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

Abstract

The invention 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, and the method comprises the steps: 1, constructing a constraint potential energy error for each actual state of an engineering controlled object; 2, constructing a composite sliding mode variable with a fixed rational index; all sliding mode variables of all actual states of the project controlled object form an integral sliding mode surface on a vector level; and 3, carrying out online approximation on the actual dynamic state of the engineering controlled object by utilizing the single-hidden-layer radial basis network, and introducing network output as a dynamic compensation item into control input, thereby ensuring that the whole sliding mode surface is driven to zero within finite time and synchronously completing real-time compensation on the actual state of the engineering controlled object. According to the invention, the finite time high-precision tracking control of the controlled object of the controlled project with unknown nonlinearity and double-side time-varying boundaries is realized.

Description

Technical Field

[0001] The present invention relates to the field of adaptive control technology, and in particular to a time-varying state constrained terminal sliding mode tracking control method based on an adaptive network. Background Art

[0002] In modern control engineering, stable control and high-precision tracking of complex nonlinear engineering objects remain hot research and application issues. This is particularly true in fields such as aerospace, high-speed rail transit, medical robotics, and high-precision manufacturing, where controlled objects often exhibit significant uncertainty, external disturbances, and strong coupling. Furthermore, engineering objects are often subject to multiple safety constraints during operation, such as physical boundaries of temperature, voltage, torque, or displacement, which in some applications can dynamically change over time. Therefore, designing a control method that simultaneously achieves high-precision tracking, strong robust control, finite-time convergence, and strict compliance with dynamic state constraints has become a crucial topic in the field of intelligent control.

[0003] Traditional control methods, such as PID control, while simple in structure and easy to implement, suffer from significant deficiencies in handling nonlinear dynamics, uncertainty modeling, and time-varying state constraints. For engineering plants with unknown nonlinear structures, PID control fails to provide a compensation mechanism for these properties, making it difficult to ensure stable operation despite disturbances or modeling errors. Furthermore, due to the fixed control gains of PID, the plant's response struggles to adapt to multiscale dynamics, making it difficult to balance convergence speed and steady-state error. For multivariable coupled plants, performance can be severely degraded due to inter-channel interference.

[0004] To address the limitations of classical control, sliding mode control has become a research hotspot in recent years. Sliding mode control utilizes the sliding motion of the controlled plant on the sliding surface by introducing sliding surface design and switching logic to achieve robust control. This approach offers excellent fault tolerance for uncertainties in the controlled plant model. However, the equal-order sliding surface employed in traditional sliding mode control often results in the controlled plant approaching the sliding surface too slowly. In particular, the approach speed decreases significantly in the small error phase, impacting the control plant's responsiveness. To improve convergence speed, terminal sliding mode control has been proposed. By incorporating a nonlinear sliding surface design based on rational powers, terminal sliding mode control enables the controlled plant's state to converge to a reference trajectory within a finite time. However, such methods can exhibit control discontinuities near zero, and their reliance on saturation functions or forced truncation for handling state constraints can easily lead to chattering. Summary of the Invention

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

[0006] In order to achieve the above objects, the technical solution adopted by the present invention is:

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

[0008] Step 1: Construct a constrained 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 vary with time;

[0009] Step 2: Construct a composite sliding mode variable with fixed rational exponents. The sliding mode variables of all actual states of the controlled plant form an overall sliding surface at the vector level. When the overall sliding surface approaches zero, all constraint potential energy errors are eliminated simultaneously, ensuring that the actual state of the controlled plant tracks the reference trajectory within a finite time and does not exceed any upper or lower bounds that vary with time.

[0010] Step 3: Use a single-hidden-layer radial basis function network to approximate the actual dynamics of the controlled object 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 finite 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 containing a quadratic term of a composite sliding mode variable and a quadratic term of a single hidden layer radial basis network weight estimation error, a gradient weight update law based on the product of the composite sliding mode variable and the single hidden layer radial basis network basis function 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 engineering controlled object achieves sliding mode arrival within a finite time, bounded convergence of the single hidden layer radial basis network weights, and maintains satisfaction of bilateral time-varying state constraints at all times.

[0012] Furthermore, step 1 specifically includes:

[0013] Step 1.1: At the beginning of each control cycle, read the upper and lower bounds corresponding to the current actual state in real time. Both are updated synchronously with time to fully reflect the instantaneous position of the bilateral 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 boundary from the reference trajectory and the reference trajectory from the lower boundary 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 an upper potential energy term; perform a natural logarithm operation on the ratio of the first deviation to the third deviation to obtain a lower potential energy term;

[0016] Step 1.4: Perform algebraic summation of the upper potential energy term and the lower potential energy term with opposite signs to obtain a constrained potential energy error; when the actual state approaches the upper boundary or the lower boundary infinitely, the amplitude of the constrained potential energy error accordingly tends to infinity within a finite time.

[0017] Furthermore, the composite sliding mode variable in step 2 is the sum of a linear normalization part, a rational exponential part, and a time integral part; the linear normalization part is stored in the first-level buffer of the register stack as a benchmark reference 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 exponential part is used to enhance the convergence rate of the error interval and ensure finite-time convergence of the engineering controlled object; the time integral part is used to compensate in real time for the effect of slow boundary movement on the amplification of the error interval and ensure symmetrical convergence behavior of the overall sliding mode surface under a dynamic constraint environment.

[0018] Furthermore, in step 2, at the beginning of the current control cycle, a fixed set of exponents that satisfy a rational number relationship is uniformly selected for all actual states. The selection of this exponent set follows the following principles: both the numerator and denominator of the exponent are positive integers to ensure that the exponent value is positive and accurately describes the finite-time convergence characteristics; different actual states share the same exponent set to eliminate the imbalance in relative convergence speed caused by exponent differences; and the exponent value remains consistent with the previous control cycle to avoid control discontinuities caused by frequent exponent adjustments. This unified selection process ensures that the subsequent composite sliding mode variables coevolve under the same order homogeneous structure, providing consistent convergence dynamics for the overall sliding mode surface.

[0019] Furthermore, in step 2, after confirming the exponent, the current constrained potential energy error is calculated in real time for each actual state. The absolute magnitude of the error is then exponentially mapped to the exponent to obtain a first sliding mode primitive. Simultaneously, the sign of the constrained potential energy error is independently extracted to form a second sliding mode primitive. The two types of sliding mode primitives work together: the first sliding mode primitive exponentially amplifies errors far from zero to ensure rapid decay; the second sliding mode primitive records the direction of the error to ensure directional ambiguity when the sliding mode approaches 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 portion of the original sliding mode primitive. When the constrained potential energy error is within a preset range, this buffer mapping does not affect the output of the original sliding mode primitive. When the constrained potential energy error approaches infinity, the buffer mapping is gradually and smoothly truncated within a finite amplitude, achieving soft clipping of the control gain to limit the control amplitude and avoid high-frequency chattering near the overall sliding surface.

[0020] Furthermore, in step 2, after obtaining the two types of sliding mode primitives after buffering, the two types of sliding mode primitives are spliced ​​in a weighted algebraic sum manner with fixed rational exponents as weights to form composite sliding mode variables; an odd-even cancellation design is adopted during splicing: the amplitude primitives and the sign primitives maintain the same direction of superposition in terms of signs, but are staggered in the order of exponents, thereby taking into account both high-gain convergence and low-gain steady-state performance; to synchronously process multi-dimensional actual states, all composite sliding mode variables are arranged in state order to form equal-dimensional vectors; then, the overall sliding surface is defined in the form of column vectors with equal-dimensional vectors; the zero vector of the overall sliding surface means that all composite sliding mode variables are simultaneously zero, indicating that all constrained potential energy errors are completely canceled out at the same time; when the overall sliding surface is far away from the zero vector, its components characterize the convergence distance of the corresponding state; when the overall sliding surface approaches the zero vector, it indicates that all actual states have approached the reference trajectory and remain within the safe interval of the bilateral time-varying constraints.

[0021] Furthermore, in step 2, when the Euclidean norm of the overall sliding mode surface drops below the preset sliding mode threshold and remains stable for several 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: the exponential update logic of the composite sliding mode variable is frozen, and the current exponential set is kept unchanged to avoid reactivating the high-gain area; the network learning rate is switched from high-frequency online update to low-frequency progressive calibration to reduce computing resource consumption.

[0022] Furthermore, in step 3, a set of non-overlapping single-hidden layer radial basis network basis function centers are constructed for each actual state, and the initial width is uniformly set for all single-hidden layer radial basis network basis functions; the network weights are initialized with a zero vector to avoid unknown bias of the network output on 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 within a limited window based on the historical difference between the actual state and the reference trajectory, thereby ensuring that the initial network structure covers the main dynamic modes.

[0023] Furthermore, in step 3, the component error signal of the overall sliding surface on the composite sliding mode variable is used as the driving force, and its product with the basis function of the single hidden layer radial basis network is calculated and weighted to fix the network learning rate to generate a weight increment; the weight increment is superimposed on the original weight vector, and is restricted to a predefined bounded area through a projection operator to ensure that the network weights always remain bounded.

[0024] Compared with existing technologies, the present invention offers the following advantages: By constructing a constrained potential error with a logarithmic barrier characteristic, this method achieves natural embedded control of bilateral time-varying state constraints, avoiding the discontinuous control issues associated with traditional projection methods while improving response sensitivity as the state approaches its boundaries. Secondly, by constructing a composite sliding mode variable consisting of a linear normalization term, a rational exponential term, and a time integral term, this method balances steady-state accuracy in small error regions with convergence speed in large error regions, achieving the control objective of approximating the reference trajectory of the engineering plant within a finite time. Thirdly, the present invention introduces a single-hidden-layer radial basis function network as an online modeler to approximate the unknown nonlinear dynamics of the engineering plant. The sliding mode error signal drives the network weight update, enabling the controller to possess dynamic compensation capabilities and self-learning characteristics during operation. Furthermore, after the overall sliding mode surface converges, the controller automatically switches the network learning rate and freezes the exponential parameter, effectively reducing computational resource consumption and the risk of control chattering, accommodating the natural transition of the engineering plant from the dynamic phase to the steady-state phase. At the same time, network weight updates employ projection operator constraints to ensure that network parameters remain bounded, further enhancing the closed-loop stability of the engineering controlled object. Overall, this invention provides a solution for engineering controlled objects in terms of handling time-varying state constraints, improving the response speed and anti-interference capability of controlled engineering objects, and optimizing network training efficiency. It is suitable for a wide range of complex, nonlinear engineering controlled object control scenarios requiring high requirements, high precision, and strong constraints. BRIEF DESCRIPTION OF THE DRAWINGS

[0025] Figure 1 This is a flow chart of the method of the time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network proposed by the present invention;

[0026] Figure 2Schematic diagram of the change process of the constraint potential energy error in an embodiment of the present invention;

[0027] Figure 3 Schematic diagram of the convergence characteristics of the composite sliding mode variable in an embodiment of the present invention. DETAILED DESCRIPTION

[0028] The following description is intended to disclose the present invention so that those skilled in the art can implement the present invention. The preferred embodiments described below are merely examples, and those skilled in the art may conceive of other obvious variations.

[0029] Reference Figure 1 As shown, a time-varying state constrained terminal sliding mode tracking control method based on an adaptive network includes:

[0030] Step 1: Construct a constrained 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 vary with time;

[0031] Step 2: Construct a composite sliding mode variable with fixed rational exponents. The sliding mode variables of all actual states of the controlled plant form an overall sliding surface at the vector level. When the overall sliding surface approaches zero, all constraint potential energy errors are eliminated simultaneously, ensuring that the actual state of the controlled plant tracks the reference trajectory within a finite time and does not exceed any upper or lower bounds that vary with time.

[0032] Step 3: Use a single-hidden-layer radial basis function network to approximate the actual dynamics of the controlled object 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 finite time and the real-time compensation of the actual state of the controlled object is completed synchronously.

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

[0034] For motor servo control systems (PMSM / BLDC / DC servo), the states can be angle / speed (outer loop) and d–q current (inner loop). Bilateral time-varying constraints include angular travel and mechanical limits, upper / lower speed limits, current and voltage amplitude limits, and temperature-related derating. Constraints can be changed online based on bus voltage, load disturbances, temperature, and efficiency optimization strategies. Matching disturbances are primarily load torque, while non-matching disturbances include cogging torque, friction, and modeling residuals. The described method introduces constrained potential energy errors and composite sliding mode variables in the outer loop or velocity loop, and uses an RBF network to approximate the total uncertainty online. The output is applied to the motor via the servo drive / inverter, achieving finite-time tracking and avoiding over-bounds. A 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 bilateral constraints on angle, velocity, acceleration, and joint torque, and is also affected by coupled rigid-flexible effects, gravity, friction, and end-load variations. The task layer (trajectory, collision avoidance) often provides time-varying safety corridors and velocity curves. The proposed method constructs componentized composite sliding mode variables in the joint space and forms an overall sliding mode surface at the vector level. A unified set of rational exponents prevents convergence imbalances between joints. An RBF network online compensates for coupling and uncertain dynamics, achieving coordinated convergence of multiple joints within a finite timeframe while maintaining strict adherence to the bilateral time-varying bounds specified by the plan.

[0036] For a quadrotor UAV's attitude / position channels (cascaded control architecture), attitude channel states (roll, pitch, yaw, and angular velocity) are subject to small angles / angular rates and rotor saturation boundaries. The position channel is bounded by altitude, no-fly zones, and safety corridors, and changes online with terrain, geofences, and wind conditions. The proposed method can be applied to either the inner attitude loop or the outer position loop: By constraining the potential energy error to meet safety corridors (e.g., altitude windows and XY channel corridors), a RBF network is used to compensate for aerodynamic uncertainties under gusts and parameter drift. A composite sliding mode ensures finite-time error convergence and avoids overshoot. Buffer mapping reduces buffeting caused by thrust saturation and distribution.

[0037] For a vehicle's drive-by-wire chassis (steering / drive / brake-by-wire), lateral deviation, heading error, and longitudinal velocity / acceleration are subject to bilateral time-varying constraints, typically defined by lane boundaries, road curvature, and stability bounds (e.g., yaw rate / slip angle windows) imposed by tire-road adhesion conditions. These bounds vary dynamically with speed, road adhesion coefficient μ, wind, and traffic regulations. The proposed method maps the "lane safety corridor / stability window" to time-varying boundaries and utilizes a composite sliding model and RBF network to perform online compensation for 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 maintaining the state within the bilateral bounds throughout the entire process.

[0038] For continuous-time nonlinear engineering controlled objects with unknown disturbances and structural uncertainties, it is difficult to simultaneously meet the three conflicting goals of "fast tracking", "state not out of bounds" and "anti-model mismatch" by relying solely on traditional proportional-integral-differential strategies or conventional sliding mode methods. To resolve this comprehensive contradiction, the time-varying state constraint terminal sliding mode tracking control method based on adaptive network proposed in this paper follows the principle of "embedding constraints, compensating unknowns and shortening time convergence" in terms of overall thinking: First, in the process of constructing the constraint potential energy error for each actual state of the engineering controlled object, a set of upper and lower boundaries that are updated synchronously with time are introduced to compare the actual state with respect to the actual state. Deviations from the reference trajectory are simultaneously mapped to the logarithmic potential energy interval between its time-varying upper and lower boundaries. The key to this mapping is to logarithmically map the originally limited deviation distance into a rapidly increasing potential energy level near the boundary. This allows the control law to numerically "sense" the degree of danger of the state approaching the boundary, thereby increasing the corrective force in advance while still within the upper and lower boundary safety zones and generating a steep braking effect when approaching the extreme position. This ensures that boundary crossings do not occur under conditions of limited computational accuracy, while eliminating the need for repeated judgments and rigid truncation of the control variable as in traditional projection methods, avoiding the loss of robustness and frequent switching chatter caused by projection.

[0039] Next, in constructing composite sliding mode variables with fixed rational exponents, the method uses the normalized constraint potential energy error as the foundation layer. A rational exponential term with nonlinear amplification characteristics and a time integral term specifically designed to track boundary movement trends are further superimposed. This allows the sliding mode variables of all actual states of the controlled plant to form an overall sliding surface at the vector level. This overall sliding surface simultaneously possesses fine adjustment capabilities in linear segments, accelerated convergence capabilities in nonlinear segments, and real-time tracking capabilities when boundaries drift slowly. When the overall sliding surface approaches zero, all constraint potential energy errors are simultaneously eliminated. This result not only indicates that the static deviation between the actual state and the reference trajectory is compressed to zero, but also implies that even if the upper and lower boundaries are changing in a slow or fluctuating manner, the actual state is always precisely locked in the bilateral safety zone. The actual state of the controlled plant tracks the reference trajectory within a finite time and does not exceed any of the time-varying upper and lower boundaries. Because the convergence order of the rational exponent is less than one, the convergence time can be directly calculated by designing the numerator and denominator of the exponent, eliminating the need for empirical tuning of the open-loop gain.

[0040] Finally, in the online approximation of the actual dynamics of the controlled engineering object using a single-hidden-layer radial basis network, the method selects a radial basis whose center covers the entire feasible state space to form a basis function cluster. When initializing the weights, a rough fit to the unknown dynamics is obtained through least-squares projection. Then, continuous fine-tuning is performed in the gradient direction during closed-loop operation. The network output is introduced into the control input as a dynamic compensation term. Combined with the boundary velocity compensation signal and the terminal sliding mode arrival gain, on the one hand, the net error caused by model mismatch and external disturbances is continuously weakened, and on the other hand, the residual energy in the overall sliding mode surface is rapidly decayed 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 scenarios with saturation of the actuator, limited sensor bandwidth, or occasional packet loss in the communication link, the method can still maintain finite-time convergence, and after convergence, the error maintains bounded oscillation rather than drift. Overall, the present invention organically integrates three interlocking links: constructing a constrained potential energy error for each actual state of the engineering controlled object, constructing a composite sliding mode variable with a fixed rational exponent, and using a single hidden layer radial basis network to perform online approximation of the actual dynamics of the engineering controlled object. The controller is equipped with an adaptive defense mechanism of "the closer the boundary, the stronger the suppression", a terminal time-reduction mechanism of "the larger 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 repeated trial and error of the empirical adjustment coefficient, stable, high-speed and high-precision tracking control of nonlinear engineering controlled objects constrained by bilateral time-varying states is achieved. The control is particularly suitable for application scenarios with strict requirements on safety margins and rapid responses, such as aircraft re-entry attitude maintenance, robot path following, flexible arm end positioning, and extreme operation of chemical processes.

[0041] Furthermore, the method also includes: step 4: by selecting a Lyapunov function containing a quadratic term of a composite sliding mode variable and a quadratic term of a single hidden layer radial basis network weight estimation error, a gradient weight update law based on the product of the composite sliding mode variable and the single hidden layer radial basis network basis function 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 engineering controlled object achieves sliding mode arrival within a finite time, bounded convergence of the single hidden layer radial basis network weights, and maintains satisfaction of bilateral time-varying state constraints at all times.

[0042] This step first introduces a candidate Lyapunov function based on an analysis of the overall sliding surface dynamics. This Lyapunov function consists of two components: a quadratic term of the composite sliding mode variable, which measures the deviation energy of the controlled plant's trajectory from the overall sliding mode surface; and a quadratic term of the weight estimation error of a single-hidden-layer radial basis network, which measures the difference energy between the real-time network weights and the ideal weights. Because both energies appear in squared form, they can geometrically unify the uncertainty of the controlled plant's state and parameters. Matrix positive definiteness directly guarantees that the function reaches a unique minimum at the origin. Subsequently, while maintaining the basis function cluster unchanged, a gradient-based weight update law based entirely on the current observation is constructed by performing a gradient descent operation on the product of the composite sliding mode variable and the network basis function. This update law uses the product term as the driving signal and a fixed diagonal learning rate as the adjustment coefficient, avoiding the common spectral norm calculation or high-order matrix inversion. This reduces the real-time computational burden of embedded implementations while ensuring robustness of the update direction. More importantly, the weight update law is executed synchronously with the control input, so that the network output can continuously adapt to the latest changing trends of unknown nonlinear dynamics with almost no lag effect.

[0043] Under the aforementioned weight update law, the time derivative of the Lyapunov function can be expressed as the difference between a set of complementary energy terms and a negative semidefinite dissipative term. Since the dissipative term is composed of both the square of the composite sliding mode variable and the square of the weight estimation error, and the complementary term is completely offset by a fixed diagonal learning rate, the overall time derivative remains non-positive at all times. This property implies that the Lyapunov function is monotonically non-increasing over time, directly implying that the energy of the composite sliding mode variable and the energy of the weight estimation error decay in a non-increasing manner. Furthermore, combined with the rational exponential convergence property introduced in the second step, it can be shown that the above energy decreases to zero within a finite time, thereby ensuring that the overall sliding surface reaches zero within a finite time, while the network weights simultaneously converge to a set of bounded constants. Since the first step maps the actual state to a logarithmic potential energy interval between the time-varying upper and lower bounds, the constraint potential energy errors are simultaneously eliminated when the overall sliding surface approaches zero, thus ensuring that the controlled system strictly satisfies the bilateral time-varying state constraints at all times.

[0044] Furthermore, step 1 specifically includes:

[0045] Step 1.1: At the beginning of each control cycle, read the upper and lower bounds corresponding to the current actual state in real time. Both are updated synchronously with time to fully reflect the instantaneous position of the bilateral time-varying state constraint.

[0046] Step 1.2: Subtract the actual state from the reference trajectory to obtain the first deviation. At the same time, subtract the upper boundary from the reference trajectory and the reference trajectory from the lower boundary to obtain the second deviation and the third deviation respectively.

[0047] Step 1.3: Perform a natural logarithm operation on the ratio of the second deviation to the first deviation to obtain an upper potential energy term; perform a natural logarithm operation on the ratio of the first deviation to the third deviation to obtain a lower potential energy term;

[0048] Step 1.4: Perform algebraic summation of the upper potential energy term and the lower potential energy term with opposite signs to obtain a constrained potential energy error; when the actual state approaches the upper boundary or the lower boundary infinitely, the amplitude of the constrained potential energy error accordingly tends to infinity within a finite time.

[0049] By reading the upper and lower boundaries in real time, the controller obtains a time-varying safe zone width. If the distance difference is used directly to measure the state position, the error scale will drift as the boundary slowly drifts, forcing frequent adjustments to the sliding mode gain and leading to unpredictable convergence time. To mitigate this risk, the method uses the ratio of the "first deviation," "second deviation," and "third deviation" and then injects a natural logarithm operation to convert the original linear metric into a logarithmic one. The physical meaning of the logarithm operation is to re-encode distance information into 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 inverse of the distance. This way, the "upper potential energy term" and the "lower potential energy term" have the properties of barrier functions: they are continuously differentiable when the state enters the safe zone, rise rapidly as the state approaches the boundary, and theoretically form an insurmountable infinite barrier at the boundary. The "constrained potential energy error" then obtained by summing with opposite signs not only retains directional information but also superimposes the potential energy of the bilateral barriers. Its sign determines the direction to which the actual state deviates, while its amplitude characterizes the strength of the deflection. The amplitude's extreme sensitivity to distance ensures that a significant sliding surface amplification effect is triggered when the state approaches either boundary. Because the terminal sliding mode convergence law is driven by the overall sliding surface amplitude, the barrier characteristics of the constrained potential energy error are seamlessly transferred to subsequent control inputs, achieving "soft collision" protection of the boundary within the continuous domain: the state experiences a strong pullback force before reaching the boundary, theoretically reducing the probability of actually reaching the boundary to zero. Importantly, this mapping process maintains consistent timestamps, ensuring that the potential energy calculation is synchronized with the reference trajectory and boundary updates, avoiding false signals caused by historical data misalignment. Furthermore, the logarithmic mapping prevents uncontrollable numerical jumps during large boundary changes or sudden trajectory changes, as the ratio always changes continuously before entering the logarithmic amplification phase. Ultimately, this potential energy mapping is connected with the online compensation of a single-hidden-layer radial basis function network, so that the network weights are only responsible for the unknown dynamics of the engineering controlled object without having to share the boundary maintenance task, significantly reducing the burden of weight updates and learning oscillations.

[0050] For each actual state There are dual time-varying constraints: . Construct the logarithmic potential energy map:

[0051] ;

[0052] in, For the project controlled object an actual state; To constrain the potential energy error, ; is the reference trajectory; is the upper boundary, continuous and differentiable; is the lower boundary, which is continuously differentiable; hour , naturally "packages" the time-varying constraints into the subsequent control law, avoiding explicit projection.

[0053] Furthermore, the composite sliding mode variable in step 2 is the sum of a linear normalization part, a rational exponential part, and a time integral part; the linear normalization part is stored in the first-level buffer of the register stack as a benchmark reference 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 exponential part is used to enhance the convergence rate of the error interval and ensure finite-time convergence of the engineering controlled object; the time integral part is used to compensate in real time for the effect of slow boundary movement on the amplification of the error interval and ensure symmetrical convergence behavior of the overall sliding mode surface under a dynamic constraint environment.

[0054] Defining rational exponents ,in (Fixed constant, guaranteed to converge in finite time). Construction :

[0055] ;

[0056] in, ; ; ; is the terminal amplification coefficient, which is used to adjust the nonlinear amplification strength of the rational exponential part on the error. It is a dummy variable of the integration interval, representing the historical time points from 0 to the current time t. It is used to traverse historical data for integration and has no independent physical quantity meaning.

[0057] When the constraint potential energy error arrives from the previous link, the controlled object immediately multiplies the error by the normalization coefficient in state index order and writes the result to the first-level buffer to generate the linear normalization component. Because this buffer is located at the forefront of the on-chip memory hierarchy, with the lowest read and write latency, it ensures that the linear normalization component can be retrieved as quickly as possible within each control cycle for subsequent superposition operations. At the same time, it provides a constant proportional channel, ensuring that the composite sliding mode variable has a strict first-order linear response when the error amplitude is small, which is crucial for avoiding small-signal jitter and quantization error amplification. The hard real-time processing unit then calls the rational exponential transformation logic embedded in the read-only memory to perform nonlinear amplitude amplification on the linear normalization component to form the rational exponential component. The reason why rational exponents whose numerator and denominator are both smaller than the linear growth order are chosen here instead of general integer powers is that the rational exponent part can provide superlinear gain when the error enters the medium and high amplitude range, but still remain continuous and differentiable when the error approaches zero. Therefore, it can ensure the finite time convergence of the engineering controlled object without introducing unpredictable differential singularities.

[0058] The controller then calculates the current boundary movement trend based on the net change in the upper and lower boundaries recorded in the previous cycle. This is then synchronized with the constraint potential energy error and the normalization coefficient in a fixed-point accumulator operation to produce the time-integrated component. This component, weighted by the sign and speed of the boundary drift, injects a progressive hedging channel into the composite sliding mode variable, ensuring symmetric convergence of the overall sliding mode surface under dynamic constraints and effectively suppressing error range amplification caused by slow boundary movement. Finally, the three components are algebraically summed according to established sign rules to form a composite sliding mode variable. This variable is then written back to the high register stack, awaiting vector assembly to form the overall sliding mode surface. Through this partitioned storage and pipelined computation, the linear normalization component provides smooth and precise low-amplitude error correction, the rational exponential component provides rapid mid- and high-amplitude error reduction, and the time-integrated component provides adaptive boundary tracking compensation. These three components are independent yet tightly coupled, ensuring that at all times there is at least one effective energy path for forced energy dissipation on the sliding mode surface. At the same time, the generation of composite sliding mode variables is transparent to upper-level logic. A single-hidden-layer radial basis function network only needs to read the product of the overall sliding mode surface and the basis function to complete weight updates, eliminating additional structural complexity. This design not only achieves a dynamic balance between the three couplings of error scale, convergence rhythm, and boundary drift in the functional dimension, but also balances cache reentrancy and pipeline parallelism in the hardware implementation dimension. This provides a stable, predictable, and computationally balanced input signal for subsequent control input calculations, fundamentally ensuring that the controlled object converges within a finite time and continuously satisfies bilateral time-varying state constraints.

[0059] Furthermore, in step 2, at the beginning of the current control cycle, a fixed set of exponents that satisfy a rational number relationship is uniformly selected for all actual states. The selection of this exponent set follows the following principles: both the numerator and denominator of the exponent are positive integers to ensure that the exponent value is positive and accurately describes the finite-time convergence characteristics; different actual states share the same exponent set to eliminate the imbalance in relative convergence speed caused by exponent differences; and the exponent value remains consistent with the previous control cycle to avoid control discontinuities caused by frequent exponent adjustments. This unified selection process ensures that the subsequent composite sliding mode variables coevolve under the same order homogeneous structure, providing consistent convergence dynamics for the overall sliding mode surface.

[0060] To avoid convergence speed imbalances due to differences in exponent selection across different actual states, while ensuring that the exponent values ​​theoretically provide a strict finite-time convergence upper bound, this scheme implements a unified exponent selection process during the initialization phase of each control cycle. The controller first queries a pre-stored "index candidate table" in the parameter area of ​​the controlled plant. This table lists pairs of rational numbers consisting of coprime positive integer numerators and denominators, each of which corresponds to a pre-calculated convergence time marker. The controller then performs a consistency check against the rational exponent set used in the previous control cycle. If the exponent set from the previous cycle is still valid in the candidate table, it is retained to ensure consistency with the exponent value from the previous control cycle. If a new exponent set is required due to external configuration changes or dynamic task switching, the controller selects the first set of exponents from the candidate table that meets the duration requirement and has both positive integer numerators and denominators, subject to the constraint that the convergence time must not exceed the remaining task duration. The set is then written to the high-speed register area. After this selection, all actual states share the same exponent set, ensuring the coevolution of the composite sliding mode variables within the same homogeneous structure of order. This approach offers two advantages: First, both the numerator and denominator of the exponent are positive integers, ensuring that the exponent value is always positive and can be directly mapped to the theoretical exponential order of the finite-time convergence law, without introducing unphysical gains due to sign changes. Second, using the same set of exponents for different actual states eliminates convergence rate imbalances between sliding mode surface components, helping the entire sliding mode surface simultaneously approach zero in a radially symmetric manner in multidimensional space. Finally, because the exponent selection mechanism ensures consistency between control cycles and avoids frequent exponential adjustments, the power amplification factor of the control law does not jump, and the sliding mode surface evolution remains continuous, avoiding control discontinuities caused by sudden exponential changes. Through this unified selection process, the nonlinear amplification component of the composite sliding mode variable forms a consistent convergence dynamics across all states, providing a stable, predictable, and synchronized finite-time descent path for the entire sliding mode surface, further consolidating the overall performance of the control method in terms of coordinated constraint satisfaction and fast tracking.

[0061] Furthermore, in step 2, after confirming the exponent, the current constrained potential energy error is calculated in real time for each actual state. The absolute magnitude of the error is then exponentially mapped to the exponent to obtain a first sliding mode primitive. Simultaneously, the sign of the constrained potential energy error is independently extracted to form a second sliding mode primitive. The two types of sliding mode primitives work together: the first sliding mode primitive exponentially amplifies errors far from zero to ensure rapid decay; the second sliding mode primitive records the direction of the error to ensure directional ambiguity when the sliding mode approaches 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 portion of the original sliding mode primitive. When the constrained potential energy error is within a preset range, this buffer mapping does not affect the output of the original sliding mode primitive. When the constrained potential energy error approaches infinity, the buffer mapping is gradually and smoothly truncated within a finite amplitude, achieving soft clipping of the control gain to limit the control amplitude and avoid high-frequency chattering near the overall sliding surface.

[0062] The controlled object of the engineering project calls the exponential value stored in the register area in the previous step, performs a power mapping operation on the absolute amplitude of the constrained potential energy error, and the result is marked as the first sliding mode primitive. The core function of the power mapping is to use the property of the exponent being less than one but greater than zero to nonlinearly amplify the error far from zero, forming a convergence driving force that is superlinearly related to the error size, thereby quickly releasing energy when the sliding surface is still in the medium and high amplitude stage, ensuring that the overall sliding surface can decay rapidly within a finite time. At the same time, the controller independently extracts and packages the sign information of the same constrained potential energy error to form the second sliding mode primitive. The second sliding mode primitive only carries single-bit directional information. It does not perform amplification functions, but provides indispensable directional judgment when the sliding mode approaches zero, ensuring that the composite sliding mode variable will not slip or vibrate at high frequencies due to directional ambiguity at the moment of sign switching. The two types of sliding mode primitives exist in parallel logically and check and balance each other numerically: the first sliding mode primitive compresses the energy of the error amplitude, and the second sliding mode primitive locates the attitude of the error direction. The synchronization of the two enables each actual state to obtain a pair of original sliding mode primitives.

[0063] To strike a balance between high-performance actuators and finite-resolution sensors, this solution adds a continuous, monotonic, and saturating buffer mapping interval to the amplitude portion of the original sliding mode primitive. Specifically, within the normal operating range where the constrained potential energy error is within a preset range, the buffer mapping output maintains linear consistency with the original sliding mode primitive amplitude, ensuring control sensitivity in the small error range. As the constrained potential energy error approaches theoretical infinity with extreme state excursions, the slope of the buffer mapping gradually decreases and eventually transitions to a flat region, smoothly truncating the amplitude of the first sliding mode primitive within a finite upper bound. This soft clipping mechanism limits the control gain to within the physical limits of the actuator and significantly suppresses high-frequency chattering near the overall sliding surface caused by over-amplification. The buffer mapping is continuous and monotonic, introducing no discontinuous transitions. Therefore, it does not violate the finite-time convergence properties of the terminal sliding mode and does not activate foldback conditions in discretely sampled engineering systems. With the introduction of buffer mapping, the first and second sliding mode primitives collaboratively output a pair of safely modulated sliding mode signals, which are written to the upper register stack and participate in the aggregation of the composite sliding mode variable for that week. Throughout this process, any state component, regardless of whether it is rapidly approaching the upper or lower boundary, receives a controlled, well-defined sliding mode driving force under the protection of the buffer mapping. This provides a strong and robust convergence impetus for the subsequent control law, while ensuring that the overall sliding mode surface maintains a symmetrical and continuous energy decay trajectory within a dynamic constraint environment.

[0064] Furthermore, in step 2, after obtaining the two types of sliding mode primitives after buffering, the two types of sliding mode primitives are spliced ​​in a weighted algebraic sum manner with fixed rational exponents as weights to form composite sliding mode variables; an odd-even cancellation design is adopted during splicing: the amplitude primitives and the sign primitives maintain the same direction of superposition in terms of signs, but are staggered in the order of exponents, thereby taking into account both high-gain convergence and low-gain steady-state performance; to synchronously process multi-dimensional actual states, all composite sliding mode variables are arranged in state order to form equal-dimensional vectors; then, the overall sliding surface is defined in the form of column vectors with equal-dimensional vectors; the zero vector of the overall sliding surface means that all composite sliding mode variables are simultaneously zero, indicating that all constrained potential energy errors are completely canceled out at the same time; when the overall sliding surface is far away from the zero vector, its components characterize the convergence distance of the corresponding state; when the overall sliding surface approaches the zero vector, it indicates that all actual states have approached the reference trajectory and remain within the safe interval of the bilateral time-varying constraints.

[0065] Under the premise that the two types of sliding mode primitives have been buffered and have achieved stable output, the controller first calls the fixed rational exponents from the unified exponent set for the current control cycle and injects them as weight coefficients into the sliding mode splicing logic. Specifically, the amplitude primitives perform the function of error amplification, and the sign primitives perform the function of direction judgment. The two need to complete a weighted algebraic sum in the numerical domain to be combined into a single-channel composite sliding mode variable. In order to ensure that this weighted process simultaneously meets the dual requirements of "strong response when far from zero and smooth response when approaching zero," the splicing link adopts an odd-even cancellation design: the amplitude primitives and the sign primitives maintain the same direction of superposition in terms of signs to ensure that the control direction does not undergo phase reversal; however, the order of the exponents is deliberately staggered, so that high-order exponents in the item arrangement take precedence over the amplitude primitives, and low-order exponents take precedence over the sign primitives. In this way, once the constrained potential energy error is in the medium and high amplitude range, the amplitude primitives will output significant gains first due to the high-order weights, pulling the overall sliding mode surface down rapidly, reflecting high-gain convergence; and after the error enters the small amplitude range, the symbol primitives dominated by low-order weights gradually become the main contributing items, making the composite sliding mode variable exhibit near-linear low-gain steady-state performance for small errors, avoiding over-amplification and resulting in chattering.

[0066] After the splicing is complete, the controlled object arranges all composite sliding mode variables in state index order to synchronously process the multi-dimensional actual state, forming an equal-dimensional vector. The dimension of this vector strictly matches the number of controlled states, and each element corresponds one-to-one, achieving structural homogeneity between the state components. The controller then maps this equal-dimensional vector directly into the overall sliding mode surface according to the column vector definition rules. The overall sliding mode surface occupies a continuous memory section in the high-speed register area in the data structure, allowing subsequent control law generation to obtain all sliding mode information through a single vector read operation.

[0067] In closed-loop operation semantics, the zero vector of the overall sliding surface represents the simultaneous zero of all composite sliding mode variables. At this point, all constraint potential energy errors are completely offset, indicating that each actual state has accurately fallen on the reference trajectory and is located in the center of the safe interval of the bilateral time-varying constraints. If the overall sliding surface is far from the zero vector, the absolute value of its components directly characterizes the convergence distance of the corresponding state: larger values ​​indicate that the state is farther from the safe center and requires a higher control gain; smaller values ​​indicate that the fine-tuning phase has begun. Driven by the control law, the overall sliding surface converges monotonically to the zero vector. When its norm approaches zero, the controlled system can determine that all actual states have generally approximated the reference trajectory and do not violate any time-varying state constraints, thus theoretically achieving finite-time convergence and achieving high-precision, low-jitter steady-state operation in engineering.

[0068] Furthermore, in step 2, when the Euclidean norm of the overall sliding mode surface drops below the preset sliding mode threshold and remains stable for several 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: the exponential update logic of the composite sliding mode variable is frozen, and the current exponential set is kept unchanged to avoid reactivating the high-gain area; the network learning rate is switched from high-frequency online update to low-frequency progressive calibration to reduce computing resource consumption.

[0069] During the real-time execution of the adaptive network-based time-varying state-constrained terminal sliding mode tracking control method, the controller continuously monitors the Euclidean norm of the overall sliding mode surface. When this norm first drops below a preset sliding mode threshold and remains stable for several consecutive sampling periods, the controlled plant determines that the overall sliding mode surface has effectively approached the zero vector. At this point, the internal logic enters the convergence determination branch, first freezing the exponential update logic of the composite sliding mode variable. Specifically, a "lock flag" is placed in the index management module, disabling any index set replacement requests from external scheduling or adaptive tuning, maintaining the current index set unchanged. This freezing mechanism prevents the reactivation of high-order weights after the index set has reached steady state, thereby preventing the composite sliding mode variable from being amplified momentarily in the small error range, thereby preventing the reactivation of high-gain regions and the attendant chattering risk. Next, the engineering controlled object cascades the learning rate of the single-hidden-layer radial basis network, converting the original high-frequency online update mode to a low-frequency progressive calibration mode: the controller writes a new smaller step size to the learning rate register and adjusts the trigger period of the weight update interrupt so that the network weights are only slightly adjusted within a longer sampling window. On the one hand, this strategy significantly reduces the real-time computing load of the processor during steady-state operation, extending the available life of embedded resources. On the other hand, it also makes the weight change process smoother and avoids high-frequency weight jitter from interfering with the converged control channel. Through the two coordinated actions of freezing the exponential set and reducing the network learning rate, this method achieves a seamless transition from the fast response stage to the steady-state energy-saving stage while maintaining the terminal sliding mode convergence results. It not only maintains the strict satisfaction of the bilateral time-varying state constraints, but also significantly optimizes the long-term operating efficiency and hardware reliability of the engineering controlled object.

[0070] Furthermore, in step 3, a set of non-overlapping single-hidden layer radial basis network basis function centers are constructed for each actual state, and the initial width is uniformly set for all single-hidden layer radial basis network basis functions; the network weights are initialized with a zero vector to avoid unknown bias of the network output on 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 within a limited window based on the historical difference between the actual state and the reference trajectory, thereby ensuring that the initial network structure covers the main dynamic modes.

[0071] When using a single-hidden-layer radial basis function network to perform online approximation of the actual dynamics of an engineering controlled object, a set of non-overlapping single-hidden-layer radial basis function centers is first established for each actual state. The fundamental purpose of ensuring non-overlap is to prevent multiple basis functions from redundantly covering the same local state domain, which would lead to gradient competition and resource waste during the weight update process. Each basis function center occupies an independent state interval, allowing the network weights to focus on capturing specific dynamic features during training, thereby improving approximation efficiency and stability. To further simplify the workload of hyperparameter debugging, this solution uniformly sets the initial width for all basis functions. This approach avoids the computational burden of searching for the optimal width for each basis function in the high-dimensional state space, while ensuring that the network still has sufficient global perception capabilities in the early stages. Its local resolution can be implicitly corrected by gradient information during subsequent weight fine-tuning. The network weights are initialized with a zero vector to ensure that the initial network output is always zero. This ensures that during the initial sampling period of closed-loop operation, the size of the control input is completely determined by the known portion of the controlled object and the sliding mode arrival gain, preventing the sudden superposition of unknown biases on the control signal, causing shock or distortion. To ensure that the initial network structure covers the main dynamic modes of the controlled object and avoids the omission of certain important state intervals due to center distribution bias, this solution collects differential data between the actual state and the reference trajectory within a finite-length historical window and feeds these differential samples into a density-based fast clustering algorithm 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 clusters is equal to the number of clusters, and the centers are naturally located in the area with the highest frequency of historical deviations, probabilistically ensuring that the most representative dynamic changes are covered when the network is established. In this way, the network can adjust the weights to the valid range without consuming too many iterations in the subsequent online training phase, and the convergence speed is faster. Combined with the finite time convergence characteristics of the terminal sliding mode, it can complete compensation for unknown dynamics within a very short transition period and maintain strict compliance with the bilateral time-varying state constraints.

[0072] Furthermore, in step 3, the component error signal of the overall sliding surface on the composite sliding mode variable is used as the driving force, and its product with the basis function of the single hidden layer radial basis network is calculated and weighted to fix the network learning rate to generate a weight increment; the weight increment is superimposed on the original weight vector, and is restricted to a predefined bounded area through a projection operator to ensure that the network weights always remain bounded.

[0073] During the online training of a single-hidden-layer radial basis function network (RBFN), the controller first extracts the component error signals corresponding to the composite sliding mode variables from the overall sliding surface and treats them as the driving force for network weight adjustment. This driving force is then element-wise multiplied with the single-hidden-layer RBFN basis function at the current sampling time. The resulting product is weighted by a fixed network learning rate to produce a new set of weight increments. The weight increments reflect the direction and magnitude of the immediate compensation that the network output should make to the unknown engineering controlled object under the current error conditions. To prevent unbounded weight growth during long iterations, the method introduces a projection operator before adding the weight increments to the original weight vector. This operator first checks whether the candidate weights after addition fall within a predefined bounded region. If they do, they are accepted directly. If they do, they are projected back onto the boundary surface according to the principle of minimum Euclidean distance, thus ensuring that the updated weights remain within a safe and controllable range. Through this three-step closed loop of "error-driven multiplication - fixed-step weighting - boundary projection", the network weights can not only respond quickly to new dynamic changes, but also not drift out of the reasonable numerical range 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, ensure that the controlled object of the engineering maintains consistent satisfaction of the bilateral time-varying state constraints throughout the entire operating period.

[0074] Define the dynamics of the engineering plant: ,in Known to be reversible. Use a single hidden layer radial basis network to approximate the unknown nonlinearity:

[0075] ;

[0076] ;

[0077] in, is a basis function column vector composed of 25 Gaussian basis functions, with uniform coverage at the center , width 0.2; for The first derivative of ; Depend on The actual state vector of the engineering controlled object composed of elements is the controlled quantity of closed-loop control; is a known and reversible control influence matrix, also known as the input gain matrix; For dimension and Same desired trajectory vector; is the boundary velocity compensation vector, which is used to offset the additional deviation caused by the time-varying upper and lower boundary movement; each component Specifically equal to ; is a rational exponent constant (fixed to be ); is the overall sliding surface vector composed of composite sliding mode variables; It is a diagonal arrival gain matrix that adaptively adjusts the arrival speed; is the number of neural network basis functions, fixed at 25; for Dimensional online adjustable weight matrix; For control instructions; It is the total number of controlled states of the controlled objects of the project.

[0078] Take the Lyapunov function:

[0079] ;

[0080] in is the ideal weight. Give the weight update:

[0081] ;

[0082] in, The value is time , is the fixed diagonal adaptation rate; is the 25th-order unit matrix;

[0083] ;

[0084] And because and , which satisfies the finite time convergence theorem; combined with the potential energy mapping (1), it can be proved that for all Maintain

[0085] .

[0086] Figure 2 This section details the dynamic characteristics of the constraint potential energy error during the control process. The horizontal axis represents time, and the vertical axis represents the magnitude of the constraint potential energy error. The zero error line, indicated by a dotted line, serves as the target benchmark for the constraint potential energy error to converge.

[0087] The constraint potential error curve, represented by a solid line, shows the complete evolution from the start of control to the complete elimination of the error. In the initial stages of control, the constraint potential error has a large positive value, reflecting the significant deviation of the actual state from the reference trajectory and the error amplitude generated by mapping this deviation through the logarithmic potential energy under the bilateral time-varying constraints.

[0088] As the control algorithm continues to operate, the constraint potential error exhibits a clear attenuation trend. The curve begins at a high amplitude and then experiences a rapid decline, demonstrating the enhanced effect of the rational exponential component of the compound sliding mode variable on the error convergence rate. As the error curve approaches the zero error line, it exhibits smooth convergence, avoiding the chattering phenomenon that can occur in traditional sliding mode control.

[0089] The control start point and error elimination point are specifically marked, clearly indicating the time interval from the generation of the constraint potential error to its complete cancellation. The finite-time convergence region is indicated by a dashed box, emphasizing that once the constraint potential error enters this region, the controlled plant can achieve complete convergence within the predetermined finite time. This entire process demonstrates the superior performance of the proposed control method in handling time-varying constraints.

[0090] Figure 3 This diagram comprehensively illustrates the composition and convergence characteristics of a composite sliding mode variable. The horizontal axis represents time, while the vertical axis shows the values ​​of the composite sliding mode variable and its components. The zero line, represented by a dotted line, serves as a reference for the convergence target.

[0091] The linear normalization component, represented by the dashed line, represents the fundamental component of the composite sliding mode variable that ensures linear response. This component exhibits good linearity when the error amplitude is small, providing a stable baseline reference for the entire composite sliding mode variable. Its relatively smooth convergence process demonstrates the gradual convergence of the linear control component.

[0092] The rational exponent portion is represented by a thick solid line, highlighting its key role in rapid convergence. This portion uses a fixed rational exponent for power mapping, effectively enhancing the convergence rate within the error interval and ensuring finite-time convergence for the engineering controlled object. The curve exhibits significant nonlinear convergence behavior, with a strong convergence-driving capability far from zero.

[0093] The composite sliding mode variable, represented by a plain solid line, reflects the combined effects of the linear normalization component, the rational exponential component, and the time integration component. This curve combines the strengths of each component, maintaining a linear response while also enabling rapid convergence. Furthermore, the time integration component provides real-time compensation for slow boundary movement.

[0094] The vertical dashed line marks the dividing point between the rapid convergence phase and the precise tracking phase, and a preset sliding mode threshold is set. When the Euclidean norm of the composite sliding mode variable falls below this threshold, the controlled plant enters precise tracking mode, reflecting the staged optimization feature of the control strategy.

[0095] Here is a The specific numerical example demonstrates the time-varying state constraint terminal sliding mode tracking control method based on adaptive network in two discrete sampling periods ( 、 ) within the implementation process.

[0096] Controlled nonlinear dynamics: .

[0097] Corresponding to a known reversible input gain .

[0098] Sampling period: .

[0099] Reference trajectory: .

[0100] Bilateral time-varying state constraints: .

[0101] Initial state: .

[0102] Mapping and sliding mode design constants:

[0103] ;

[0104] Radial Basis Network: ;

[0105] Constrain boundary velocity: ;

[0106] In this example, two cycles , so the time integral term is zero at the initial stage.

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

[0108] Composite sliding mode variables: ;

[0109] Overall sliding surface: .

[0110] Reach Gain: .

[0111] Neural Network Compensation: .

[0112] Control law: .

[0113] Weight update: .

[0114] Discretization: .

[0115] State update (Eulerian approximation):

[0116] .

[0117] Sampling time

[0118] Constraint potential error: .

[0119] Composite sliding mode variables: ;

[0120] Overall sliding surface: .

[0121] Control input:

[0122] .

[0123] Weight update: .

[0124] Status Update: .

[0125] After two sampling cycles, the norm of the overall sliding surface changes from down to , the monotonically decreasing process indicates that the finite time convergence process has started; at the same time, the network weight Gradually adjust away from zero to the unknown dynamic compensation direction, and each update amount is affected by and projection operator constraints remain bounded. After continuous iteration, when Below threshold After stabilizing for several cycles, the exponential set is frozen, the learning rate is decelerated, and the engineering controlled object enters a steady state, and all moments satisfy .

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

[0127] The above shows and describes the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The above embodiments and descriptions merely illustrate the principles of the present invention. Various changes and modifications may be made to the present invention without departing from the spirit and scope of the present invention. Such changes and modifications are intended to fall within the scope of the present invention. The scope of protection claimed by the present invention is defined by the appended claims and their equivalents.

Claims

1. A time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network, the method comprising: Step 1: Construct a constrained 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 vary with time; Step 2: Construct a composite sliding mode variable with fixed rational exponents. The sliding mode variables of all actual states of the controlled plant form an overall sliding surface at the vector level. When the overall sliding surface approaches zero, all constraint potential energy errors are eliminated simultaneously, ensuring that the actual state of the controlled plant tracks the reference trajectory within a finite time and does not exceed any upper or lower bounds that vary with time. Step 3: Use a single-hidden-layer radial basis function network to approximate the actual dynamics of the controlled object 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 finite time and the real-time compensation of the actual state of the controlled object is completed synchronously.

2. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network according to claim 1, characterized in that: The method further includes: step 4: providing a gradient weight update law based on the product of the composite sliding mode variable and the single hidden layer radial basis network basis function by selecting a Lyapunov function including 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; 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 engineering controlled object achieves sliding mode arrival within a finite time, bounded convergence of the weight of the single hidden layer radial basis network, and maintains satisfaction of the bilateral time-varying state constraint at all times.

3. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 2, characterized in that: Step 1 specifically includes: Step 1.1: At the beginning of each control cycle, read the upper and lower bounds corresponding to the current actual state in real time. Both are updated synchronously with time to fully reflect the instantaneous position of the bilateral time-varying state constraint. Step 1.2: Subtract the actual state from the reference trajectory to obtain the first deviation. At the same time, subtract the upper boundary from the reference trajectory and the reference trajectory from the lower boundary to obtain the second deviation and the third deviation respectively. Step 1.3: Perform a natural logarithm operation on the ratio of the second deviation to the first deviation to obtain an upper potential energy term; perform a natural logarithm operation on the ratio of the first deviation to the third deviation to obtain a lower potential energy term; Step 1.4: Perform algebraic summation of the upper potential energy term and the lower potential energy term with opposite signs to obtain a constrained potential energy error; when the actual state approaches the upper boundary or the lower boundary infinitely, the amplitude of the constrained potential energy error accordingly tends to infinity within a finite time.

4. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 3, characterized in that: The composite sliding mode variable in step 2 is the sum of a linear normalization part, a rational exponential part, and a time integral part. The linear normalization part is stored in the first-level buffer of the register stack and serves as a benchmark reference 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 exponential part is used to enhance the convergence rate of the error interval and ensure finite-time convergence of the engineering controlled object. The time integral part is used to compensate in real time for the effect of slow boundary movement on the amplification of the error interval, ensuring symmetrical convergence behavior of the overall sliding mode surface under a dynamic constraint environment.

5. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 4, characterized in that: In step 2, at the beginning of the current control cycle, a fixed set of exponents that satisfy a rational number relationship is uniformly selected for all actual states. The selection of the exponent set follows the following principle: both the numerator and denominator of the exponent are positive integers to ensure that the exponent value is positive and can accurately describe the finite time convergence characteristics; Different actual states share exactly the same set of exponents to eliminate the imbalance in relative convergence speed caused by exponent differences; the exponent values ​​are kept consistent with the previous control cycle to avoid control discontinuity caused by frequent exponent adjustments; through this unified selection process, the subsequent composite sliding mode variables are ensured to co-evolve under the same order homogeneous structure, providing consistent convergence dynamics for the overall sliding surface.

6. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 5, characterized in that: In step 2, after confirming the exponent, the current constrained potential energy error is calculated in real time for each actual state. The absolute magnitude of this error is then subjected to a power mapping with the exponent to obtain a first sliding mode primitive. Simultaneously, the sign of this constrained potential energy error is independently extracted to form a second sliding mode primitive. The two types of sliding mode primitives work together: the first sliding mode primitive amplifies errors far from zero by a power, ensuring rapid decay; the second sliding mode primitive records the direction of the error to ensure directional ambiguity when the sliding mode approaches zero. Thus, each actual state corresponds to a pair of original sliding mode primitives. A continuous, monotonic, and saturated buffer mapping interval is appended to the amplitude portion of the original sliding mode primitive. When the constrained potential energy error is within a preset range, this buffer mapping does not affect the output of the original sliding mode primitive. When the constrained potential energy error approaches infinity, the buffer mapping is gradually and smoothly truncated within a finite amplitude, achieving soft clipping of the control gain, limiting the control amplitude and avoiding high-frequency chattering near the overall sliding surface.

7. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 6, characterized in that: In step 2, after obtaining the two types of sliding mode primitives after buffering, the two types of sliding mode primitives are spliced ​​in a weighted algebraic sum manner with fixed rational exponents as weights to form composite sliding mode variables; an odd-even cancellation design is used during splicing: the amplitude primitives and the sign primitives maintain the same direction of superposition in terms of signs, but are staggered in the order of exponents, thereby taking into account both high-gain convergence and low-gain steady-state performance; to synchronously process multi-dimensional actual states, all composite sliding mode variables are arranged in state order to form equal-dimensional vectors; then, the overall sliding surface is defined in the form of column vectors of equal-dimensional vectors; the zero vector of the overall sliding surface means that all composite sliding mode variables are simultaneously zero, indicating that all constrained potential energy errors are completely canceled out at the same time; when the overall sliding surface is far away from the zero vector, its components characterize the convergence distance of the corresponding state; when the overall sliding surface approaches the zero vector, it indicates that all actual states have approached the reference trajectory and remain within the safe interval of the bilateral time-varying constraints.

8. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 7, characterized in that: In step 2, when the Euclidean norm of the overall sliding mode surface drops below the preset sliding mode threshold and remains stable for several 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: the exponential update logic of the composite sliding mode variable is frozen to keep the current exponential set unchanged to avoid reactivating the high-gain area; and the network learning rate is switched from high-frequency online update to low-frequency progressive calibration to reduce computing resource consumption.

9. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 8, characterized in that: In step 3, a set of non-overlapping single-hidden layer radial basis network basis function centers are constructed for each actual state, and the initial width is uniformly set for all single-hidden layer radial basis network basis functions; the network weights are initialized with a zero vector to avoid unknown bias of the network output on 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 within a limited window based on the historical difference between the actual state and the reference trajectory, thereby ensuring that the initial network structure covers the main dynamic modes.

10. The time-varying state-constrained terminal sliding mode tracking control method based on an adaptive network as claimed in claim 9, characterized in that: In step 3, the component error signal of the overall sliding surface on the composite sliding mode variable is used as the driving force, and its product with the basis function of the single hidden layer radial basis network is calculated and weighted to fix the network learning rate to generate a weight increment; the weight increment is superimposed on the original weight vector and restricted to a predefined bounded area through a projection operator to ensure that the network weight always remains bounded.

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