuring that the overall sliding surface is driven to zero in a finite time and the real-time compensation of the actual state of the engineering controlled object is simultaneously completed.

[0033] Specifically, the adaptive network-based time-varying state constraint terminal sliding mode tracking control method is used for trajectory tracking control of an engineering controlled object with double-sided time-varying state constraints, and the engineering controlled object is a nonlinear system (which can be a second-order or higher-order system), preferably a motor servo control system, an industrial robot joint actuator, a four-rotor unmanned aerial vehicle attitude / position channel, or a vehicle line control chassis.

[0034] For motor servo control system (PMSM / BLDC / DC servo): states are angular position / speed (outer loop) and d-q current (inner loop); double-sided time-varying constraints include: angular stroke and mechanical limits, speed upper / lower limits, current and voltage amplitude limits, temperature-related derating, etc.; constraints can change online due to bus voltage, load disturbance, temperature, and efficiency optimization strategies. Matching disturbances are mainly load torque, and non-matching disturbances include cogging torque, friction, and modeling residuals. The method introduces constraint potential error and composite sliding mode variables in the outer loop or speed loop, and uses RBF network to online approximate total uncertainty, and outputs are applied to the motor through servo driver / inverter, achieving tracking in finite time and avoiding over-limit; buffer mapping suppresses high-frequency chattering and current / voltage saturation.

[0035] For industrial robot joint actuators (multi-joint MIMO): each joint state is subject to double-sided constraints of angle, speed, acceleration, and joint torque, and is affected by coupled rigid-flexible effects, gravity, friction, and changes in end load; the task layer (trajectory, collision avoidance) often gives time-varying safety corridors and speed curves. The method constructs quantized composite sliding mode variables in joint space and forms an overall sliding surface on the vector level, and a unified set of rational exponents avoids convergence imbalance among joints; RBF network online compensates for coupled and uncertain dynamics, achieving multi-joint collaborative convergence in finite time and strictly maintaining within the given double-sided time-varying boundaries.

[0036] For quadrotor UAV attitude / position channels (cascade control architecture): attitude channel states (roll / pitch / yaw angle and angular velocity) have small angle / angle rate and rotor saturation boundaries; position channel has double-sided boundaries of height, no-fly zone / safety corridor, which changes online with terrain, geographic fence, and wind field. The method can be used for attitude inner loop or position outer loop: through constraint potential error, it interfaces with safety corridors (such as height window, XY channel corridor), relies on RBF network to compensate aerodynamic uncertainty under gust and parameter drift, and composite sliding mode makes error converge in finite time and avoids over-limit, and buffer mapping reduces chattering caused by thrust saturation and distribution.

[0037] For vehicle-by-wire chassis (steering / drive / brake-by-wire): lateral deviation, heading error, and longitudinal speed / acceleration are subject to double-sided time-varying constraints, typically lane boundaries, road curvature, and tire-road adhesion conditions defined stability boundaries (such as yaw rate / side slip angle window), whose upper and lower limits change dynamically with speed, road adhesion coefficient μ, wind, and traffic rules. The method maps "lane safety corridor / stability window" into time-varying boundaries, and uses composite sliding mode and RBF network to online compensate tire-road uncertainty and disturbances such as crosswind; control inputs correspond to EPS angle / torque and drive / brake torque commands, ensuring finite-time tracking of the reference path and keeping the state within the double-sided boundaries.

[0038] For a continuous-time nonlinear engineering controlled object with unknown disturbance and structure uncertainty, it is difficult to simultaneously meet the three conflicting goals of "fast tracking", "state not crossing the boundary" and "anti-model mismatch" by simply relying on traditional proportional integral derivative strategy or conventional sliding mode method. To solve this comprehensive contradiction, the time-varying state-constrained terminal sliding mode tracking control method based on adaptive network proposed in the application follows the principle of "embedding constraints, compensating unknowns and shortening time convergence" in the overall idea. Firstly, in the link of constructing the constraint potential error of each actual state of the engineering controlled object, a group of upper and lower boundaries updated synchronously with time is introduced, and the deviation of the actual state from the reference trajectory is simultaneously mapped into the logarithmic potential interval between the upper and lower boundaries changing with time. The key of this mapping is to map the originally limited deviation distance into the potential energy level rapidly increasing near the boundary by logarithmic form, so that the control law can "perceive" the degree of danger of the state approaching the boundary in numerical value, thereby increasing the correction strength in advance when still in the safety zone of the upper and lower boundaries, and generating a steep braking effect when approaching the limit position. In this way, it not only ensures that the boundary crossing does not occur under the condition of limited calculation precision, but also avoids the robustness decline and frequent switching chattering caused by repeated judgment and hard truncation of the control amount as in the traditional projection method.

[0039] Then, in the process of constructing the fixed rational exponential composite sliding variable, the method takes the normalized constraint potential error as the basis layer, and further superimposes the rational exponential term with nonlinear amplification characteristics and the time integral term specially tracking the boundary movement trend, so that all the sliding variables of the actual states of the engineering controlled object form an overall sliding surface in the vector plane. The overall sliding surface has the fine adjustment ability of the linear segment, the accelerated convergence ability of the nonlinear segment and the real-time following ability when the boundary slowly drifts. When the overall sliding surface tends to zero, all the constraint potential errors are eliminated at the same time. This result not only indicates that the static deviation of the actual state from the reference trajectory is compressed to zero, but also means that even if the upper and lower boundaries are changing in a slow or fluctuating manner, the actual state is always accurately locked in the double-sided safety zone. The actual state of the engineering controlled object tracks the reference trajectory within a limited time and does not exceed any upper and lower boundaries changing with time. Since the convergence order of the rational exponential is less than one, the convergence time can be directly calculated by designing the numerator and denominator of the exponential, and no longer depends on the experience-based tuning of open-loop gain.

[0040] Finally, in the online approximation of the actual dynamics of the engineering controlled object by using a single-hidden-layer radial basis network, the method selects radial basis functions covering the entire feasible state space as the base function cluster, and obtains a rough fitting of the unknown dynamics through least squares projection during weight initialization, and then continuously fine-tunes in the gradient direction during closed-loop operation. The network output is introduced into the control input as a dynamic compensation term, and the boundary velocity compensation signal and the terminal sliding mode reaching gain are used to continuously weaken the net error caused by model mismatch and external disturbance, and to quickly decay the residual error energy in the overall sliding mode surface to zero according to the preset terminal convergence law. Since the approximation error of the radial basis network can be planned in advance according to an arbitrarily small upper bound by increasing the number of nodes or adjusting the width parameter, combined with the strong robustness of the sliding mode surface, even in the presence of actuator saturation, limited sensor bandwidth or occasional packet loss of the communication link, the method can still maintain finite-time convergence and maintain bounded oscillation of the error after convergence rather than drift. Overall, by organically integrating the three interlocking links of constructing a constrained potential error for each actual state of the engineering controlled object, constructing a fixed rational exponential composite sliding mode variable, and using a single-hidden-layer radial basis network to online approximate the actual dynamics of the engineering controlled object, the controller has an adaptive defense mechanism of "the closer the boundary, the stronger the suppression", a terminal time reduction mechanism of "the greater the error, the faster the convergence", and an online learning mechanism of "the less accurate the model, the more timely the compensation". Therefore, without relying on the accuracy of the prior physical model and the repeated trial-and-error adjustment of the empirical coefficient, the stable, high-speed and high-precision tracking control of the nonlinear engineering controlled object with bilateral time-varying state constraints is realized, especially suitable for applications such as aircraft re-entry attitude holding, robot path following, flexible arm end positioning, and chemical process limit operation, which have strict requirements on both safety boundaries and rapid response.

[0041] Further, the method further comprises: step 4: by selecting a Lyapunov function containing a quadratic form of the composite sliding mode variable and a quadratic form of the single-hidden-layer radial basis network weight estimation error, a gradient-type weight update law based on the product of the composite sliding mode variable and the single-hidden-layer radial basis network base function is given; under the action of a fixed diagonal learning rate, the time derivative of the Lyapunov function is always non-positive and simultaneously decays to zero with the composite sliding mode variable and the weight estimation error, thereby ensuring that the engineering controlled object realizes sliding mode reaching, single-hidden-layer radial basis network weight bounded convergence, and satisfies all bilateral time-varying state constraints at all times within a finite time.

[0042] The step firstly introduces a Lyapunov function candidate based on the analysis of the dynamics of the overall sliding surface. The Lyapunov function is composed of two parts: one is the quadratic term of the composite sliding variable, which is used to measure the deviation energy of the trajectory of the engineering controlled object from the overall sliding surface; the other is the quadratic term of the weight value estimation error of the single hidden layer radial basis network, which is used to measure the gap energy between the real-time network weight value and the ideal weight value. Since both types of energy appear in the form of square, the state of the engineering controlled object and the parameter uncertainty can be measured in a geometric sense, and the function is directly guaranteed to have a unique minimum at the origin through matrix positive definiteness. Subsequently, while keeping the basis function cluster unchanged, a gradient-type weight value updating law based entirely on the current observation is constructed by performing gradient descent operation on the product of the composite sliding variable and the network basis function. The updating law is driven by the product term in the implementation, and the fixed diagonal learning rate is used as the adjustment coefficient, avoiding the common spectral norm calculation or high-order matrix inversion, which not only reduces the real-time computing burden of embedded implementation, but also guarantees the robustness of the updating direction. More importantly, the weight value updating law is executed synchronously with the control input, so that the network output can continuously adapt to the latest trend of the unknown nonlinear dynamic without introducing significant lag.

[0043] Under the action of the above weight value updating law, the time derivative of the Lyapunov function can be expressed as the difference between a set of complementary energy terms and a negative semi-definite dissipation term. Since the dissipation term is composed of the square sum of the composite sliding variable and the square of the weight value estimation error, and the complementary term is completely offset by the fixed diagonal learning rate, the overall time derivative remains non-positive at all times. This property means that the Lyapunov function is monotonically non-increasing over time, thus directly deducing that the energy of the composite sliding variable and the weight value estimation error energy are both decaying in a non-increasing manner. Further, combined with the rational exponential convergence property introduced in the second step, it can be proved that the above energy decreases to zero in finite time, thereby ensuring that the overall sliding surface reaches zero in finite time, while the network weight value converges to a set of bounded constants. Since the first step has mapped the actual state to the logarithmic potential energy interval between the upper and lower boundaries that vary with time, when the overall sliding surface tends to zero, the constraint potential energy error is also eliminated simultaneously, thus ensuring that the engineering controlled object strictly satisfies the double-sided time-varying state constraints at any time.

[0044] Further, step 1 specifically includes:

[0045] Step 1.1: At the beginning of each control period, the upper and lower boundaries corresponding to the current actual state are read in real time, both of which are updated synchronously with time to fully reflect the immediate position of the double-sided time-varying state constraints;

[0046] Step 1.2: The actual state is subtracted from the reference trajectory to obtain a first deviation; meanwhile, the upper boundary is subtracted from the reference trajectory and the reference trajectory is subtracted from the lower boundary to obtain a second deviation and a third deviation respectively;

[0047] Step 1.3: A natural logarithm operation is performed on the ratio of the second deviation and the first deviation to obtain an upper side potential energy term; a natural logarithm operation is performed on the ratio of the first deviation and the third deviation to obtain a lower side potential energy term;

[0048] Step 1.4: The upper side potential energy term and the lower side potential energy term are algebraically summed with opposite signs to obtain a constraint potential energy error; when the actual state approaches the upper boundary or the lower boundary infinitely, the amplitude of the constraint potential energy error correspondingly tends to infinity in a finite time.

[0049] During real-time reading of the upper and lower boundaries, the controller obtains a safe zone width that varies over time. If the distance difference is directly used to measure the state position, the error scale will drift as the boundary slowly shifts, forcing frequent adjustments to the sliding mode gain and leading to unpredictable convergence time. To eliminate this risk, the method uses the ratio of the "first deviation," "second deviation," and "third deviation," and then injects a natural logarithmic operation to transform the original linear metric into a logarithmic one. The physical meaning of the logarithmic operation is to re-encode distance information as potential energy information: the smaller the distance, the higher the potential energy, and as the distance approaches zero, the potential energy increases explosively within a finite time, with this explosive rate being exponentially related to the reciprocal of the distance. In this way, the "upper potential energy term" and the "lower potential energy term" possess the properties of barrier functions: they are continuously differentiable when the state enters the safe zone, rise rapidly when the state approaches the boundary, and theoretically form an insurmountable infinitely high barrier at the boundary. The "constrained potential energy error," obtained by summing with opposite signs, not only retains directional information but also incorporates the potential energy of both sides of the barrier. Its sign determines which side the actual state leans towards, its amplitude characterizes the intensity of the lean, and the extreme sensitivity of the amplitude to distance ensures that a significant sliding surface increase effect is triggered when the state approaches either boundary. Since the terminal sliding mode approach law uses the overall sliding surface amplitude as the driving force, the barrier characteristics of the constrained potential energy error are seamlessly transmitted to subsequent control inputs, achieving "soft collision" protection of the boundary in the continuous domain: the state experiences a strong pull-back force before reaching the boundary, theoretically compressing the possibility of actually reaching the boundary to zero. More importantly, this mapping process maintains a consistent timestamp, ensuring that the potential energy calculation is synchronized with the reference trajectory and boundary updates, avoiding false signals caused by historical data misalignment; at the same time, logarithmic mapping ensures that no uncontrollable numerical jumps occur when the boundary changes significantly or the trajectory abruptly changes, because the ratio always changes continuously before entering the logarithmic amplification stage. Ultimately, this potential energy mapping is connected to the online compensation of the single hidden layer radial basis network, so that the network weights are only responsible for the unknown dynamics of the controlled object in the project, without having to share the task of boundary preservation, which significantly reduces the burden of weight updates and learning oscillations.

[0050] For each actual state There are upper and lower time-varying constraints: Assume Construct a logarithmic potential mapping:

[0051] ;

[0052] in, For the controlled object of the project One actual state; To constrain potential energy error, ; For reference trajectory; It is continuously differentiable with the upper boundary as its boundary. It is continuously differentiable at the lower boundary; hour This naturally incorporates time-varying constraints into subsequent control laws, avoiding explicit projection.

[0053] Furthermore, the composite sliding mode variable in step 2 is the sum of a linearly normalized part, a rational exponent part, and a time integral part. The linearly normalized part is stored in the first buffer of the register stack as a reference value for subsequent superposition operations to ensure that the composite sliding mode variable has linear response characteristics when the error amplitude is small. The rational exponent part is used to enhance the convergence rate of the error interval and ensure that the controlled object of the project converges in a finite time. The time integral part is used to compensate in real time for the influence of slow boundary movement on the amplification of the error interval, ensuring the symmetrical convergence behavior of the overall sliding surface under dynamic constraints.

[0054] Define rational exponent ,in (With fixed constants, convergence in finite time is guaranteed). Construction :

[0055] ;

[0056] in, ; ; ; This is the terminal amplification factor, used to adjust the nonlinear amplification intensity of the error by the rational exponent. It is a dummy variable in the integration interval, representing a historical time point from 0 to the current time t. It is used to integrate historical data and has no independent physical quantity meaning.

[0057] When the constraint potential error is delivered from the previous stage, the engineering controlled object immediately multiplies the error with the normalization coefficient in the order of state index, writes the result into the first-level buffer, and generates the linear normalization part. Since this buffer position is located at the front end of the in-chip memory hierarchy, the read-write delay is the lowest, so it can ensure that the linear normalization part can be taken out to participate in the subsequent superposition operation in each control cycle; at the same time, it provides a constant proportional channel, so that the compound sliding mode variable has a strict first-order linear response when the error amplitude is small, which is crucial to avoid small signal chattering and amplify quantization error. Then, the hard real-time processing unit calls the rational exponential transformation logic solidified in the read-only memory to perform amplitude nonlinear amplification on the linear normalization part to form the rational exponential part. The reason why the rational exponent with a numerator and a denominator smaller than the linear growth order is chosen instead of the general integer power is that the rational exponential part can provide super-linear gain when the error enters the medium-high amplitude interval, but it remains continuous and derivable when the error approaches zero, so it can guarantee the finite-time convergence of the engineering controlled object, and it will not introduce unpredictable differential singularities.

[0058] Subsequently, the controller calculates the current boundary moving trend according to the net change of the upper and lower boundaries recorded in the previous cycle, and participates in the fixed-point accumulator operation with the constraint potential error and the normalization coefficient to obtain the time integral part. This part is weighted by the sign and speed of boundary drift, and injects an asymptotic hedging channel into the compound sliding mode variable, so that the overall sliding mode surface still maintains a symmetric convergence behavior in a dynamic constraint environment, and effectively suppresses the error interval amplification caused by slow boundary movement. Finally, the three parts complete algebraic summation according to the established sign rule to form the compound sliding mode variable, and write it back to the high bit of the register stack, waiting to be spliced into the overall sliding mode surface. Through this partition storage and pipeline calculation, the linear normalization part provides smooth and delicate low-amplitude error correction, the rational exponential part provides fast pressure reduction for medium-high amplitude error reduction, and the time integral part provides adaptive boundary tracking compensation. The three are independent and tightly coupled, ensuring that at least one effective energy path is available at any time to force dissipate the sliding mode surface. At the same time, the generation of the compound sliding mode variable is transparent to the upper logic, and the single hidden layer radial basis network only needs to read the overall sliding mode surface and the basis function product to complete the weight update, eliminating additional structural complexity. This design not only realizes the dynamic balance of the coupling of error scale, convergence rhythm and boundary drift in the functional dimension, but also takes into account the reentrancy of the cache and the pipeline parallelism in the hardware implementation dimension, providing stable, predictable and balanced computational load for the subsequent control input calculation. input signal, which fundamentally guarantees the finite-time convergence of the engineering controlled object and continuously meets the double-sided time-varying state constraints.

[0059] Further, in step 2, at the beginning of the current control period, a set of fixed and rational number relationship satisfying indexes is selected for all actual states; the selection of the index set follows the following principles: the numerator and denominator of the index are positive integers to ensure that the index value is positive and can accurately describe the finite time convergence characteristics; different actual states share the same index set to eliminate the imbalance of the relative convergence speed caused by the difference in the index; the index value remains the same as the last control period to avoid control discontinuity caused by frequent adjustment of the index. Through this uniform selection process, the subsequent composite sliding mode variables evolve in the same order homogeneous structure, providing consistent convergence power for the overall sliding mode surface.

[0060] To avoid the imbalance of convergence speed caused by the difference in the index selection of different actual states, while ensuring that the index value can theoretically give a strict finite time convergence upper bound, the scheme performs a uniform index selection process in the initialization phase of each control period. The controller first queries the pre-stored "index candidate table" in the engineering controlled object parameter area, which lists several rational numbers composed of a pair of mutually prime positive integer numerators and denominators, and each pair corresponds to a pre-calculated convergence time mark. Then, the controller performs consistency check according to the rational index set used in the last control period: if the index set in the last period is still valid in the candidate table, the set is directly used to ensure that the index value remains the same as the last control period; if the index set needs to be selected again due to external configuration changes or dynamic task switching, the controller selects the first set of index sets that meet the time length requirement and have positive integer numerators and denominators from the candidate table according to the constraint that "the convergence time must not exceed the remaining time length of the task", and writes it into the high-speed register area. After the selection is completed, all actual states share the same index set, ensuring that the composite sliding mode variables evolve in the same order homogeneous structure. This approach has two advantages: first, the numerators and denominators of the index are positive integers, which makes the index value always positive and can be directly mapped to the theoretical index order of the finite time convergence law, without introducing non-physical gain due to sign changes; second, different actual states use the same index set, eliminating the imbalance of convergence speed between the sliding mode surface components, which helps the overall sliding mode surface to simultaneously approach the zero point in a radially symmetric manner in the multi-dimensional space. Finally, since the index selection mechanism ensures consistency between control periods, the power amplification factor of the control law does not jump, and the sliding mode surface evolution also remains continuous, avoiding the problem of control discontinuity caused by index mutation. Through this uniform selection process, the nonlinear amplification part of the composite sliding mode variable forms a consistent convergence power among the states, providing a stable, predictable and synchronous finite time descent channel for the overall sliding mode surface, thereby further consolidating the comprehensive performance of the overall control method in meeting the cooperative constraints and fast tracking.

[0061] Further, in step 2, after confirming the index, the current constraint potential energy error of each actual state is calculated in real time, and the absolute amplitude of the error is mapped with the index to obtain a first sliding mode primitive; at the same time, the sign information of the constraint potential energy error is independently extracted to form a second sliding mode primitive. The two types of sliding mode primitives cooperate with each other: the first sliding mode primitive amplifies the error far from zero through power mapping, ensuring rapid attenuation; the second sliding mode primitive records the error direction, ensuring that the switching does not produce directional ambiguity when the sliding mode tends to zero; thus, each actual state corresponds to a pair of original sliding mode primitives; a continuous, monotonic and saturated buffer mapping interval is added to the amplitude part of the original sliding mode primitive; when the constraint potential energy error is in the preset range, the buffer mapping does not affect the output of the original sliding mode primitive; when the constraint potential energy error tends to infinity, the buffer mapping is gradually smoothed and truncated within a limited amplitude, realizing soft clipping of the control gain to limit the control amplitude and avoid high-frequency chattering near the overall sliding mode surface.

[0062] The engineering controlled object calls the index value stored in the register in the previous step, and performs a power mapping operation on the absolute amplitude of the constraint potential energy error, and the result is marked as the first sliding mode primitive. The core function of power mapping is to use the property that the index is less than one but greater than zero to amplify the error far from zero, forming a super-linear relationship between the error size and the convergence driving force, so that the energy can be quickly released when the sliding mode surface is still in the medium-high amplitude stage, ensuring that the overall sliding mode surface can be quickly attenuated in a limited time. At the same time, the controller independently extracts and packages the sign information of the same constraint potential energy error to form the second sliding mode primitive. The second sliding mode primitive only carries single-bit direction information, and it does not perform amplification, but provides indispensable direction criteria when the sliding mode tends to zero, ensuring that the composite sliding mode variable does not slip or high-frequency chattering due to directional ambiguity at the moment of sign switching. The two types of sliding mode primitives exist logically side by side and balance each other numerically: the first sliding mode primitive compresses the error amplitude, and the second sliding mode primitive locates the error direction, and the synchronization of the two enables each actual state to obtain a pair of original sliding mode primitives.

[0063] To balance the high-performance actuator and the limited resolution sensor, a continuous, monotonic and saturation characteristic buffer mapping interval is added to the amplitude part of the original sliding mode element. Specifically, in the normal working interval where the constraint potential energy error is in the preset range, the buffer mapping output is linearly consistent with the amplitude of the original sliding mode element, ensuring the control sensitivity of small error section. When the constraint potential energy error tends to be theoretically infinite due to the extreme deviation of the state, the slope of the buffer mapping gradually decreases and finally transitions to a flat area, so that the amplitude of the first sliding mode element is smoothly truncated within a finite upper limit. Through this soft clipping mechanism, the control gain is limited within the physical limit of the actuator, while the high-frequency chattering near the overall sliding surface due to excessive amplification is greatly suppressed. The buffer mapping is continuous and monotonic, and does not introduce discontinuous jumps, so it does not destroy the finite time convergence property of the terminal sliding mode, nor does it activate the discrete sampling engineering controlled object in the return working condition. With the introduction of the buffer mapping, the first sliding mode element and the second sliding mode element cooperatively output a pair of safety-modulated sliding mode signals, which are written into the high-order register stack and participate in the aggregation of the current composite sliding mode variable. In the whole process, any state component approaching the upper or lower boundary quickly can obtain a controlled size and clear direction sliding mode driving force under the protection of the buffer mapping, providing strong and stable convergence power for the subsequent control law, while ensuring that the overall sliding surface maintains a symmetric and continuous energy decay trajectory under dynamic constraint environment.

[0064] Further, in step 2, after obtaining the buffered two types of sliding mode elements, the two types of sliding mode elements are spliced in the form of weighted algebraic sum with fixed rational exponents as weights to form a composite sliding mode variable; when splicing, the odd-even cancellation design is adopted: the amplitude element and the sign element are added in the same direction in sign, but are staggered in exponential order, so as to balance the high-gain convergence and low-gain steady-state performance; in order to synchronously process the multi-dimensional actual state, all the composite sliding mode variables are arranged in the order of state to form an equal-dimensional vector; then, the overall sliding surface is defined in the form of equal-dimensional vector as column vector; the zero vector of the overall sliding surface represents that all the composite sliding mode variables are zero at the same time, indicating that all the constraint potential energy errors are completely offset at the same time; when the overall sliding surface is far from the zero vector, each component thereof describes the convergence distance of the corresponding state; when the overall sliding surface approaches the zero vector, it indicates that all the actual states have approached the reference trajectory and are kept within the safety interval of the double-sided time-varying constraint.

[0065] Under the premise that the two types of sliding mode primitives have been stabilized after buffering, the controller first calls the fixed rational exponent in the uniform exponential set for the current control period, and injects it as a weight coefficient into the sliding mode splicing logic. Specifically, the amplitude primitive undertakes the function of error amplification, and the sign primitive undertakes the function of direction criterion. Both of them need to complete a weighted algebraic sum in the numerical domain before being synthesized into a single channel of composite sliding mode variable. In order to make this weighting process meet the dual requirements of "strong response far from zero point and smooth response close to zero point", the splicing link adopts odd-even cancellation design: the amplitude primitive and the sign primitive are superimposed in the same direction to ensure that the control direction does not reverse phase; but deliberately mispositioned in the order of exponents, so that high-order exponents preferentially act on the amplitude primitive, and low-order exponents preferentially act on the sign primitive. In this way, once the constraint potential error is in the medium-high amplitude range, the amplitude primitive will first output significant gain due to the high-order weight, pulling the overall sliding mode surface to drop rapidly and reflecting high gain convergence; and when the error enters the small amplitude range, the sign primitive dominated by low-order weight gradually becomes the main contributor, making the composite sliding mode variable exhibit linear low gain steady-state performance to small errors, avoiding overshoot caused by over-amplification.

[0066] After splicing, the engineering controlled object processes multi-dimensional actual states synchronously, arranges all composite sliding mode variables in the order of state index to form an equal-dimension vector. The dimension of this vector is strictly consistent with the number of controlled states, and each element corresponds one by one, realizing the structural homogeneity between state components. Subsequently, the controller directly maps this equal-dimension vector into the overall sliding mode surface according to the column vector definition rule; the overall sliding mode surface occupies a continuous memory in the high-speed register area in terms of data structure, so that the subsequent control law generation can obtain all sliding mode information through a single vector read operation.

[0067] In the closed-loop running semantics, the zero vector of the overall sliding mode surface represents that all composite sliding mode variables are zero at the same time, at which point all constraint potential errors are completely canceled out, indicating that each actual state has accurately fallen on the reference trajectory and is located in the central safe interval of the bilateral time-varying constraints. If the overall sliding mode surface is far from the zero vector, the absolute value of each component directly depicts the convergence distance of the corresponding state: the larger the value, the farther the state is from the safe center, requiring higher control gain; the smaller the value, the more it indicates that it has entered the fine adjustment stage. With the driving of the control law, the overall sliding mode surface monotonically converges to the zero vector, and when its norm approaches zero, the engineering controlled object can determine that all actual states have approached the reference trajectory as a whole and do not violate any time-varying state constraints, thus theoretically completing the finite-time convergence and achieving high-precision and low-oscillation steady-state operation in engineering.

[0068] Further, in step 2, when the Euclidean norm of the overall sliding surface falls below the preset sliding threshold and remains stable for several consecutive sampling periods, it is determined that the overall sliding surface has effectively tended to the zero vector; at this time, the following process is triggered: freezing the index update logic of the composite sliding variable, keeping the current index set from changing, to avoid reactivating the high-gain region; switching the network learning rate from high-frequency online update to low-frequency gradual calibration, reducing the consumption of computing resources.

[0069] In the real-time execution process of the time-varying state constraint terminal sliding mode tracking control method based on adaptive network, the controller continuously monitors the Euclidean norm of the overall sliding surface, and when the norm first falls below the preset sliding threshold and remains stable for several consecutive sampling periods, the engineering controlled object is determined to have effectively tended to the zero vector. At this time, the internal logic enters the convergence judgment branch, and first the index update logic of the composite sliding variable is frozen: specifically, a "lock flag" is placed in the index management module to prohibit any index set replacement request from external scheduling or adaptive tuning, keeping the current index set completely unchanged. This freezing mechanism can prevent the reactivation of high-order weights in the index set in the steady state, thereby preventing the instantaneous amplification of the composite sliding variable in the small error range, and preventing the reactivation of the high-gain region and the resulting chattering risk. Next, the engineering controlled object performs cascading switching on the learning rate of the single-hidden-layer radial basis network, converting the original high-frequency online update mode to a low-frequency gradual calibration mode: the controller writes a new smaller step size in the learning rate register area, and adjusts the trigger period of the weight update interruption, so that the network weights are only adjusted slightly in a longer sampling window. This strategy not only significantly reduces the real-time computing load of the processor during steady-state operation, prolonging the service life of the embedded resources, but also makes the weight change process smoother, avoiding interference from high-frequency weight jitter in the converged control channel. Through the two coordinated actions of freezing the index set and reducing the network learning rate, the method not only maintains the convergence results of the terminal sliding mode, but also realizes a seamless transition from the fast response stage to the steady-state energy-saving stage, maintaining the strict satisfaction of the time-varying state constraints on both sides, and significantly optimizing the long-term running efficiency and hardware reliability of the engineering controlled object.

[0070] Further, in step 3, a set of single-hidden-layer radial basis network basis function centers that do not overlap each other are constructed for each actual state, and an initial width is uniformly set for all single-hidden-layer radial basis network basis functions; the network weights are initialized to zero vectors to avoid unknown bias in the network output affecting the control input in the initial stage; at the same time, to prevent outliers from affecting the convergence speed, all single-hidden-layer radial basis network basis function centers are clustered and selected according to the historical difference between the actual state and the reference trajectory within a limited window, thereby ensuring that the initial network structure covers the main dynamic mode.

[0071] In the online approximation of the actual dynamics of the engineering controlled object using a single-hidden-layer radial basis network, a set of mutually non-overlapping single-hidden-layer radial basis network basis function centers is first established for each actual state. The fundamental purpose of ensuring mutual non-overlapping is to prevent multiple basis functions from producing redundant coverage in the same local state domain, resulting in gradient competition and resource waste in the weight value update process; each basis function center occupies an independent state interval, and the network weight value can focus on capturing specific dynamic characteristics during training, thereby improving approximation efficiency and stability. In order to further simplify the workload of super parameter adjustment, the initial width of all basis functions is uniformly set, which avoids the computational burden of finding the best width for each basis function in high-dimensional state space, while ensuring that the network still has sufficient global perception ability in the early stage, and its local resolution can be implicitly corrected by gradient information in the subsequent weight value fine-tuning process. The network weight value is initialized to zero vector, which is to make the initial network output always zero, so that in the first sampling period of closed-loop operation, the size of the control input is completely determined by the known engineering controlled object part and the sliding mode reaching gain, avoiding the impact or distortion caused by the sudden superposition of unknown bias on the control signal. In order to make the initial network structure cover the main dynamic mode of the engineering controlled object, and avoid the neglect of some important state intervals due to center distribution bias, this scheme collects the difference data of actual state and reference trajectory in a limited length of history window, and sends these difference samples into the fast clustering algorithm based on density for automatic clustering; the cluster centers output by the algorithm are directly used as the basis function centers of the single-hidden-layer radial basis network, the number of which is the number of clusters, and the center position is naturally located in the area with the highest frequency of historical deviation, which ensures that the network covers the most representative dynamic changes when it is established. In this way, the network does not need to spend too much iteration in the subsequent online training stage to adjust the weight value to the effective interval, and the convergence speed is faster, which can complete the compensation of unknown dynamics and maintain the strict compliance with the bilateral time-varying state constraints in a very short transition period, in cooperation with the finite time convergence characteristics of the terminal sliding mode.

[0072] Further, in step 3, the component error signal of the overall sliding surface on the composite sliding variable is used as the driving force to calculate the product of the single-hidden-layer radial basis network basis function and the weight increment, and the weight increment is added to the original weight vector and limited in the pre-defined bounded region through the projection operator to ensure that the network weight value is always bounded.

[0073] In the online training session of the single-hidden-layer radial basis network, the controller first extracts the component error signal corresponding to the composite sliding mode variable from the overall sliding surface, which is regarded as the driving quantity for network weight adjustment; then, the driving quantity is multiplied element by element with the single-hidden-layer radial basis network basis function at the current sampling time, and a new weight increment is obtained by weighting the product result with a fixed network learning rate. The weight increment reflects the immediate compensation direction and amplitude that the network output should make to the unknown engineering controlled object dynamics under the current error condition. In order to prevent the weight from unbounded growth in the long-term iteration process, the method introduces a projection operator before adding the weight increment to the original weight vector: the operator first detects whether the candidate weight after superposition falls within the pre-defined bounded region, and if it falls within the region, it is directly accepted, and if it exceeds the boundary, it is projected back to the boundary surface according to the principle of minimum Euclidean distance, so as to ensure that the updated weight is always within the safe and controllable range. Through this "error-driven product-fixed step weighting-boundary projection" three-step closed loop, the network weight can quickly respond to new dynamic changes, and will not deviate from the reasonable numerical interval due to the cumulative effect of the learning rate, ultimately achieving robust compensation for unknown nonlinear dynamics, and together with the terminal sliding mode convergence mechanism, ensuring that the engineering controlled object consistently satisfies the bilateral time-varying state constraints throughout the entire operation period.

[0074] Define the dynamics of the engineering controlled object: where is known and reversible. The single-hidden-layer radial basis network is used to approximate the unknown nonlinearity:

[0075] ;

[0076] ;

[0077] where, is a basis function column vector composed of 25 Gaussian basis functions, with the centers uniformly covering , and the width being 0.2; is the first derivative of ; is the actual state vector of the engineering controlled object composed of elements, which is the controlled quantity of closed-loop control; is the known and reversible control influence matrix, also known as the input gain matrix; is the desired trajectory vector with the same dimension as ; is the boundary velocity compensation vector used to offset the additional deviation caused by the movement of the time-varying upper and lower boundaries; each component is equal to ; is a rational exponential constant (fixed at ); is the overall sliding mode surface vector composed of composite sliding mode variables; is the diagonal reaching gain matrix, which is self-adaptively adjusted to reaching speed; is the number of neural network basis functions, which is fixed as 25; is the dimensional online adjustable weight matrix; is the control command; is the total number of controlled states of the engineering controlled object.

[0078] Take Lyapunov function:

[0079] ;

[0080] wherein is the ideal weight. The weight update is given as:

[0081] ;

[0082] wherein is taken as time is the fixed diagonal adaptive rate; is the 25-order unit matrix; the simultaneous equations can be obtained as

[0083] ;

[0084] and because and , the finite time convergence theorem is satisfied; in combination with the potential energy mapping (1), it can be proved that for all , the

[0085] .

[0086] Figure 2 The dynamic change characteristics of the constraint potential energy error in the control process are described in detail. The abscissa is the time axis, and the ordinate represents the numerical value of the constraint potential energy error. The zero error line in it is indicated by a dotted line, which is the target benchmark for the convergence of the constraint potential energy error.

[0087] The constraint potential energy error curve is represented by a solid line, which shows the complete evolution process from the beginning of control to the complete elimination of the error. In the initial stage of control, the constraint potential energy error has a large positive value, reflecting the significant deviation of the actual state from the reference trajectory, as well as the error amplitude generated by the logarithmic potential energy mapping under the constraints of the double-sided time-varying constraints.

[0088] With the continuous action of the control algorithm, the constraint potential energy error presents a clear attenuation trend. The curve starts from a high amplitude area and experiences a rapid decline phase, embodying the enhancing effect of the rational exponential part of the composite sliding mode variable on the error convergence rate. In the process of approaching the zero error line, the error curve shows a smooth convergence characteristic, avoiding the chattering phenomenon that may occur in traditional sliding mode control.

[0089] The control starting point and error elimination point are specially marked, clearly indicating the time interval from the generation to the complete cancellation of the constraint potential energy error. The finite time convergence area is marked with a dashed line box, highlighting that the engineering controlled object can achieve complete convergence within a predetermined finite time when the constraint potential energy error enters this area. The entire change process verifies the superior performance of the proposed control method in handling time-varying constraints.

[0090] Figure 3 The composition and convergence characteristics of the composite sliding mode variable are fully displayed. The horizontal coordinate represents the time process, and the vertical coordinate represents the numerical value of the composite sliding mode variable and its components. The zero line is included as a reference benchmark for convergence, represented by a dotted line.

[0091] The linear normalization part is represented by a dashed line, which shows the basic component of the composite sliding mode variable that ensures linear response characteristics. This part has good linear characteristics when the error amplitude is small, providing a stable reference quantity for the entire composite sliding mode variable. Its convergence process is relatively smooth, embodying the gradual convergence characteristics of the linear control component.

[0092] The rational exponential part is represented by a thick solid line, highlighting its key role in rapid convergence. This part uses a fixed rational exponent for power mapping, effectively enhancing the convergence rate of the error interval and ensuring the finite time convergence characteristics of the engineering controlled object. Its curve shows a significant nonlinear convergence behavior, with strong convergence driving ability far from the zero point.

[0093] The composite sliding mode variable is represented by a solid line, embodying the comprehensive effect of the linear normalization part, the rational exponential part, and the time integral part. This curve combines the advantages of each component, maintaining linear response characteristics and having rapid convergence capability, while achieving real-time compensation for the slow movement of the boundary through the time integral part.

[0094] The vertical dashed line indicates the dividing point between the rapid convergence stage and the precise tracking stage, and a preset sliding mode threshold line is set. When the Euclidean norm of the composite sliding mode variable falls below this threshold, the engineering controlled object enters the precise tracking mode, embodying the phased optimization characteristics of the control strategy.

[0095] An example is given below of specific numerical examples, the implementation process of the adaptive network-based time-varying state constraint terminal sliding mode tracking control method in two discrete sampling periods (T1, T2) 、 ) is demonstrated.

[0096] Controlled nonlinear dynamics: .

[0097] Corresponding known reversible input gain .

[0098] Sampling period: .

[0099] Reference trajectory: .

[0100] Bilateral time-varying state constraint: .

[0101] Initial state: .

[0102] Mapping and sliding mode design constants:

[0103] ;

[0104] Radial basis network: ;

[0105] Constraint boundary velocity: ;

[0106] In this example, the time integral term is initially zero within the two periods .

[0107] Sampling time , constraint potential energy error: ;

[0108] Composite sliding mode variable: ;

[0109] Overall sliding surface: .

[0110] Reaching gain: .

[0111] Neural network compensation: .

[0112] Control law: .

[0113] Weight update: .

[0114] Discretization: .

[0115] State update (Euler approximation):

[0116] .

[0117] Sampling instant

[0118] Constraint potential error: .

[0119] Composite sliding variable: ;

[0120] Integral sliding surface: .

[0121] Control input:

[0122] .

[0123] Weight update: .

[0124] State update: .

[0125] After two sampling periods, the integral sliding surface norm decreases from to , and the monotonic decrease indicates that the finite-time convergence process has started; at the same time, the network weights gradually move away from the compensation direction of the unknown dynamics, and the update amount at each time is bounded by and the projection operator. After continuous iteration, when is lower than the threshold and stable for several periods, the exponential set is frozen, the learning rate is downshifted, and the engineering controlled object enters the steady state, and all times satisfy .

[0126] The control loop continues to execute, and the sliding surface norm monotonically decreases. When is first less than the threshold 0.05 and remains for 5 consecutive sampling periods, the exponential is locked and the learning rate is switched to a low-frequency mode, and the engineering controlled object enters the steady state.

[0127] The above shows and describes the basic principles, main features and advantages of the present application. Those skilled in the art should understand that the present application is not limited by the above examples, and the above examples and descriptions in the specification are only the principles of the present application. Without departing from the spirit and scope of the present application, various changes and improvements can be made to the present application, and these changes and improvements all fall within the scope of the claimed present application. The scope of protection claimed by the present application is defined by the appended claims and their equivalents.

Claims

1. A time-varying state constraint terminal sliding mode tracking control method based on adaptive network, the method comprising: Step 1: constructing a constraint potential error for each actual state of an engineering controlled object with bilateral time-varying state constraints, and mapping the deviation of the actual state from a reference trajectory to a logarithmic potential interval between its upper bound and lower bound that varies with time; Step 2: constructing a composite sliding variable with fixed rational exponent; the sliding variables of all actual states of the engineering controlled object form an overall sliding surface in the vector plane, and when the overall sliding surface tends to zero, all constraint potential errors are simultaneously eliminated, thereby ensuring that the actual states of the engineering controlled object track the reference trajectory in a finite time and do not exceed any upper bound or lower bound that varies with time; Step 3: using a single-hidden-layer radial basis network to online approximate the actual dynamics of the engineering controlled object, and introducing the network output as a dynamic compensation term into the control input, thereby ensuring that the overall sliding surface is driven to zero in a finite time and real-time compensation of the actual states of the engineering controlled object is simultaneously completed.

2. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 1, wherein, The method further comprises: Step 4: by selecting a Lyapunov function containing a quadratic term of the composite sliding variable and a quadratic term of the weight estimation error of the single-hidden-layer radial basis network, a gradient-type weight update law based on the product of the composite sliding variable and the basis function of the single-hidden-layer radial basis network is given; under the action of a fixed diagonal learning rate, the time derivative of the Lyapunov function is always non-positive and simultaneously decays to zero with the composite sliding variable and the weight estimation error, thereby ensuring that the engineering controlled object realizes sliding mode reaching in a finite time, the weight of the single-hidden-layer radial basis network converges to a bound, and all time constraints are satisfied.

3. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 2, wherein, Step 1 specifically comprises: Step 1.1: at the beginning of each control cycle, the upper bound and the lower bound corresponding to the current actual state are read in real time, both of which are updated synchronously with time to fully reflect the instantaneous position of the bilateral time-varying state constraints; Step 1.2: the actual state is subtracted from the reference trajectory to obtain a first deviation; the upper bound is subtracted from the reference trajectory, and the reference trajectory is subtracted from the lower bound to obtain a second deviation and a third deviation, respectively; Step 1.3: performing a natural logarithm operation on the ratio of the second deviation to the first deviation to obtain an upper potential item; performing a natural logarithm operation on the ratio of the first deviation to the third deviation to obtain a lower potential item; Step 1.4: algebraically summing the upper potential item and the lower potential item with opposite signs to obtain a constraint potential error; when the actual state approaches the upper bound or the lower bound infinitely, the amplitude of the constraint potential error correspondingly tends to infinity in a finite time.

4. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 3, wherein, The composite sliding mode variable in step 2 is the sum of a linear normalized part, a rational exponential part and a time integral part; the linear normalized part is stored in the first buffer area of the register stack as a reference quantity for subsequent superposition operations, to ensure that the composite sliding mode variable has linear response characteristics when the error amplitude is small; the rational exponential part is used to enhance the convergence rate of the error interval and ensure the finite time convergence of the engineering controlled object; and the time integral part is used to compensate in real time for the influence of the slow movement of the boundary on the expansion of the error interval, to ensure the symmetrical convergence behavior of the overall sliding mode surface in a dynamic constraint environment.

5. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 4, wherein, In step 2, a set of fixed and rational exponentials is uniformly selected for all actual states at the beginning of the current control period; the selection of the exponential set follows the following principles: the numerator and denominator of the exponent are positive integers, to ensure that the exponential value is positive and can accurately describe the finite time convergence characteristics; The same set of exponents is shared by different actual states to eliminate the imbalance in the relative convergence speed caused by the difference in exponents; the exponent value remains the same as that in the previous control period to avoid control discontinuity caused by frequent adjustment of the exponent; through this uniform selection process, the subsequent composite sliding mode variables are ensured to evolve cooperatively under the same order homogeneous structure, providing consistent convergence power for the overall sliding mode surface.

6. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 5, wherein, In step 2, after confirming the exponents, the current constraint potential error is calculated in real time for each actual state, and the absolute amplitude of the error is mapped with the exponent to obtain a first sliding mode primitive; at the same time, the sign information of the constraint potential error is extracted independently to form a second sliding mode primitive; the two types of sliding mode primitives cooperate with each other: the first sliding mode primitive amplifies the error far from zero through power to ensure rapid decay; the second sliding mode primitive records the error direction to ensure that the switching direction is not ambiguous when the sliding mode tends to zero; thus, a pair of original sliding mode primitives is obtained for each actual state; a continuous, monotonic and saturated buffer mapping interval is added to the amplitude part of the original sliding mode primitive; when the constraint potential error is within the preset range, the buffer mapping does not affect the output of the original sliding mode primitive; when the constraint potential error tends to infinity, the buffer mapping is gradually smoothed and truncated within a limited amplitude, to realize soft clipping of the control gain, limit the control amplitude, and also avoid high-frequency chattering near the overall sliding mode surface.

7. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 6, wherein, In step 2, after obtaining the two types of sliding mode elements after buffering, the two types of sliding mode elements are spliced in the form of weighted algebraic sum with fixed rational exponents as weights to form composite sliding mode variables; when splicing, the odd-even cancellation design is adopted: the amplitude element and the sign element are superimposed in the same direction in sign, but are staggered in the order of exponents, so as to balance the high-gain convergence and low-gain steady-state performance; in order to synchronously process multi-dimensional actual states, all composite sliding mode variables are arranged in order of states to form an equal-dimension vector; then, the overall sliding surface is defined in the form of column vector with the equal-dimension vector; the zero vector of the overall sliding surface represents that all composite sliding mode variables are zero at the same time, indicating that all constraint potential energy errors are completely offset at the same time; when the overall sliding surface is far from the zero vector, its components depict the convergence distance of the corresponding state; when the overall sliding surface is close to the zero vector, it indicates that all actual states have approached the reference trajectory and are kept within the safety interval of the double-sided time-varying constraint.

8. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 7, wherein, In step 2, when the Euclidean norm of the overall sliding surface falls below the preset sliding mode threshold and remains stable for several consecutive sampling periods, it is determined that the overall sliding surface has effectively tended to the zero vector; at this time, the following processes are triggered: freezing the exponent update logic of the composite sliding mode variable, keeping the current exponent set from changing, to avoid reactivating the high-gain area; switching the network learning rate from high-frequency online update to low-frequency gradual calibration, reducing the consumption of computing resources.

9. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 8, wherein, In step 3, a group of single-hidden-layer radial basis network basis function centers that do not overlap each other are constructed for each actual state, and an initial width is uniformly set for all single-hidden-layer radial basis network basis functions; the network weights are initialized to zero vector to avoid unknown bias of network output on control input in the initial stage; at the same time, in order to prevent outliers from affecting the convergence speed, all single-hidden-layer radial basis network basis function centers are selected by clustering according to the historical difference between the actual state and the reference trajectory within a limited window, so as to ensure that the initial network structure covers the main dynamic mode.

10. The adaptive network-based time-varying state-constrained terminal sliding mode tracking control method of claim 9, wherein, In step 3, the component error signal of the overall sliding surface on the composite sliding mode variable is used as the driving force to calculate the product of the single-hidden-layer radial basis network basis function and the product weighted with a fixed network learning rate to generate weight increments; the weight increments are added to the original weight vector, and the projection operator is used to limit them within the predefined bounded region, ensuring that the network weights are always bounded.

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