Wireless communication power control method based on fractional calculus theory

Abstract

The invention discloses a wireless communication power control method based on a fractional calculus theory, which comprises the following steps of: firstly, constructing a continuous time stochastic differential equation channel model driven by a symmetric-stable Levy process, and respectively depicting heavy tail jump behaviors in long-term path loss and short-term multipath fading; then, on the basis of the channel model, a generalized Riesz fractional order derivative operator is introduced, a class of fractional order HJB (FHJB) control equations with non-local operators are deduced under the dynamic programming principle, and the fractional order HJB (FHJB) control equations are used for describing an optimal power control strategy under the infinite variance interference condition; and finally, in a multi-base-station multi-user same-frequency interference scene, constructing a fractional order FHJB power control solution framework, obtaining an optimal transmitting power distribution strategy by using a value function iteration method, and verifying the robustness and communication quality assurance capability of the optimal transmitting power distribution strategy under a jump fading channel through numerical simulation. According to the invention, adaptive control of the transmitting power in a non-stationary and heavy-tailed fading environment can be realized.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of communication, and particularly relates to a wireless communication power control method based on fractional calculus theory. BACKGROUND

[0002] With the rapid development of the sixth generation (6G) wireless communication technology, Space-Air-Ground Integrated Networks (SAGINs) are widely considered as one of the key architectures to achieve ubiquitous communication coverage. In this system, communication nodes such as unmanned aerial vehicles, orbital transportation, and high-speed mobile terminals are often in a highly dynamic environment, facing problems such as channel state mutation, multipath reflection, and strong interference, which result in obvious non-stationarity and non-Gaussianity of the channel, constituting an important challenge for 6G communication modeling and optimization.

[0003] In complex terrain environments such as mines and mine shafts, due to the significant effects of rock layer reflection, metal shielding, and tunnel refraction, mobile terminals are prone to encounter phenomena such as signal strength jumps, interference power surges, and link interruptions during movement, which severely affect communication stability and reliability. In particular, in unmanned vehicle or unmanned aerial vehicle systems, communication quality is highly dependent on the ability to dynamically perceive complex channel environments and real-time power regulation.

[0004] Traditional wireless channel modeling methods are mostly based on the assumption of stationary evolution of channel state, and usually divide fading into two categories: long-term (such as lognormal path loss) and short-term (such as Rayleigh or Rician multipath fading). Under this framework, classical models such as the Ornstein-Uhlenbeck (O-U) process and the Rician power statistical model have been introduced by existing research, which have good fitting performance in stationary scenarios, but are difficult to accurately capture the non-continuous jump changes and heavy-tailed statistical properties caused by shielding, scattering, or irregular reflection in mine environments.

[0005] In addition, the interference in mine environments has high intermittency and randomness, often exhibiting sudden signal strength drops and interference power surges, significantly weakening the robustness of traditional power control models. Most current optimization methods rely on the Hamilton-Jacobi-Bellman (HJB) optimal control framework, which usually assumes that the state evolution process is continuously differentiable and the noise term is a Brownian motion with finite variance. This modeling assumption is no longer applicable when facing non-Gaussian disturbances with infinite variance and jump characteristics.

[0006] In view of the above problems, in recent years, symmetric - Stable (Symmetric - Stable, S S) Lévy processes are gradually introduced into the modeling of communication systems to characterize the jump propagation, heavy-tailed interference, and infinite variance characteristics. This class of processes has been widely validated in the field of financial engineering and signal processing, and its potential application in wireless communication is being continuously explored. S Lévy processes have natural discontinuity and impulse response capabilities, which can effectively capture the complex multi-scale, non-Gaussian, and non-stationary fading characteristics in mine channels, providing a new paradigm for communication optimization in extreme environments. SUMMARY

[0007] To overcome the shortcomings of the prior art, the present application provides a wireless communication power control method based on fractional calculus theory. First, combined with the scene characteristics of mine mobile communication, a channel model driven by a stable Lévy process is constructed, which describes the heavy-tailed jump behavior in long-term path loss and short-term multipath fading. - stable Lévy process, respectively; then, based on the channel model, a generalized Riesz fractional derivative operator is introduced, and a class of fractional HJB (FHJB) control equations with non-local operators is derived under the dynamic programming principle, which is used to describe the optimal power control strategy under the condition of infinite variance interference; finally, in the multi-base station multi-user co-channel interference scenario, a fractional FHJB power control solving framework is constructed, and the optimal transmit power allocation strategy is obtained using the value function iteration method, and its robustness and communication quality guarantee capability under the jump fading channel are verified through numerical simulation. The present application can realize adaptive control of the transmit power in a non-stationary, heavy-tailed distribution fading environment, and under the condition of coupling between the jump channel and interference, it can effectively reduce the system energy consumption while maintaining the quality of service constraint, and is suitable for power scheduling and interference management in 6G communication systems in complex wireless communication environments such as mines and tunnels.

[0008] The technical solution adopted by the present application to solve its technical problems is as follows:

[0009] Step 1: Construct a non-Gaussian multi-scale channel model suitable for mine wireless environment, introduce a symmetric - stable (S S) Lévy process describes the jump and heavy-tailed characteristics in the channel, and fully reflects the complex dynamics of long-term and short-term fading;

[0010] Step 2: Based on the channel model, an optimization control framework with fractional non-local diffusion operator is established, and a fractional Hamilton-Jacobi-Bellman (FHJB) equation is proposed to model the transmit power control problem;

[0011] Step 3: In the scenario of multi-base station and multi-user co-frequency communication, construct system performance indicators and solve control equations to obtain a class of adaptive power scheduling strategies, thereby achieving joint optimization of communication quality and energy consumption cost.

[0012] Furthermore, step 1 specifically includes:

[0013] Step 1-1: Construct a non-Gaussian long-term fading model;

[0014] Define long-term decay Satisfy the following by S S-Lévy process-driven stochastic differential equations (SDEs):

[0015]

[0016] In the formula, The mean recovery intensity, Control the location of the stable point. For noise intensity, For S SLévy process, Initial conditions This represents the initial state of the fading process;

[0017] Step 1-2: Construct a non-Gaussian short-term fading channel model;

[0018] Definition of the first In-phase components of the path Orthogonal components The evolutionary form is as follows:

[0019]

[0020]

[0021] In the formula, The damping coefficient is... Control the intensity of the jump. and For mutually independent S SLévy process; initial values and The initial state is defined;

[0022] The instantaneous power of each path is thus defined. With signal envelope for:

[0023]

[0024]

[0025] Substituting Itô's lemma and considering Lévy jumps, we get The dynamic evolution equation:

[0026]

[0027] In the formula, Indicates the process in time Left limit at the point, capture The value before any potential jump occurs; These represent the in-phase and quadrature components of the signal at a specific time. The amplitude of the jump; for processing To address the problem of non-integrability of the squared terms of time jumps, a small jump truncation mechanism is introduced, with a truncation amplitude of... The Lévy-Itô decomposition of the jump terms is performed using the compensated Poisson measure, ultimately yielding the expected representation of the cumulative squared jump terms:

[0028]

[0029] In the formula, The deterministic second-order moment contributions of all Lévy jumps were aggregated, thus fixing the drift intensity; process It is a square-integrable martingale with zero expectation, capturing the cumulative square jump in Random fluctuations around a specified mean; where, and Corresponding to and Compensated Poisson random measure; This represents the magnitude of the jump when a Lévy process undergoes a transition at a certain moment. This represents the cutoff threshold used to distinguish between large and small transitions;

[0030] Define the normalized Lévy process:

[0031]

[0032] The instantaneous power process is represented as a generalized Cox-Ingersoll-Ross, i.e., a CIR-type SDE:

[0033]

[0034] in, Represents a normalized symmetry Stable semimartingar process;

[0035] Steps 1-3: Construct a complete channel transmission process through cascading;

[0036] Considering transmit power control input Treating short-term fading as an input-driven output power process By combining the long-term and short-term models, the following complete cascaded channel model is obtained:

[0037]

[0038]

[0039] In the formula, This indicates the avoidance of singularities and reflects the low power noise floor; A fast timescale representing short-term decline relative to long-term changes; It represents the noise intensity coefficient of the short-term fading process, controlling the severity of random fluctuations and jumps; This represents the instantaneous power ultimately received by the user equipment in a cascaded channel. Indicates the base station's transmission power; This represents the independent normalized S that drives the short-term fading process. S Lévy process.

[0040] Furthermore, step 2 specifically includes:

[0041] Step 2-1: Construct the system state equation, objective function, and value function, and perform Bellman expansion using the dynamic programming principle (DPP).

[0042] Consider the following control system with state variables as follows: It is described by the following stochastic differential equation:

[0043]

[0044]

[0045] In the formula, Given the initial state, For the current moment, Termination time; control variables For permissible control; diffusion coefficient The noise intensity is state-dependent. This represents the set of all permissible controls, where the control strategies must satisfy adaptability and measurability.

[0046] Noise item The dimension is S The Lévy process is measured by the following Lévy measure:

[0047]

[0048] in Indicates a single jump during the Lévy process 3D jump magnitude vector, normalization constant It is given by the following formula:

[0049]

[0050] in, Represents the gamma function;

[0051] The optimization objective of the system is to minimize the expected cost functional of the following form:

[0052]

[0053] In the formula, This refers to operating costs in general form. It is the terminal cost function;

[0054] Define value function From time and state The minimum cost that can be obtained by starting:

[0055]

[0056] For any Based on DPP, the following Bellman relation can be obtained:

[0057]

[0058] in, Represents the mathematical expectation;

[0059] Step 2-2: Construct the FHJB governing equations;

[0060] To accommodate spatial non-uniformity, a generalized Riesz fractional operator is introduced, which is defined in two equivalent forms:

[0061] Nuclear form: acting on Above, the function itself and its first and second partial derivatives are globally bounded, and its generalized fractional derivative is defined as:

[0062]

[0063] In the formula, Represents the principal value integral; the kernel function explicitly depends on... The expression is:

[0064]

[0065] in, Representation matrix The determinant of;

[0066] Lévy-Khintchine form: The infinitesimal generator of the Lévy measure form is expressed as:

[0067]

[0068] In the formula, for Lévy metric; index function Used to adjust small jump terms to ensure integral convergence;

[0069] The two operator forms mentioned above are equivalent;

[0070] Apply Itô's formula to the Lévy process. The increment can be expressed as:

[0071]

[0072] In the formula, The nonlocal terms resulting from Lévy jumps correspond to the generator or kernel definitions above;

[0073] Substituting the above expansion into the Bellman equation, rearranging the terms, and dividing by... ,have to:

[0074]

[0075] In the formula, when hour, ; Represents a standard infinitesimal;

[0076] Introduce the Legendre transform and define Hamiltonian as follows:

[0077]

[0078] In the formula, ; This represents the drift term in a stochastic differential equation;

[0079] Finally, the FHJB equation is obtained:

[0080]

[0081] Terminal conditions are .

[0082] Furthermore, step 3 specifically includes:

[0083] Step 3-1: Construct a multi-base station, multi-user co-frequency system and define a state evolution model;

[0084] Build a containing Each base station BS and A co-frequency downlink communication system for user equipment (UE) This indicates the total number of base stations in the system. This represents the total number of user equipment; in this system, each base station... For multiple user devices For broadcast signals, the instantaneous power received by user equipment is affected by both long-term and short-term fading; the specific modeling is as follows:

[0085] Long-term decay: base station To user equipment Long-term fading gain It follows the following OU-type stochastic differential equation:

[0086]

[0087] In the formula, S represents S Lévy: The jumping process;

[0088] Short-term fading and receive power: User equipment From base station Received instantaneous power The dynamic evolution is as follows:

[0089]

[0090] In the formula, Indicates base station In control strategy The transmission power under the action, That is, the effective transmission power after long-term attenuation; This represents the independent normalized S that drives the short-term fading process. S Lévy process; The noise intensity coefficient represents the noise intensity during short-term fading and controls the magnitude of random fluctuations.

[0091] The two state variables in the above process are combined to form a joint state vector:

[0092]

[0093] Its corresponding state space SDE is:

[0094]

[0095] In the formula, ;

[0096] Step 3-2: Define a channel performance evaluation method using SINR and construct an objective function;

[0097] The signal-to-interference-plus-noise ratio (SINR) of the received signal is used as an evaluation criterion for base stations. For user equipment The core metrics of QoS (Quality of Service) for communication are defined as follows:

[0098]

[0099] In the formula, For noise power, For other base stations For users The resulting interference power;

[0100] Constructing a cost function for mining scenarios based on SINR Taking into account communication speed, QoS penalty for breach of contract, and energy consumption cost, the design is as follows:

[0101]

[0102] In the formula, Indicates base station Transmit power control, Including base stations Control of all other base stations, It is the penalty intensity coefficient. Indicates if , then Otherwise , This indicates the cost of controlling transmission power consumption; This represents a predefined SINR threshold used to determine whether QoS standards are met. Indicates the base station's transmission power;

[0103] Step 3-3: Solve the FHJB equations using the numerical discretization method to obtain the power control strategy;

[0104] Based on the established state evolution and cost functions, the FHJB governing equations with generalized Riesz space derivatives constructed in steps 2-3 are used as the optimization framework, in the following form:

[0105]

[0106] The equation is solved numerically using an explicit grid discretization and value iteration algorithm, specifically including: solving the state space... Establish a grid; approximate the nonlocal derivative using the numerical integral form of the generalized Riesz operator; iteratively update the value function at each time step. For each grid point, enumerate the control strategy. And calculate the minimum cost to obtain optimal control;

[0107] The final output control strategy is:

[0108] .

[0109] An electronic device includes: a processor and a memory; the memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory to cause the electronic device to perform the above-described wireless communication power control method.

[0110] A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the above-described wireless communication power control method.

[0111] A chip includes a processor for retrieving and running a computer program from a memory, causing a device on which the chip is installed to perform the aforementioned wireless communication power control method.

[0112] A computer program product includes a computer storage medium storing a computer program, the computer program including instructions executable by at least one processor, which, when executed by the at least one processor, implement the above-described wireless communication power control method.

[0113] The beneficial effects of this invention are as follows:

[0114] This invention is the first to use symmetry This invention combines a stable Lévy process with a fractional-order Hamilton-Jacobi-Bellman (FHJB) control framework to accurately characterize channel interruption, hopping interference, and non-Gaussian non-stationary fading behavior. By constructing a generalized state-space system and using nonlocal fractional-order SDE modeling, this invention achieves an adaptive control strategy for transmit power in response to hopping fading and time-varying interference. Under the framework of fractional-order dynamic programming, a value iteration algorithm is used to solve for the optimal power trajectory, effectively improving the SINR robustness and power efficiency of the communication system. The method of this invention can still ensure that the signal quality of multiple users meets QoS requirements under complex Lévy interference, and features low power consumption, high robustness, and good convergence. It can provide theoretical basis and engineering support for the adaptive control design of future 6G high-reliability, low-latency mining wireless communication systems. Attached Figure Description

[0115] Figure 1 shows the time evolution of the transmit power and control actions of each base station when M = N = 3: (a) initial state; (b) 50 iterations; (c) 100 iterations.

[0116] Figure 2 It is a graph showing the evolution of the value function over time under different iteration numbers.

[0117] Figure 3 This is a signal-to-interference-plus-noise ratio (SIR) chart for different user devices after 50 iterations.

[0118] Figure 4 This is a signal-to-interference-plus-noise ratio (SIR) graph for different user devices after 100 iterations. Detailed Implementation

[0119] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0120] This invention proposes a wireless communication power control method based on fractional calculus theory. This method is based on symmetry. A multi-scale channel model driven by a stable Lévy process and a fractional-order Hamilton-Jacobi-Bellman (FHJB) control framework were constructed to establish a state evolution model and system cost function suitable for non-Gaussian jumping interference environments. Based on this, an adaptive transmit power control strategy calculation method combining numerical iteration and viscous solution theory was designed. This method effectively reduces transmit power consumption and system interruption risk while ensuring communication quality, thereby achieving optimal power control performance under complex mining area communication conditions.

[0121] Step S1: Construct a system based on symmetry - A continuous-time wireless channel model driven by a stable Lévy process. The long-term fading component is modeled using a Lévy-driven Ornstein-Uhlenbeck (OU) process, reflecting mean reversion characteristics and jump-tailed fluctuations. The short-term fading component is described by a Lévy-type Cox-Ingersoll-Ross (CIR) process, depicting the non-Gaussian evolution of received power and forming a multi-scale jump channel structure, reflecting the discontinuous fluctuations and local multipath transient disturbances in the channel state within the mining area.

[0122] Step S101: Introduce symmetry - A stable Lévy noise-driven OU process is used to construct a non-Gaussian long-term fading model.

[0123] In enclosed and complex environments such as mines and tunnels, path loss exhibits a jump-mean regression phenomenon characterized by sudden increases and slow recovery due to factors such as rock wall reflection and metal interference. To characterize this behavior, long-term fading is defined. Satisfy the following by S Stochastic Differential Equation (SDE) driven by S-Lévy processes:

[0124]

[0125] In the formula, The mean recovery intensity, Control the location of the stable point. For noise intensity, For S SLévy process, Initial conditions This represents the initial state of the fading process. This model breaks through the stationarity assumption of traditional log-normal path loss, supports abrupt change modeling, and is particularly suitable for the path evolution process of "rock wall penetration—sudden descent—recovery" in mines;

[0126] Step S102: Introduce the Lévy-type Cox-Ingersoll-Ross (CIR) process to construct a non-Gaussian short-term fading channel model.

[0127] To reflect the transient fluctuation characteristics of signals in local multipath scattering under non-Gaussian conditions in mining areas, a method based on S is introduced. S Lévy noise-driven signal component model. Define the first... In-phase components of the path Orthogonal components The evolutionary form is as follows:

[0128]

[0129]

[0130] In the formula, The damping coefficient is... Control the intensity of the jump. and For mutually independent S SLévy process. Initial values and The initial state is defined;

[0131] This allows us to define the instantaneous power of each path. With signal envelope for:

[0132]

[0133]

[0134] Substituting the above components into Itô's lemma and considering Lévy jumps, we obtain The dynamic evolution equation:

[0135]

[0136] In the formula, Indicates the process in time Left limit at the point, capture The value before any potential jump occurs. Because the underlying model is a Lévy process, the signal is allowed to have these instantaneous, discontinuous changes. These represent the in-phase and quadrature components of the signal at a specific time. The magnitude of the jump. To handle... To address the problem of non-integrability of the squared terms of time jumps, a small jump truncation mechanism is introduced, with a truncation amplitude of... The Lévy-Itô decomposition of the jump terms is performed using the compensated Poisson measure, ultimately yielding the expected representation of the cumulative squared jump terms:

[0137]

[0138] In the formula, The deterministic second-order moment contributions of all Lévy jumps were aggregated, thus fixing the drift intensity; process It is a square-integrable martingale with zero expectation, capturing the cumulative square jump in Random fluctuations around a specified mean; where, and Corresponding to and The compensated Poisson stochastic metric. It represents the magnitude of the jump when the Lévy process undergoes a transition at a certain moment. In the integral of the Lévy-Itô decomposition, it is an integral variable representing all possible jump magnitudes. This represents the cutoff threshold used to distinguish between large and small jumps. This threshold is used to define the jump amplitude. The jumps are treated as “small jumps” so that their second moments can be mathematically handled, ensuring the convergence of the integral;

[0139] Define the normalized Lévy process:

[0140]

[0141] The instantaneous power process can be represented as a generalized Cox-Ingersoll-Ross (CIR) type SDE:

[0142]

[0143] in, Represents a normalized symmetry A stable semimartingar process, which is achieved by focusing on the phase ( ) and orthogonal ( The Lévy process of the components is weighted and normalized to combine two independent noise sources into one, while preserving the original heavy tail. Stable statistical properties.

[0144] This model reflects the non-Gaussian, non-stationary, and sporadic strong disturbances of multipath signals, which is far superior to the exponential distribution assumption of the traditional Rayleigh model.

[0145] Step S103: Through the cascading of long-term and short-term fading and the separation of fast and slow scales, a complete channel transmission process is constructed.

[0146] To capture both long-term and short-term propagation effects, the wireless channel is modeled as a cascade of two fading processes. Transmit power control input is considered. Treating short-term fading as an input-driven output power process By combining the long-term and short-term models, the following complete cascaded channel model is obtained:

[0147]

[0148]

[0149] In the formula, This indicates the avoidance of singularities and reflects the low power noise floor; A fast timescale representing short-term decay relative to long-term changes; for a more physically meaningful notation, here we use... When placed into a complete cascaded channel model, change to . It represents the noise intensity coefficient of the short-term fading process, controlling the severity of random fluctuations and jumps. This represents the instantaneous power ultimately received by the user equipment in a cascaded channel. This indicates the base station's transmission power. This represents the independent normalized S that drives the short-term fading process. S Lévy process.

[0150] Through a multi-scale SDE coupling mechanism, this model comprehensively characterizes the transmit power in a mining environment. After large-scale path loss With small-scale short-term fading Then, the output power of the receiver. The system can dynamically sense and predict changes in power transmission paths in the mining environment, realizing a modeling framework with high robustness and adaptability to jumps, laying the foundation for subsequent power optimization strategies.

[0151] Step S2: Based on the constructed Lévy channel state equation, the generalized Riesz fractional derivative is introduced to establish the Hamilton-Jacobi-Bellman control equation with nonlocal diffusion term. Under the principle of dynamic programming, the value function evolution process is defined, and the fractional HJB (FHJB) optimization equation is derived for the calculation of transmit power control strategy under nonstationary interference conditions.

[0152] Step S201: Construct with S The system state equations are driven by the S Lévy process, and the objective function and value function are defined for minimization. The Bellman expansion is performed using the Dynamic Programming Principle (DPP) to obtain the recursive form.

[0153] Consider the following control system with state variables as follows: It is described by the following stochastic differential equation:

[0154]

[0155]

[0156] In the formula, Given the initial state, For the current moment, The termination time. Control variables. For permissible control, it adapts to natural filtering, satisfying the requirements of measurability and integrability. Drift term. Regarding state variables It is Lipschitz continuous, concerning control. It is measurable. Diffusion coefficient Let be the state-dependent noise intensity, satisfying the Lipschitz condition, and its determinant be uniformly bounded, satisfying . ;

[0157] Noise item The dimension is S The Lévy process is measured by the following Lévy measure:

[0158]

[0159] Where the normalization constant It is given by the following formula:

[0160]

[0161] The optimization objective of the system is to minimize the expected cost functional of the following form:

[0162]

[0163] In the formula, This refers to operating costs in general form. It is the terminal cost function;

[0164] Define value function From time and state The minimum cost that can be obtained by starting:

[0165]

[0166] For any Based on DPP, the following Bellman relation can be obtained:

[0167]

[0168] Step S202: Introduce fractional generators and generalized Riesz operators, and construct the FHJB equations using nonlocal Taylor expansion and Itô-Lévy formula.

[0169] Because it uses S S Lévy process (when) (When the noise has infinite variance), the traditional Itô formula and second-order expansion are no longer applicable. However, for symmetric, isotropic Lévy processes with zero drift and no Gaussian components, an infinitesimal generator can still be derived using the Lévy-Khintchine formula. In the case of uniform noise with constant intensity, this generator simplifies to a fractional Laplace operator. Also known as the Riesz fractional derivative;

[0170] In contrast, system states involve state-dependent noise figures. This leads to a non-uniform process. In this setup, there is a core. The standard Riesz operator is no longer valid. To accommodate spatial non-uniformity, a generalized Riesz fractional operator must be introduced, which can be defined in two equivalent forms:

[0171] From the kernel's perspective, it acts on Above, the function itself and its first and second partial derivatives are globally bounded, and its generalized fractional derivative is defined as:

[0172]

[0173] In the formula, Represents the principal value integral. The kernel function explicitly depends on... The expression is:

[0174]

[0175] From the Lévy-Khintchine perspective, the infinitesimal generator of the Lévy measure form is expressed as:

[0176]

[0177] In the formula, for Lévy's metric. Index function Used to adjust small jump terms to ensure integral convergence. In principle, and Equivalent descriptions of the same fractional operator;

[0178] Assumption Sufficiently smooth, and taking advantage of the independent incremental property of the Lévy process, for A nonlocal Taylor expansion is applied. This extension explains the jump effect through the Lévy metric-weighted integral term. Itô's formula is then applied to the Lévy process. The increment can be expressed as:

[0179]

[0180] In the formula, The nonlocal terms arising from Lévy jumps correspond to the generator or kernel definitions mentioned above. From the generator's perspective, Indicates the infinitesimal operator in Linear principal terms within the range;

[0181] Substituting the above expansion into the Bellman equation, rearranging the terms, and dividing by... ,have to:

[0182]

[0183] In the formula, when hour, ;

[0184] Introduce the Legendre transform and define Hamiltonian as follows:

[0185]

[0186] In the formula, ; This represents the drift term in a stochastic differential equation, which describes the deterministic evolution trend of the system state in the absence of random noise.

[0187] Finally, the FHJB equation is obtained:

[0188]

[0189] Terminal conditions are Therefore, a rigorous derivation and structured construction of the FHJB equations, from non-Gaussian Lévy jump modeling and the definition of nonlocal generators, was completed, providing a theoretical basis for solving the transmit power optimization strategy in subsequent steps.

[0190] Step S3: In a multi-base station, multi-user co-frequency communication system, a system cost function is constructed based on state-space evolution and interference structure. The signal-to-interference-plus-noise ratio (SINR) is used as the standard for judging communication quality (QoS). The FHJB equation is discretized and solved using numerical methods to obtain an executable adaptive transmit power control strategy that ensures both communication quality and reduces communication costs.

[0191] Step S301: Construct a multi-base station, multi-user co-frequency system and define a state evolution model.

[0192] To verify the effectiveness and robustness of the proposed fractional-order HJB power control optimization method in non-Gaussian interference channels, a system containing... Each base station (BS) and A co-frequency downlink communication system for individual user equipment (UE). In this system, each base station... For multiple user devices For broadcast signals, the instantaneous power received by user equipment is affected by both long-term and short-term fading. The specific modeling is as follows:

[0193] Long-term fading channel modeling, base station To users Long-term fading gain It follows the following OU-type stochastic differential equation:

[0194]

[0195] In the formula, S represents S Lévy's jump process, Control the intensity of jump disturbances.

[0196] Short-term fading and receiver power modeling, user equipment From base station Received instantaneous power The dynamic evolution is as follows:

[0197]

[0198] In the formula, Indicates base station In control strategy The transmission power under the action, This refers to the effective transmission power after a long period of attenuation.

[0199] The two state variables in the above process can be combined into a joint state vector:

[0200]

[0201] Its corresponding state space SDE is:

[0202]

[0203] In the formula, ;

[0204] Step S302: Define a channel performance evaluation method using SINR and construct an objective function.

[0205] The signal-to-interference-plus-noise ratio (SINR) of the received signal is used as an evaluation metric for base stations. For user equipment The core metrics for Quality of Service (QoS) are defined as follows:

[0206]

[0207] In the formula, For noise power, For other base stations For user equipment The resulting interference power.

[0208] To formalize the control objective, a phased cost function is defined to balance communication performance and power efficiency. This formula comprises three key components: (i) a rate utility term that rewards higher SINR to improve throughput; (ii) a penalty clause that prevents quality of service (QoS) violations when SINR falls below a predefined threshold; and (iii) a power cost term that penalizes excessive transmission power. A cost function for mining scenarios is constructed based on SINR. Taking into account communication speed, QoS penalty for breach of contract, and energy consumption cost, the design is as follows:

[0209]

[0210] In the formula, Indicates base station Transmit power control, Including base stations Control of all other base stations, It is the penalty intensity coefficient. Indicates if , then Otherwise , This indicates the cost of controlling transmission power consumption.

[0211] This method establishes a quantitative trade-off standard between QoS and power consumption in practical mine communication from a system perspective;

[0212] Step S303: Solve the FHJB equations using the numerical discretization method to generate an executable adaptive power control strategy.

[0213] Based on the established state evolution and cost function, the FHJB governing equations with generalized Riesz space derivatives constructed in step S203 are used as the optimization framework, in the following form:

[0214]

[0215] The equation is solved numerically using an explicit grid discretization and value iteration algorithm, specifically including: solving the state space... Establish a grid; approximate the nonlocal derivative using the numerical integral form of the generalized Riesz operator; iteratively update the value function at each time step. For each grid point, enumerate the control strategy. And calculate the minimum cost to obtain optimal control;

[0216] The final output control strategy is:

[0217]

[0218] This strategy demonstrates stability and energy efficiency in high-interference, non-Gaussian, and non-stationary communication environments such as mines. It can adjust the base station transmit power in real time according to the channel status, meet QoS requirements, and significantly reduce communication energy consumption.

[0219] Example:

[0220] The proposed power control optimization strategy is evaluated through numerical simulations using the Monte Carlo method under multiple random channel states. Specifically, a typical multi-cell co-channel communication scenario is constructed in the simulation, including... Each base station and The system features a one-to-one connection structure for each user equipment, with all base stations operating in the same frequency band and experiencing mutual interference. The system channel state consists of two parts: long-term fading and short-term fading, each satisfying the following model:

[0221] Non-Gaussian long-term fading: using S S Lévy jump process-driven OU stochastic differential equation, where the stability exponent is The jump intensity is The average power factor is set as the diagonal path. and off-diagonal paths .

[0222] Non-Gaussian short-term fading: The short-term fading at the user equipment receiver is modeled as a generalized CIR process, with the hopping perturbation strength being... The response time scale parameter is Truncation constant of the disturbance term .

[0223] The system employs a fractional-order Hamilton-Jacobi-Bellman (FHJB) control framework, aiming to minimize the communication cost function while balancing communication quality (SINR) and energy consumption. To achieve this, dynamic programming principles are utilized, and a value iteration algorithm is employed to discretize and solve the FHJB equations in the state space. The iteration cycle is 100 rounds, and in each round, the control actions are... Enumeration from A discrete set with a step size of SINR threshold value .

[0224] Figure 1 illustrates the evolution of the transmit power trajectory and control actions of each base station in the initial state, after 50 iterations, and after 100 iterations. In the initial stage, the control actions are selected almost randomly, resulting in drastic fluctuations in power levels and a lack of coordination among them, leading to low energy utilization efficiency and strong interference.

[0225] Figure 1(b) shows that by the 50th iteration, each base station began to proactively adjust its transmit power to cope with local fading and interference from other base stations. At this point, the power trajectory showed significantly fewer abrupt fluctuations compared to the initial stage, and the control actions exhibited a more structured trend. Notably, a dynamic pattern of "competition and cooperation" gradually emerged: when one base station increased its power, other base stations often correspondingly decreased their power. This behavior reflects an implicit cooperative learning process driven by a dynamic programming mechanism. Through iterative optimization, base stations gradually learned to avoid simultaneous high-power transmission, thereby effectively mitigating the SINR degradation problem caused by mutual interference.

[0226] Figure 1(c) shows the state after 100 iterations, at which point the control strategy has essentially converged. The transmit power trajectory tends to stabilize and exhibits a bounded fluctuation range, while also demonstrating a clear peak-valley alternation coordination characteristic. This indicates that each base station has been able to effectively absorb information from multi-step channel dynamics and interference feedback, and the control actions consistently show regularity over time, aiming to minimize power costs while ensuring that the SINR remains above the threshold. This type of power balancing behavior reflects a distributed balancing strategy, where each base station adaptively adjusts its transmit strategy based on the local channel state and the interference levels generated by other base stations, thereby achieving global coordinated control of the network.

[0227] Figure 2 Demonstrates value functions The cost function gradually decreases with increasing iteration count. Since the cost function consists of multiple non-negative cost terms, including SINR interruption penalties, power consumption costs, and fairness violation penalties, the goal of the optimal control strategy is to minimize the cumulative cost while ensuring communication quality. In the backward iteration of dynamic programming, the value function at any given time represents the expected total cost from that time onwards. As time gradually approaches the terminal time... As the remaining controllable time of the system decreases, the impact of the current action on future performance also weakens. Therefore, the value function naturally converges to zero at the terminal time and satisfies the terminal boundary conditions. This reflects the physical boundary meaning that there is no further accumulative cost.

[0228] Figure 3 and Figure 4 The SINR evolution trajectory of each user equipment after dynamic programming optimization is shown. Due to the influence of hop-driven Lévy fading on the channel process, there are abrupt changes in signal quality in the early stages, manifested as sudden drops or rises. This discontinuous change reflects the S The proposed optimized control strategy effectively models non-Gaussian impulse interference and heavy-tailed fading environments. Despite this, the strategy can adjust transmit power in real time, successfully suppressing the severe disturbances caused by channel transitions. As the number of iterations increases from 50 to 100, the SINR curves of all user equipment gradually stabilize, with significantly reduced volatility. Even in the presence of heavy-tailed interference and non-Gaussian channels, the SINR values ​​of all user equipment remain stably within the set threshold. The above fully verifies the stability and robustness of the proposed control method.

[0229] The results show that even under non-Gaussian and non-stationary conditions, the system can still stably converge to the optimal value function and output a dynamically adjusted transmit power trajectory. The SINR of each user equipment remains above the preset threshold, verifying the effectiveness and engineering applicability of the wireless communication power optimization method based on fractional-order control theory proposed in this invention under non-Gaussian heavy-tailed fading environments.

Claims

1. A wireless communication power control method based on fractional calculus theory, characterized in that, Includes the following steps: Step 1: Construct a non-Gaussian multi-scale channel model suitable for wireless environments in mining areas by introducing symmetry. - Stable (S) The S) Lévy process characterizes the transition and heavy-tailed characteristics in the channel, comprehensively reflecting the complex dynamics of long-term and short-term fading. Step 2: Based on the channel model, an optimization control framework with fractional-order nonlocal diffusion operators is established, and the fractional-order Hamilton-Jacobi-Bellman (FHJB) equation is proposed to model the transmit power control problem. Step 3: In the scenario of multi-base station and multi-user co-frequency communication, construct system performance indicators and solve control equations to obtain a class of adaptive power scheduling strategies, thereby achieving joint optimization of communication quality and energy consumption cost.

2. The wireless communication power control method based on fractional calculus theory according to claim 1, characterized in that, Step 1 specifically includes: Step 1-1: Construct a non-Gaussian long-term fading model; Define long-term decay Satisfy the following by S S-Lévy process-driven stochastic differential equations (SDEs): ； In the formula, The mean recovery intensity, Control the location of the stable point. For noise intensity, For S S Lévy process, Initial conditions This represents the initial state of the fading process; Step 1-2: Construct a non-Gaussian short-term fading channel model; Definition of the first In-phase components of the path Orthogonal components The evolutionary form is as follows: ； ； In the formula, The damping coefficient is... Control the intensity of the jump. and For mutually independent S S Lévy process; initial value and The initial state is defined; The instantaneous power of each path is thus defined. With signal envelope for: ； ； Substituting Itô's lemma and considering Lévy jumps, we get The dynamic evolution equation: ； In the formula, Indicates the process in time Left limit at the point, capture The value before any potential jump occurs; These represent the in-phase and quadrature components of the signal at a specific time. The amplitude of the jump; for processing To address the problem of non-integrability of the squared terms of time jumps, a small jump truncation mechanism is introduced, with a truncation amplitude of... The Lévy-Itô decomposition of the jump terms is performed using the compensated Poisson measure, ultimately yielding the expected representation of the cumulative squared jump terms: ； In the formula, The deterministic second-order moment contributions of all Lévy jumps were aggregated, thus fixing the drift intensity; process It is a square-integrable martingale with zero expectation, capturing the cumulative square jump in Random fluctuations around a specified mean; where, and Corresponding to and Compensated Poisson random measure; This represents the magnitude of the jump when a Lévy process undergoes a transition at a certain moment. This represents the cutoff threshold used to distinguish between large and small transitions; Define the normalized Lévy process: ； The instantaneous power process is represented as a generalized Cox-Ingersoll-Ross, i.e., a CIR-type SDE: ； in, Represents a normalized symmetry Stable semimartingar process; Steps 1-3: Construct a complete channel transmission process through cascading; Considering transmit power control input Treating short-term fading as an input-driven output power process By combining the long-term and short-term models, the following complete cascaded channel model is obtained: ； ； In the formula, This indicates the avoidance of singularities and reflects the low power noise floor; A fast timescale representing short-term decline relative to long-term changes; It represents the noise intensity coefficient of the short-term fading process, controlling the severity of random fluctuations and jumps; This represents the instantaneous power ultimately received by the user equipment in a cascaded channel. Indicates the base station's transmission power; This represents the independent normalized S that drives the short-term fading process. S Lévy process.

3. The wireless communication power control method based on fractional calculus theory according to claim 2, characterized in that, Step 2 specifically includes: Step 2-1: Construct the system state equation, objective function, and value function, and perform Bellman expansion using the dynamic programming principle (DPP). Consider the following control system with state variables as follows: It is described by the following stochastic differential equation: ； ； In the formula, Given the initial state, For the current moment, Termination time; control variables For permissible control; diffusion coefficient The noise intensity is state-dependent. This represents the set of all permissible controls, where the control strategies must satisfy adaptability and measurability. Noise item The dimension is S The Lévy process is measured by the following Lévy measure: ； in Indicates a single jump during the Lévy process 3D jump magnitude vector, normalization constant It is given by the following formula: ； in, Represents the gamma function; The optimization objective of the system is to minimize the expected cost functional of the following form: ； In the formula, This refers to operating costs in general form. It is the terminal cost function; Define value function From time and state The minimum cost that can be obtained by starting: ； For any Based on DPP, the following Bellman relation can be obtained: ； in, Represents the mathematical expectation; Step 2-2: Construct the FHJB governing equations; To accommodate spatial non-uniformity, a generalized Riesz fractional operator is introduced, which is defined in two equivalent forms: Nuclear form: acting on Above, the function itself and its first and second partial derivatives are globally bounded, and its generalized fractional derivative is defined as: ； In the formula, Represents the principal value integral; the kernel function explicitly depends on... The expression is: ； in, Representation matrix The determinant of; Lévy-Khintchine form: The infinitesimal generator of the Lévy measure form is expressed as: ； In the formula, for Lévy metric; index function Used to adjust small jump terms to ensure integral convergence; The two operator forms mentioned above are equivalent; Apply Itô's formula to the Lévy process. The increment can be expressed as: ； In the formula, The nonlocal terms resulting from Lévy jumps correspond to the generator or kernel definitions above; Substituting the above expansion into the Bellman equation, rearranging the terms, and dividing by... ,have to: ； In the formula, when hour, ; Represents a standard infinitesimal; Introduce the Legendre transform and define Hamiltonian as follows: ； In the formula, ; This represents the drift term in a stochastic differential equation; Finally, the FHJB equation is obtained: ； Terminal conditions are .

4. The wireless communication power control method based on fractional calculus theory according to claim 3, characterized in that, Step 3 specifically involves: Step 3-1: Construct a multi-base station, multi-user co-frequency system and define a state evolution model; Build a containing Each base station BS and A co-frequency downlink communication system for user equipment (UE) This indicates the total number of base stations in the system. Indicates the total number of user devices; In this system, each base station For multiple user devices For broadcast signals, the instantaneous power received by user equipment is affected by both long-term and short-term fading; the specific modeling is as follows: Long-term decay: base station To user equipment Long-term fading gain It follows the following OU-type stochastic differential equation: ； In the formula, S represents S Lévy: The jumping process; Short-term fading and receive power: User equipment From base station Received instantaneous power The dynamic evolution is as follows: ； In the formula, Indicates base station In control strategy The transmission power under the action, That is, the effective transmission power after long-term attenuation; This represents the independent normalized S that drives the short-term fading process. S Lévy process; The noise intensity coefficient represents the noise intensity during short-term fading and controls the magnitude of random fluctuations. The two state variables in the above process are combined to form a joint state vector: ； Its corresponding state space SDE is: ； In the formula, ; Step 3-2: Define a channel performance evaluation method using SINR and construct an objective function; The signal-to-interference-plus-noise ratio (SINR) of the received signal is used as an evaluation criterion for base stations. For user equipment The core metrics of QoS (Quality of Service) for communication are defined as follows: ； In the formula, For noise power, For other base stations For users The resulting interference power; Constructing a cost function for mining scenarios based on SINR Taking into account communication speed, QoS penalty for breach of contract, and energy consumption cost, the design is as follows: ； In the formula, Indicates base station Transmit power control, Including base stations Control of all other base stations, It is the penalty intensity coefficient. Indicates if , then Otherwise , This indicates the cost of controlling transmission power consumption; This represents a predefined SINR threshold used to determine whether QoS standards are met. Indicates the base station's transmission power; Step 3-3: Solve the FHJB equations using the numerical discretization method to obtain the power control strategy; Based on the established state evolution and cost functions, the FHJB governing equations with generalized Riesz space derivatives constructed in steps 2-3 are used as the optimization framework, in the following form: ； The equation is solved numerically using an explicit grid discretization and value iteration algorithm, specifically including: solving the state space... Establish a grid; approximate the nonlocal derivative using the numerical integral form of the generalized Riesz operator; iteratively update the value function at each time step. For each grid point, enumerate the control strategy. And calculate the minimum cost to obtain optimal control; The final output control strategy is: 。 5. An electronic device, characterized in that, include: Processor and memory; The memory is used to store a computer program, and the processor is used to execute the computer program stored in the memory to cause the electronic device to perform the method as described in any one of claims 1 to 4.

6. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method as described in any one of claims 1 to 4.

7. A chip, characterized in that, include: A processor for retrieving and running a computer program from memory, causing a device on which the chip is mounted to perform the method as described in any one of claims 1 to 4.

8. A computer program product, characterized in that, The computer program product includes a computer storage medium storing a computer program, the computer program including instructions executable by at least one processor, which, when executed by the at least one processor, implement the method as described in any one of claims 1 to 4.

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