A power and bit rate joint scheduling method to minimize convergence rate and energy consumption

Abstract

A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption comprises: designing a switching control unit including a prediction controller and a feedback controller for a Markov jump system affected by data packet loss and quantization coding; designing a time-varying quantization coding rule and a gradient value of a scalar function for different data packet loss situations; obtaining an explicit expression for the time-varying convergence rate based on a correlation function of the scalar function; proposing a concept of exponential convergence in the sense of the mean value that depends on the time-varying convergence rate, and setting a quadratic cost function that depends on the time-varying convergence rate and transmission energy consumption; introducing a constant that satisfies a restricted inequality to obtain a feasible power set; iteratively obtaining a power and bit rate joint scheduling set that ensures exponential convergence in the sense of the system mean and minimizes the quadratic cost function, and determining the time-varying quantization level at each moment; while ensuring the stable operation of the Markov jump system, the present invention improves the convergence speed of the system and reduces the energy consumption of the sensor.

Description

Technical Field

[0001] The present invention relates to the technical field of transmission energy consumption control and bit rate scheduling, and in particular to a power and bit rate joint scheduling method for minimizing convergence rate and energy consumption. Background Art

[0002] Markov jump systems have attracted widespread attention from scholars due to their wide applications in robotic arms, traffic flow control systems, aircraft systems, etc. In order to achieve remote monitoring of the system, sensors need to transmit the collected data through wireless networks. Considering that the power supply of sensors in the monitoring system is always provided by batteries, the sensing, processing and communication information of the sensors are supported by limited energy. In addition, in large-scale remote monitoring systems, due to the large amount of sensor data, the number of bits that can be used by each sensor is limited. Limited transmission power and limited bit rate may lead to unreliable communication links, thereby causing information loss in a random manner. The present invention aims to seek the optimal power and bit scheduling method under a given time-varying channel gain to improve the convergence speed and minimize energy consumption while ensuring the stability of the closed-loop system.

[0003] The main results of the joint scheduling of transmission power and bit rate are as follows: Quevedo, DE, Ahlen, A., Ostergaard, J. (2010). Energy efficient state estimation with wireless sensors through the use of predictive power control and coding, IEEE Transactions on Signal Processing, 58(9), 4811-4823 (Quevedo, DE, Ahlen, A., Ostergaard, J. (2010). Energy efficient state estimation with wireless sensors through the use of predictive power control and coding, IEEE Transactions on Signal Processing, 58(9), 4811-4823) and Quevedo, DE, Ostergaard, J., Ahlen, A. (2014). Power control and coding formulation for state estimation with wireless sensors, IEEE Transactions on Control Systems Technology, 22(2), 413-427(Quevedo, DE, Ostergaard, J., Ahlen, A. (2014). Power control and coding for wireless sensor state estimation, IEEE Transactions on Control Systems Technology, 22(2), 413-427), Reference 3: Zhou, J., Luo, Y., Liu, Y., Yang, W. (2023) Eavesdropping strategies for remote stateestimation under communication constraints, IEEE Transactions on Information Forensics and Security, 18, 2250-2261(Zhou, J., Luo, Y., Liu, Y., Yang, W. (2023). Decryption strategies for remote state estimation under communication constraints, IEEE Transactions on Information Forensics and Security, 18, 2250-2261).To balance energy consumption and state estimation accuracy, Reference 1 developed a predictive control method to ensure the power and bit rate used by each node online. Reference 2 studied the impact of power control and coding on the Kalman filter, implementing a decision-making process to balance energy consumption and state estimation accuracy. Reference 3 proposed a decryption scheduling scheme to minimize estimation error. It can be seen that the aforementioned articles design joint scheduling strategies for general systems from the perspective of open-loop system performance (estimation accuracy). To date, there have been no reports on joint scheduling of Markov jump systems from the perspective of closed-loop system performance (convergence speed). However, robotic arms, traffic flow control systems, aircraft systems, etc. are all modeled as closed-loop Markov jump systems. Therefore, it is crucial to design a joint scheduling method for transmission power and bit rate, using convergence speed and transmission energy consumption as performance indicators. Summary of the Invention

[0004] The purpose of the present invention is to provide a method for jointly scheduling transmission power and transmission bit rate that can minimize the time-varying convergence rate (to improve the convergence speed) and transmission energy consumption (to save battery power), thereby ensuring the stable operation of the Markov jump system while improving the system's convergence speed and reducing the consumption of transmission energy.

[0005] To achieve the above objectives, the present invention adopts the following technical solution: a power and bit rate joint scheduling method that minimizes convergence rate and energy consumption, comprising the following steps:

[0006] Step 1: Design a switching control unit consisting of a predictive controller and a feedback controller for a Markov jump system affected by data packet loss and quantization coding. The Markov jump system is described as follows:

[0007] x(k+1)=A σ(k) x(k)+B σ(k) u(k)

[0008] in are the system state and control input respectively, (A σ(k) ,B σ(k) ) is the system matrix at time k, represents the system mode at time k, satisfying:

[0009]

[0010] Step 2: Design a time-varying quantization encoding rule and a gradient value of a scalar function that depends on the time-varying quantization level to be timed for different data packet loss situations.

[0011] Step 3: Based on the scalar function at time 0 and Correlation function of time scalar function Get time-varying convergence rate Explicit expressions of ;

[0012] Step 4: Propose the concept of exponential convergence in the mean sense that depends on the time-varying convergence rate, and set a quadratic cost function that depends on the time-varying convergence rate and transmission energy consumption;

[0013] Step 5: Introduce a constant χ∈(0,1) that satisfies the restricted inequality and obtain the feasible power set that depends on the constant;

[0014] Step 6: Iterate to obtain a power and bit rate joint scheduling set that guarantees exponential convergence in the sense of the system mean and minimizes the quadratic cost function; according to the functional relationship between the bit rate b(k) and the quantization level N(k):

[0015]

[0016] The time-varying quantization level N(k) at each moment is determined offline.

[0017] Furthermore, the switching control unit including the prediction controller and the feedback controller in step 1 is:

[0018]

[0019] Where u(k) is the output of the switching control unit, and its value is based on the data transmission packet loss situation (i.e. the value of θ(k)) in the feedback control input u f (k) and the predicted control input Switch between K σc(k) and are dependent on the controller mode σ c (k) feedback control gain and predictive control gain;

[0020] c in the above formula k is a quantitative value, and its acquisition and transmission rules are: at any time Defining a polyhedron in and E k S k The center and radius of k ,Right now The quantizer converts the polyhedron S k Divide into N(k) n sub-regions, each with N(k) dimensions (where n is the dimension of the Markov jump system and N(k) is the time-varying quantization level to be determined); assuming An odd number to ensure is one of the quantized values; the center of the subregion containing x(k) is defined as the quantized value of x(k), denoted as c k ; Number the sub-regions i in a specific wayk ∈{1,2,...,N(k) n}; The encoder record contains c k The number of the box is transmitted to the controller together with σ(k); it is assumed that the decoder knows how the boxes are numbered; if the transmission is successful, the decoder can decode the quantized value c k , the controller uses c k and σ(k) to design the control algorithm;

[0021] In the above formula The moment of the most recent successful transmission The predicted state at time k satisfies:

[0022]

[0023] and boundary conditions The modal The system matrix.

[0024] Furthermore, the time-varying quantization coding rule in step 2 that depends on the time-varying quantization level to be determined is:

[0025]

[0026] Among them (A σ(k) ,B σ(k) )and The system modes are σ(k) and The system matrix of

[0027]

[0028] Furthermore, the correlation function in step 3 for:

[0029]

[0030] in

[0031]

[0032] And φ(i) is the probability of successful data transmission at time i, And π pp ,π pq are the probabilities of mode p turning into mode p, and mode p turning into mode q. r1(k), r2(k), r3, and r4 are the gradient values ​​of the scalar function under different packet loss and switching conditions at time k:

[0033] 1) The value of r1(k) is:

[0034]

[0035] Among them G p From the G in the time-varying encoding rule in step 2 σ(k) ,σ(k)=pdetermine,P p ,Q p To satisfy A positive definite matrix, ζ1 is any given positive constant, function λ min (·),λ max (·) represent the minimum and maximum eigenvalues ​​of the matrix, δ p In order to make r 1p (k) < 1, a sufficiently large positive constant, Λ p :=‖A p ‖and

[0036] 2) The value of r2(k) is:

[0037]

[0038] If θ(k)=1,

[0039] Where ζ2 is any given positive constant;

[0040] 3) The value of r3 is:

[0041]

[0042] in From the time-varying coding rules in step 2 Sure;

[0043] 4) The value of r4 is:

[0044]

[0045] Furthermore, the time-varying convergence rate in step 3 is The explicit expression for

[0046] Furthermore, the concept of exponential convergence in the sense of the mean of the time-varying convergence rate in step 4 is: if there is a constant and time-varying convergence rate Make

[0047]

[0048] The system is said to be exponentially convergent in the mean sense.

[0049] Furthermore, the quadratic cost function in step 4 that depends on the time-varying convergence rate and transmission energy consumption is:

[0050]

[0051] Where M is the prediction step size, α, β are given weighting coefficients, is the energy consumption function that depends on the bit rate b(l) and the transmission power ρ(l), and is defined as:

[0052]

[0053] and is the process cost, and κ is the channel bit rate.

[0054] Furthermore, the inequality defined in step 5 is:

[0055]

[0056] in

[0057] Furthermore, the feasible power set in step 5 is:

[0058]

[0059] Furthermore, obtaining the power and bit rate joint scheduling set in step 6 includes the following steps:

[0060] Step 6.1: Get the scheduling set at the initial time, that is, the optimal bit rate at time 0 and optimal power

[0061] 1) Arbitrary selection of constants Where B is the set of available bit rates; in To satisfy The probability of successful data transmission when φ(0) is the smallest positive integer of and For all elements in w (0),M w (0),Δ(0), Where r1(0) and r2(0) represent the gradient values ​​of the scalar function r1(k) and r2(k) when k=0, respectively. w (0),M w (0), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value when k-1=0;

[0062] 2) Similarly, for the available bit rate set B and the feasible power set at time i For all elements in , calculate r1(i), r2(i), Φ w (i),M w (i),Δ(i), Where r1(i) and r2(i) represent the gradient values ​​of the scalar function r1(k) and r2(k) when k=i, respectively. w (i),M w (i),Δ(i), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value when k-1=i;

[0063] 3) Get the minimized J M (0)

[0064] 4) For the current and Calculate φ(0). If φ(0) satisfies Go to step 6.2; otherwise, if φ(0) does not satisfy Let J M (0) = h, where h is any given sufficiently large positive number, and return to step 6.1-3);

[0065] Step 6.2: Get k, The scheduling set at time k, that is, the optimal bit rate at time k and optimal power

[0066] 1) For β and For all elements in , calculate r1(k), r2(k), Φ w (k),M w (k),Δ(k),

[0067] 2) For β and For all elements in , calculate r1(k+i),r2(k+i),Φ w (k+i),M w (k+i),Δ(k+i), Where r1(k+i) and r2(k+i) represent the gradient values ​​of the scalar function r1(k) and r2(k) at k+i, respectively. w (k+i),M w (k+i),Δ(k+i), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value at k+i;

[0068] 3) Get the minimized J M (k)

[0069] 4) For the current optimal power scheduling value and the optimal bit rate scheduling value Calculate φ(k); if φ(k) satisfies Then let k=k+1 and go to step 6.2-1); otherwise, if φ(k) does not satisfy Let J M (k) = h, return to step 6.2-3).

[0070] By adopting the above technical solution, the present invention has the following technical effects:

[0071] (1) A control unit consisting of a feedback controller and a predictive controller is proposed, which can improve system performance while simplifying the design of the predictive controller.

[0072] (2) An explicit expression for the time-varying convergence rate is obtained; a more intuitive quadratic cost function is defined based on the time-varying convergence rate, achieving the synergy and trade-off between convergence speed and energy consumption.

[0073] (3) A power and bit rate joint scheduling method was designed; the key constant χ was introduced to ensure that the time-varying convergence rate is less than 1, thus guaranteeing the feasibility and optimality of the power and bit rate scheduling results. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 An abstract model diagram of a double-capacity water tank provided in an embodiment of the present invention;

[0075] Figure 2 A system block diagram of a power and bit rate joint scheduling method for minimizing convergence rate and energy consumption provided by an embodiment of the present invention;

[0076] Figure 3 Optimal power scheduling value diagram provided for implementation of the present invention;

[0077] Figure 4 A performance comparison chart under dispatching power and constant power provided for the implementation of the present invention. DETAILED DESCRIPTION

[0078] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0079] The present invention provides a power and bit rate joint scheduling method for minimizing convergence rate and energy consumption, which improves the convergence speed of the system and reduces the energy consumption of the sensor. Figure 1 This is an abstract model diagram of a double-capacity water tank provided in an embodiment of the present invention. Figure 2 A system block diagram of a power and bit rate joint scheduling method for minimizing convergence rate and energy consumption provided by an embodiment of the present invention.

[0080] This embodiment utilizes an existing dual-tank multi-parameter networked measurement and control system. This system uses KingView software to adjust the control algorithm to control process liquid levels. The dual-tank test bench includes a manually adjustable valve, a liquid level differential pressure sensor, a glass rotor flowmeter, a liquid flow sensor, a universal pressure sensor, a pressure gauge, a high-pressure water pump, a solenoid valve, and a multifunctional measurement and control platform. The two water tanks measure 27 x 22 x 42 cm, the water tank measures 110 x 22 x 32 cm, and the overall dimensions are 116 x 28 x 150 cm.

[0081] In a dual-tank test rig, fluid flows through two tanks, and the state variable is the reservoir level (relative to its nominal value—the dashed line). The flow rate between the two tanks is proportional to the difference in their levels. If the sampling interval is 0.5 seconds, the experimental system can be modeled as a discrete-time Markov jump system:

[0082] x(k+1)=A σ(k) x(k)+B σ(k) u(k),σ(k)∈{1,2}

[0083] in:

[0084]

[0085] In this embodiment, the transition probability between the two modes is set to π 11 =π 22 =-0.05,π 12 =π 21 =0.05, we can calculate The set of optional quantization levels (i.e. bit rates) is B = {4, 5, 6, 7}, so N =3; let ζ1 = 0.1, δ1 = 10, δ2 = 8, ζ2 = 0.2, then corresponding to the optional quantization level set Β = {4, 5, 6, 7}, r1 and r2 are r1∈[0.2664, 0.2413, 0.2515, 0.2682] and r2∈[3.6174, 3.6174, 3.6174, 3.6174] respectively; obviously, r1 =0.2664, r2=3.6174; direct calculation yields r3=1.9346, r4=5.9414, And we can choose χ as χ = 0.4, and we can easily get and

[0086] The specific steps of the power and bit rate joint scheduling algorithm of this embodiment are as follows:

[0087] Step 1: Design a switching control unit consisting of a predictive controller and a feedback controller for a Markov jump system affected by data packet loss and quantization coding:

[0088]

[0089] Where u(k) is the output of the switching control unit, and its value is based on the data transmission packet loss situation (i.e. the value of θ(k)) in the feedback control input u f (k) and the predicted control input Switch between, K1=[-1.5271,-1.5732], K2=[1.5732,1.5271] and are dependent on the controller mode σ c (k) feedback control gain and predictive control gain, c k is the quantitative value, The moment of the most recent successful transmission The predicted state at time k.

[0090] Step 2: Design a time-varying quantization encoding rule and a gradient value of a scalar function that depends on the time-varying quantization level to be determined for different data packet loss scenarios. The time-varying quantization rule is:

[0091]

[0092] in and E k+1 are the quantization center and radius of the quantization area, (A σ(k) ,B σ(k) )and The system modes are σ(k) and The system matrix of , N(k)∈{4,5,6,7} is the time-varying quantization level to be determined, and

[0093] Step 3: Based on the scalar function at time 0 and Correlation function of time scalar function Get time-varying convergence rate Explicit expression for . Correlation function for:

[0094]

[0095] in

[0096]

[0097] r1(k)∈[0.2664,0.2413,0.2515,0.2682], r2(k)∈[3.6174,3.6174,3.6174,3.6174], r3=1.9346, r4=5.9414 are the gradient values ​​of the scalar function under different packet loss and switching conditions, respectively. φ(i) is the probability of successful data transmission at time i.

[0098] Time-varying convergence rate The explicit expression for

[0099] Step 4: Propose the concept of exponential convergence in the sense of the mean value that depends on the time-varying convergence rate, and set a quadratic cost function that depends on the time-varying convergence rate and transmission energy consumption; the quadratic cost function is:

[0100]

[0101] Where M=5 is the prediction step, α=0.5, β=0.5 / 35 are the given weighting coefficients, is the energy consumption function that depends on the bit rate b(l) and the transmission power ρ(l).

[0102] Step 5: Introduce a constant value χ=0.4∈(0,1) that satisfies the constraint inequality and obtain the feasible power set that depends on the constant value. The constraint inequality is:

[0103]

[0104] in

[0105] Time 0 is defined as

[0106] The feasible power set for

[0107] Step 6: Iterate to obtain a power and bit rate joint scheduling set that guarantees exponential convergence in the sense of the system mean and minimizes the quadratic cost function; according to the functional relationship between the bit rate b(k) and the quantization level N(k):

[0108]

[0109] Offline determination of the time-varying quantization level N(k) at each moment; obtaining the power and bit rate joint scheduling set includes the following steps:

[0110] Step 6.1: Get the schedule set at the initial time:

[0111] 1) For a given and All elements of

[0112] r1(0)=0.2413,r2(0)=3.6174,

[0113] Φ0(0)=[0.0764,0.0602,0.0476,0.0378,0.0300],

[0114] M0(0)=8.0182,

[0115] Δ(0)=[0.3810,0.3868,0.3914,0.3952,0.3981],

[0116]

[0117] 2) Similarly, for the available bit rate sets B and For all elements in

[0118] 3) Get the minimized J M (0)

[0119] 4) For the current and Calculate φ(0)=0.9546. If φ(0) satisfies Go to step 6.2; otherwise, if φ(0) does not satisfy Let J M (0) = h = 1000, return to step 6.1-3);

[0120] Step 6.2: Get k, Schedule collection of time:

[0121] 1) For β and For all elements in , calculate r1(k), r2(k), Φ w (k),M w (k),Δ(k),

[0122] 2) For β and For all elements in , calculate r1(k+i),r2(k+i),Φw (k+i),M w (k+i),Δ(k+i), Where r1(k+i) and r2(k+i) represent the gradient values ​​of the scalar function r1(k) and r2(k) at k+i, respectively. w (k+i),M w (k+i),Δ(k+i), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value at k+i;

[0123] 3) Get the minimized J M (k)

[0124] 4) For the current optimal power scheduling value and the optimal bit rate scheduling value Calculate φ(k); if φ(k) satisfies Then let k=k+1 and go to step 6.2-1); otherwise, if φ(k) does not satisfy Let J M (k)=h=1000, return to step 6.2-3).

[0125] In this embodiment, according to the above scheduling algorithm, if the given time-varying channel gain is Figure 3 As shown in (a), the optimal bit rate scheduling value is Optimal power scheduling value like Figure 3 (b) shows that the scheduling method proposed by the present invention and the constant power The performance indicators obtained are compared with Figure 4 It can be seen that the scheduling method proposed in the present invention effectively reduces the performance index, thereby increasing the convergence speed of the system and reducing energy consumption.

[0126] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present invention.

Claims

1. A power and bit rate joint scheduling method that minimizes convergence rate and energy consumption, characterized in that: The following steps are involved: Step 1: Design a switching control unit consisting of a predictive controller and a feedback controller for a Markov jump system affected by data packet loss and quantization coding. The Markov jump system is described as follows: x(k+1)=A σ(k) x(k)+B σ(k) u(k) in are the system state and control input respectively, (A σ(k) ,B σ(k) ) is the system matrix at time k, represents the system mode at time k, satisfying: Among them, π pp ,π pq are the probabilities of mode p transforming to mode p and mode p transforming to mode q respectively; Step 2: Design a time-varying quantization encoding rule and a gradient value of a scalar function that depends on the time-varying quantization level to be timed for different data packet loss situations. Step 3: Based on the scalar function at time 0 and Correlation function of time scalar function Get time-varying convergence rate Explicit expressions of ; Step 4: Propose the concept of exponential convergence in the mean sense that depends on the time-varying convergence rate, and set a quadratic cost function that depends on the time-varying convergence rate and transmission energy consumption; Step 5: Introduce a constant χ∈(0,1) that satisfies the restricted inequality and obtain the feasible power set that depends on the constant; Step 6: Iteratively obtain a power and bit rate joint scheduling set that guarantees exponential convergence in the system mean sense and minimizes the quadratic cost function; determine the time-varying quantization level N(k) at each moment offline based on the functional relationship between bit rate and quantization level.

2. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 1, characterized in that: The switching control unit including the prediction controller and the feedback controller in step 1 is: Where u(k) is the output of the switching control unit, and its value depends on the data transmission packet loss situation, that is, the value of θ(k). f (k) and the predicted control input Switch between and are dependent on the controller mode σ c (k) feedback control gain and predictive control gain, c k is the quantitative value, The moment of the most recent successful transmission The predicted state at time k.

3. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 2, characterized in that: The time-varying quantization coding rule in step 2 that depends on the time-varying quantization level to be determined is: in and E k+1 are the quantization center and radius of the quantization area, (A σ(k) ,B σ(k) )and The system modes are σ(k) and The system matrix of , N(k) is the time-varying quantization level to be determined, and The scalar function at time k is: Among them, P σ(k) is a positive definite matrix that depends on σ(k), δ σ(k) is a sufficiently large positive constant that depends on σ(k).

4. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 3, characterized in that: The correlation function in step 3 for: in, r1(k), r2(k), r3, and r4 are the gradient values ​​of the scalar function under different packet loss and switching conditions at time k, and φ(i) is the probability of successful data transmission at time i.

5. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 4, characterized in that: The time-varying convergence rate in step 3 The explicit expression for is:

6. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 5, characterized in that: The concept of exponential convergence in the sense of the mean of the time-varying convergence rate in step 4 is: if there is a constant and time-varying convergence rate Make The system is said to be exponentially convergent in the mean sense.

7. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 6, characterized in that: The quadratic cost function in step 4 that depends on the time-varying convergence rate and transmission energy consumption is: Where M is the prediction step size, α, β are given weighting coefficients, is the energy consumption function that depends on the bit rate b(l) and the transmission power ρ(l).

8. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 7, characterized in that: The inequality defined in step 5 is: in 9. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 8, characterized in that: The feasible power set at time k in step 5 is:

10. A power and bit rate joint scheduling method for minimizing convergence rate and energy consumption according to claim 9, characterized in that: Obtaining the power and bit rate joint scheduling set in step 6 includes the following steps: Step 6.1: Get the scheduling set at the initial time, that is, the optimal bit rate at time 0 and optimal power 1) Arbitrary selection of constants Where B is the set of available bit rates; make in To satisfy The probability of successful data transmission when φ(0) is the smallest positive integer of and For all elements in w (0),M w (0),Δ(0), Where r1(0) and r2(0) represent the gradient values ​​of the scalar function r1(k) and r2(k) when k=0, respectively. w (0),M w (0),Δ(0), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value when k-1=0; 2) Similarly, for the available bit rate set B and the feasible power set at time i For all elements in , calculate r1(i), r2(i), Φ w (i),M w (i),Δ(i), Where r1(i) and r2(i) represent the gradient values ​​of the scalar function r1(k) and r2(k) when k=i, respectively. w (i),M w (i),Δ(i), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value when k-1=i; 3) Get the minimized J M (0) 4) For the current and Calculate φ(0); if φ(0) satisfies Then go to step 6.2; Otherwise, if φ(0) does not satisfy Let J M (0) = h, where h is any given sufficiently large positive number, and return to step 6.1-3); Step 6.2: Get The scheduling set at time k, that is, the optimal bit rate at time k and optimal power 1) For B and For all elements in , calculate r1(k), r2(k), Φ w (k),M w (k),Δ(k), 2) For B and For all elements in , calculate r1(k+i),r2(k+i),Φ w (k+i),M w (k+i), Where r1(k+i) and r2(k+i) represent the gradient values ​​of the scalar function r1(k) and r2(k) at k+i, respectively. w (k+i),M w (k+i),Δ(k+i), Represents Φ respectively w (k-1),M w (k-1),Δ(k-1), The value at k+i; 3) Get the minimized J M (k) 4) For the current optimal power scheduling value and the optimal bit rate scheduling value Calculate φ(k); If φ(k) satisfies Then let k=k+1 and go to step 6.2-1); Otherwise, if φ(k) does not satisfy Let J M (k) = h, return to step 6.2-3).

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