CN110429999A - Extensive MIMO detection method based on lp-Box ADMM algorithm - Google Patents
Extensive MIMO detection method based on lp-Box ADMM algorithm Download PDFInfo
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B17/00—Monitoring; Testing
- H04B17/30—Monitoring; Testing of propagation channels
- H04B17/309—Measuring or estimating channel quality parameters
- H04B17/336—Signal-to-interference ratio [SIR] or carrier-to-interference ratio [CIR]
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B17/00—Monitoring; Testing
- H04B17/30—Monitoring; Testing of propagation channels
- H04B17/391—Modelling the propagation channel
- H04B17/3912—Simulation models, e.g. distribution of spectral power density or received signal strength indicator [RSSI] for a given geographic region
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B7/00—Radio transmission systems, i.e. using radiation field
- H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
- H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
- H04B7/0413—MIMO systems
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B7/00—Radio transmission systems, i.e. using radiation field
- H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
- H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
- H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
- H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
- H04B7/0842—Weighted combining
- H04B7/0848—Joint weighting
- H04B7/0854—Joint weighting using error minimizing algorithms, e.g. minimum mean squared error [MMSE], "cross-correlation" or matrix inversion
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B7/00—Radio transmission systems, i.e. using radiation field
- H04B7/02—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas
- H04B7/04—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas
- H04B7/08—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station
- H04B7/0837—Diversity systems; Multi-antenna system, i.e. transmission or reception using multiple antennas using two or more spaced independent antennas at the receiving station using pre-detection combining
- H04B7/0842—Weighted combining
- H04B7/0848—Joint weighting
- H04B7/0857—Joint weighting using maximum ratio combining techniques, e.g. signal-to- interference ratio [SIR], received signal strenght indication [RSS]
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/0202—Channel estimation
- H04L25/024—Channel estimation channel estimation algorithms
- H04L25/025—Channel estimation channel estimation algorithms using least-mean-square [LMS] method
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03006—Arrangements for removing intersymbol interference
Abstract
The invention discloses a kind of extensive MIMO detection method based on lp-Box ADMM algorithm, mainly solve the problems, such as existing detection algorithm detection performance and computation complexity can not simultaneously close to it is optimal this.Its implementation is: 1. establish extensive MIMO detection system Optimized model；2. being replaced using box set and lp norm ball intersection of sets collection to Integer constrained characteristic condition；3. introducing two auxiliary vectors to decompose the Integer constrained characteristic condition of Optimized model；3. being Augmented Lagrangian Functions form by Optimized model transformed structure；4 application ADMM algorithms are updated and are solved to transmission signal vector in Augmented Lagrangian Functions and two auxiliary variable iteration；5. the solution value of pair transmission signal vector is determined to obtain testing result, extensive MIMO detection is completed.The present invention is able to achieve low complex degree, high performance signal detection, can be used in wireless communication system.
Description
Technical field
The invention belongs to fields of communication technology, further relate to extensive multiple-input and multiple-output MIMO detection method.It can
For in wireless communication system, realizing low complex degree, high performance signal detection.
Background technique
Extensive mimo system is based on traditional MIMO technology, by more days that increase the formation of dual-mode antenna number on a large scale
Linear array communication system, it is other than the wireless transmission advantage for having traditional MIMO, additionally it is possible to provide higher spectrum efficiency and
Energy efficiency greatlys improve power system capacity.It is fixed according to big number with the increase of number of antennas in extensive mimo system
Reason, progressive will be intended between transmission channel matrix column vector it is orthogonal, meanwhile, the influence of partial noise and decline will disappear
It loses, signal interference becomes easy elimination, and can obtain nearly optimal system performance, this needs the signal detection side haveing excellent performance
Method is realized.
Compared to mimo system, number of antennas and processing signal dimension are significantly increased in extensive mimo system, and solution space refers to
Several levels expand, the meter of the loose SDR scheduling algorithm of traditional MIMO detection method such as Maximum Likelihood Detection ML, globular decoding SD, positive semidefinite
Calculation complexity is excessively high, and least mean-square error MMSE, squeeze theorem ZF scheduling algorithm and optimal detection performance gap are excessive, all not applicable
In extensive mimo system signal detection.Such as likelihood of current main extensive MIMO detection method rises search LAS, searches at random
The detection of rope RS, reactive tabu search RTS, probabilistic data association PDA, Monte Carlo MCMC, confidence spread BP scheduling algorithm
Performance and computation complexity also can not all reach close optimal simultaneously.
Academic paper " the An efficient quasi-maximum- that Zhi-Quan Luo et al. is delivered at it
likelihood decoder for PSK signals”(Proceedings of International Conference
On Acoustics, Speech and Signal Processing, 2003, volume 6) in disclose a kind of detection side MIMO
Method, referred to as PSK detection algorithm.This method is a kind of positive semidefinite relaxation SDR detection method, it is based on the non-convex positive semidefinite pine of low-rank
It relaxes to solve the problems, such as Maximum Likelihood Detection, core is to carrying out coordinate descent after feasible set relaxation, which can be
Preferably detection performance is obtained under lower computation complexity, and as problem scale and signal-to-noise ratio change steadily.This method
Existing deficiency is can only to solve phase-shift keying (PSK) modulation problems.
Academic paper " the Modified Fast Recursive that Tsung-Hsien Liu et al. people delivers at it
Algorithm for Efficient MMSE-SIC Detection of the V-BLAST System”(IEEE
Transactions on Wireless Communications, 2008, volume 7, the 10th phase) in disclose a kind of MIMO
Detection method, referred to as MMSE-SIC algorithm.This method combines the low computation complexity of least mean-square error MMSE algorithm and string
The preferable detection performance of SIC algorithm is eliminated in row interference, can be applied in extensive mimo system.Deficiency existing for this method
It is that computation complexity is higher than MMSE method, there are also gaps away from optimal detection performance for detection performance.
Summary of the invention
It is a kind of based on lp-Box ADMM algorithm it is an object of the invention in view of the above shortcomings of the prior art, propose
Extensive MIMO detection method, with solve existing detection algorithm detection performance and computation complexity can not simultaneously close to it is optimal this
One problem.Signal detection performance may be implemented and be continuously improved as number of antennas increases, not only have lower multinomial multiple
Miscellaneous degree, and there is nearly optimal detection performance.
The technical scheme of the present invention is realized as follows:
One, technical principle
Academic paper " the Distributed Optimization and that Stephen Boyd et al. is delivered at it
Statistical Learning via the Alternating Direction Method of Multipliers”
It is summarized in (Foundations and Trends in Machine Learning, 2011, volume 3, the 1st phase) and proposes friendship
For direction multiplier method ADMM, which is a kind of Computational frame of solving optimization problem, it, will be big by composition decomposition process
Global issue be decomposed into it is multiple it is smaller, be easier to solve local subproblems, and by coordination subproblem solution obtain it is big
The solution of global issue, it is suitable for large-scale distributed optimization problems.
Academic paper " the lp-Box ADMM:A Versatile Framework that Baoyuan Wu et al. is delivered at it
for Integer Programming”(IEEE Transactions on Pattern Analysis and Machine
Intelligence, DOI:10.1109/TPAMI.2018.2845842) propose the general of a referred to as lp-Box ADMM
It solves integer and optimizes Computational frame, which replaces discrete constraint with equivalent and simple continuous constraint, then calculates using ADMM
Good characteristic of the method in continuous domain solves integer optimization problem.
From the perspective of optimum theory, extensive MIMO test problems are substantially an extensive secondary integer optimization
Problem can be subject to very good solution using lp-Box ADMM algorithm.
Two, technical solutions
According to above-mentioned principle, technical scheme is as follows:
(1) it according to the transmission signal vector x and received signal vector y in extensive MIMO detection system, establishes extensive
MIMO detection system Optimized model:
Wherein,H is channel matrix,
For additive Gaussian noise vector, x, y, H, n are real number, NtAnd NrRespectively transmission antenna number and receiving antenna number, | |
||2Expression takes two norms to operate, ()TIndicate transposition operation；
(2) the binary integer constraint condition for using x vector in continuous constraint condition equivalence alternate form<1>, i.e., with two companies
Sequel closesWithIntersection replace integer setIt is expressed as follows:
Wherein (0, ∞) p ∈,It is the 2N that element is 1tDimensional vector；It is a binary integer collection
It closes；It is a box continuous collection；Be one
In lp norm space, withCentered onFor (the 2N of radiust- 1) ball continuous collection is tieed up；||·||∞Expression takes nothing
Poor norm operation, | | | |pExpression takes p norm to operate；
(3) for the constraint condition in Optimized model<1>, two auxiliary vectors are introducedWithThe constraint condition<2>replaced in applying step (2), to the constraint condition of Optimized model<1>into
Row decomposes, and obtains the expression of Optimized model are as follows:
(4) formula<3>is configured to Augmented Lagrangian Functions form:
Wherein, λ1And λ2For the dual vector that dual variable is constituted, ρ1And ρ2For penalty term parameter；
(5) vector x, z are set1,z2,λ1,λ2Initial value be
(6) it by the minimum to formula<4>Augmented Lagrangian Functions, updates and solves two auxiliary vector z1And z2,
After k+1 iteration, the value solved is respectivelyWith
(7) it by the minimum to formula<4>Augmented Lagrangian Functions, updates and solves transmission signal vector x, in kth+1
After secondary iteration, updating solution to obtain the value of transmission signal vector x is xk+1:
Wherein, ()-1Representing matrix inversion operation,WithRespectively dual vector λ after kth time iteration1And λ2More
New value；
(8) using the dual vector λ in traditional gradient rise method newer<4>1And λ2, obtain after+1 iteration of kth
Dual vector updated valueWith
(9) stop condition that alternating iteration is arranged is the number of iterations 30 times；
(10) above-mentioned steps (6)-(8) are to transmission signal vector x and two auxiliary vector z1,z2In respectively constraint item
Solution procedure under part, stop condition of alternating iteration step (6)-(8) up to reaching step (9) setting, after the completion of calculating, obtains
To x, z1,z2The solution of these three vectors；
(11) solution of transmission signal vector x is further determined as binary integer vectorThe decision content of x is
Testing result, decision criteria are as follows:
Wherein, xiIndicate i-th of element variable in transmission signal vector x.
Compared with prior art, the present invention having the effect that
First, due to the present invention to extensive MIMO inspection optimization problem carried out constraint replacement, i.e., with two continuously about
The Integer constrained characteristic in former optimization problem is replaced in the intersection that constriction closes, and enables to effectively utilize ADMM algorithm, by objective function
It is decomposed into multiple subproblems and is subject to parallel processing, high dimensional data signal in extensive MIMO detection algorithm can be enable fine
Processing, significantly improve the detection performance of detection algorithm.
Second, since former optimization problem is non-convex nonlinear optimal problem, received so that many optimization methods do not have
Holding back property, the present invention uses lp-Box ADMM algorithm, wherein the outstanding convergence of multiplier method is utilized, after number of iterations ten times
It can restrain, overcome the excessively high problem of the existing extensive MIMO detection algorithm computation complexity haveing excellent performance, significantly reduce
The computation complexity of detection algorithm.
Detailed description of the invention
Fig. 1 is the extensive MIMO detection system structure chart that the present invention uses；
Fig. 2 is implementation flow chart of the invention；
Fig. 3 is the detection performance simulation comparison figure using the present invention and existing detection algorithm；
Fig. 4 is the computation complexity simulation comparison figure using the present invention and existing detection algorithm.
Specific embodiment
The embodiment of the present invention and effect are further described with reference to the accompanying drawing:
Referring to Fig.1, the extensive MIMO detection system that the present invention uses is point-to-point mimo channel, i.e. transmitting terminal and reception
It holds while being equipped with more antennas and communicated.Initial data carries out constellation mapping according to modulation system first, by serioparallel exchange
Parallel baseband signal is formed afterwards, is sent simultaneously from multiple and different antennas respectively after modulation；It is transmitted by wireless channel
Afterwards, the signal from different transmission antennas is received simultaneously by mutiple antennas, and each receiving branch is superimposed different noises, by solution
Multiple parallel baseband signals are generated after tune.
The present invention is based on lp-Box ADMM algorithms to carry out estimation recovery to data are received, and string is formed after parallel-serial conversion
Capable recovery data.
Referring to Fig. 2, steps are as follows for realization of the invention:
Step 1, extensive MIMO detection system Optimized model is established.
In extensive MIMO detection system, the plural form of received signal vector can be indicated are as follows:
yc=Hcxc+nc 1)
Wherein,Indicate NtThe plural form of × 1 dimension transmission signal vector,
Indicate the N obtained in receiving endrThe plural form of × 1 dimension received signal vector, HcFor Nr×NtTie up the plural shape of channel matrix
Formula,Indicate NrThe additivity complex-valued Gaussian noise vector of × 1 dimension, NtAnd NrRespectively transmission antenna number and
Receiving antenna number, ()TIndicate transposition operation；
Calculated for the ease of application detection algorithm, by complex vector located formula 1) it is converted into real number form of equal value:
Y=Hx+n 2)
Wherein, x is the real number form of transmission signal vector, and y is the real number form of received signal vector, and H is channel matrix
Real number form, n is the real number form of additive Gaussian noise vector； It indicates that () ties up integer set, and has:
Functional symbolWithIt respectively represents and takes real part and imaginary part is taken to operate；
According to above-mentioned analysis, extensive MIMO detection system Optimized model is established:
Wherein, | | | |2Expression takes two norms to operate, and takes in this exampleSNR is on each receiving antenna
Average signal-to-noise ratio, SNR value range be 0~20dB, A 2Nr×2NtTie up the pseudo random number of standardized normal distribution, Nt=128,
Nr=128.
Step 2, constraint condition is replaced.
Because ADMM algorithm can only handle continuous constraint, in order to apply the algorithm, present invention continuous constraint condition equivalence
Alternate form 3) in x vector binary integer constraint condition, i.e., with two continuous collectionsWithIntersection replace integer setIt is expressed as follows:
Wherein (0, ∞) p ∈,It is the 2N that element is 1tDimensional vector；It is a binary integer collection
It closes；It is a box continuous collection；Be one
In lp norm space, withCentered onFor (the 2N of radiust- 1) ball continuous collection is tieed up；||·||∞Expression takes nothing
Poor norm operation, | | | |pExpression takes p norm to operate, and takes p=2 in this example.
Step 3, constraint condition is decomposed.
In order to Optimized model 3) constraint condition solve respectively, do not changing formula 3) in Optimized model objective function feelings
Under condition, two auxiliary vectors are introducedWithIn applying step 2
The constraint condition 4 of replacement), to Optimized model 3) constraint condition decompose, obtain the expression of Optimized model are as follows:
Step 4, Augmented Lagrangian Functions are constructed.
In order to apply ADMM algorithm, by formula 5) it is constructed in Augmented Lagrangian Functions form:
Wherein, λ1And λ2For the dual vector that dual variable is constituted, ρ1And ρ2For penalty term parameter, ρ is taken in this example1=90,
ρ2=90.
Step 5, initial vector is set.
Vector x, z are set1,z2,λ1,λ2Initial value beIt takes in this example
Step 6, two auxiliary variable z are updated1,z2。
Using ADMM algorithm, by the Augmented Lagrangian Functions formula 6 constructed by step 4) minimum, update solve
Two auxiliary vector z1And z2Value, after+1 iteration of kth, the value solved is respectivelyWith
(6a) in+1 iteration of kth, by formula 6) in the first auxiliary vector z1Solve problems be converted into following form:
To formula 7) solution is updated, obtain the first auxiliary variable z1Value after+1 iteration of kth
Wherein,It indicates to Integer constrained characteristicAnd setThe projection of intersection, a indicate vectorIn
Either element variable.
(6b) in+1 iteration of kth, by formula 6) in the second auxiliary vector z2Solve problems be converted into following form:
To formula 9) solution is updated, obtain the second auxiliary variable z2Value after+1 iteration of kth
WhereinIt indicates to setProjection, b indicate vectorIn either element variable.
Step 7, received signal vector x is updated.
Using ADMM algorithm, by the Augmented Lagrangian Functions formula 6 constructed by step 4) minimum, update solve
Transmission signal vector x, after+1 iteration of kth, updating solution to obtain the value of transmission signal vector x is xk+1:
Wherein, ()-1Representing matrix inversion operation,WithRespectively dual vector λ after kth time iteration1And λ2More
New value.
Step 8, Lagrangian dual vector λ is updated1,λ2。
The Lagrangian formula 6 constructed by step 4 is updated using traditional gradient rise method) in dual vector λ1And λ2,
Obtain the dual vector updated value after+1 iteration of kthWith
Step 9, stop condition is set.
The present invention uses lp-Box ADMM algorithm, wherein the outstanding convergence of multiplier method is utilized, at number of iterations ten times
After can restrain, in this example be arranged alternating iteration stop condition be the number of iterations 30 times.
Step 10, x, z are completed1,z2It solves and calculates.
Above-mentioned steps 6-8 is to transmission signal vector x and two auxiliary vector z1,z2Asking under respective constraint condition
Solution preocess, stop condition of the alternating iteration step 6-8 up to reaching step 9 setting, after the completion of calculating, obtains x, z1,z2These three
The solution of vector.
Step 11, testing result is completed to determine.
The solution of transmission signal vector x is further determined as binary integer vectorThe decision content of x is to detect
As a result, decision criteria is as follows:
Wherein, xiIndicate i-th of element variable in transmission signal vector x.
Effect of the invention can be further illustrated by following emulation:
Emulation 1: Matlab R2017a simulation software is used, channel model is rayleigh fading channel, and modulation system is
BPSK, transmission antenna number and receiving antenna number are 128, and signal-to-noise ratio constant interval is 0~20dB.Using the present invention and now
Some PSK algorithms and MMSE-SIC algorithm carry out detection error rate BER with Signal to Noise Ratio (SNR) under the identical system environments
The performance simulation of variation, as a result as shown in figure 3, wherein " circle " data point curve indicates bit error rate performance curve of the invention,
" star " data point curve indicates that the bit error rate performance curve of PSK algorithm, " rectangular " data point curve indicate MMSE-SIC algorithm
Bit error rate performance curve.
By the simulation result of Fig. 3 as it can be seen that under the conditions of identical signal-to-noise ratio in extensive mimo system, detection of the invention
The bit error rate is lower compared to other algorithms, and when signal-to-noise ratio is 10dB, the bit error rate is 10-5, approached ideal optimal extensive
Mimo system signal detection performance shows that the present invention has nearly optimal detection performance.
Emulation 2: Matlab R2017a simulation software is used, channel model is rayleigh fading channel, and modulation system is
BPSK, transmission antenna number and receiving antenna number of variations section are 2~120, signal-to-noise ratio 12dB.Using the present invention and now
Some PSK algorithms and MMSE-SIC algorithm carry out computation complexity and change with number of antennas under the identical system environments
Emulation, as a result as shown in figure 4, wherein " circle " data point curve indicates computation complexity curve of the invention, " star " data
Point curve indicates that the computation complexity curve of PSK algorithm, " rectangular " data point curve indicate that the calculating of MMSE-SIC algorithm is complicated
It writes music line.
By the simulation result of Fig. 4 as it can be seen that method of the invention is after number of antennas is greater than 30, i.e., in extensive mimo system
In, the average operating time for carrying out a signal detection is less than other algorithms, shows that the present invention has lower computation complexity.
In summary simulation result is able to achieve low complex degree, high performance signal detection using the present invention, is a kind of effective
Extensive MIMO detection method.
Claims (2)
1. a kind of extensive MIMO detection method based on lp-Box ADMM algorithm, which is characterized in that include the following:
(1) according to the transmission signal vector x and received signal vector y in extensive MIMO detection system, extensive MIMO is established
Detection system Optimized model:
Wherein,H is channel matrix,
For additive Gaussian noise vector, x, y, H, n are real number, NtAnd NrRespectively transmission antenna number and receiving antenna number, | |
||2Expression takes two norms to operate, ()TIndicate transposition operation；
(2) the binary integer constraint condition for using x vector in continuous constraint condition equivalence alternate form<1>, i.e., with two continuums
It closesWithIntersection replace integer setIt is expressed as follows:
Wherein (0, ∞) p ∈,It is the 2N that element is 1tDimensional vector；It is a binary integer set；It is a box continuous collection；It is one in lp model
In number space, withCentered onFor (the 2N of radiust- 1) ball continuous collection is tieed up；||·||∞Expression takes infinite model
Number operation, | | | |pExpression takes p norm to operate；
(3) for the constraint condition in Optimized model<1>, two auxiliary vectors are introducedWithThe constraint condition<2>replaced in applying step (2), to the constraint condition of Optimized model<1>into
Row decomposes, and obtains the expression of Optimized model are as follows:
(4) formula<3>is configured to Augmented Lagrangian Functions form:
Wherein, λ1And λ2For the dual vector that dual variable is constituted, ρ1And ρ2For penalty term parameter；
(5) vector x, z are set1,z2,λ1,λ2Initial value be x0,
(6) it by the minimum to formula<4>Augmented Lagrangian Functions, updates and solves two auxiliary vector z1And z2, in kth+1
After secondary iteration, the value solved is respectivelyWith
(7) it by the minimum to formula<4>Augmented Lagrangian Functions, updates and solves transmission signal vector x, repeatedly at kth+1 time
Dai Hou, updating solution to obtain the value of transmission signal vector x is xk+1:
Wherein, ()-1Representing matrix inversion operation,WithRespectively dual vector λ after kth time iteration1And λ2Updated value；
(8) using the dual vector λ in traditional gradient rise method newer<4>1And λ2, obtain pair after+1 iteration of kth
Even vector updated valueWith
(9) stop condition that alternating iteration is arranged is the number of iterations 30 times；
(10) above-mentioned steps (6)-(8) are to transmission signal vector x and two auxiliary vector z1,z2Under respective constraint condition
Solution procedure, alternating iteration step (6)-(8) until reach step (9) setting stop condition, after the completion of calculating, obtain x,
z1,z2The solution of these three vectors；
(11) solution of transmission signal vector x is further determined as binary integer vectorThe decision content of x is to detect
As a result, decision criteria is as follows:
Wherein, xiIndicate i-th of element variable in transmission signal vector x.
2. solving two auxiliary vector z the method according to claim 1, wherein updating in (6)1And z2, in kth
+ 1 iteration, the value solved are respectivelyWithIt is accomplished by
(6a) in+1 iteration of kth, by the first auxiliary vector z in formula<4>1Solve problems be converted into following form:
Formula<8>are updated and are solved, the first auxiliary variable z is obtained1Value after+1 iteration of kth
Wherein,It indicates to Integer constrained characteristicAnd setThe projection of intersection, a indicate vectorIn any
Element variable；
(6b) in+1 iteration of kth, by the second auxiliary vector z in formula<4>2Solve problems be converted into following form:
Formula<10>are updated and are solved, the second auxiliary variable z is obtained2Value after+1 iteration of kth
WhereinIt indicates to setProjection, b indicate vectorIn either element variable.
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| CN113467405A (en) * | 2021-08-02 | 2021-10-01 | 福州大学 | Industrial 4.0 demand-driven distributed dynamic edge cloud intelligent manufacturing method and system |
| CN113900377A (en) * | 2021-10-19 | 2022-01-07 | 江南大学 | Point-to-point iterative learning minimum energy control method for double-rotor pneumatic system |
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