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Design method of multivariable PI controller for turboprop engine based on equivalent transfer function
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作者 Shancheng LI Yong WANG Haibo ZHANG 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2024年第9期237-260,共24页
Traditional centralized Proportional Integral(PI)controller design methods based on Equivalent Transfer Functions(ETFs)have poor decoupling effect in turboprop engines.In this paper,a centralized PI design method base... Traditional centralized Proportional Integral(PI)controller design methods based on Equivalent Transfer Functions(ETFs)have poor decoupling effect in turboprop engines.In this paper,a centralized PI design method based on dynamic imaginary matrix and equivalent transfer function is proposed.Firstly,a method for solving equivalent transfer functions based on the dynamic imaginary matrix is proposed,which adopts dynamic imaginary matrix to describe the dynamic characteristics of the system,and obtains the equivalent transfer function based on the dynamic imaginary matrix characteristics.Secondly,for the equivalent transfer function,a central-ized PI control gain is designed using the Taylor expansion method.Meanwhile,this paper further proves that the centralized PI design method proposed in this paper has integral stability.Consid-ering the impact of altitude and Mach number on turboprop engines,a linear feedforward control method based on the transfer function matrix is further proposed based on the centralized PI con-troller,and the stability of the entire comprehensive control method is proved.Finally,to ensure the safe and effective operation of the turboprop engine,a temperature and torque limiting protection controller is designed for the turboprop engine.Simulation results show that the centralized PI con-troller design method and linear feedforward control method proposed can effectively improve the control quality of turboprop engine control systems. 展开更多
关键词 multivariable control sys-tems Equivalent transfer function Disturbance compensation control Control parameter design Turboprop engine control
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HIGHER-ORDER MULTIVARIABLE EULER'S POLYNOMIALAND HIGHER-ORDER MULTIVARIABLEBERNOULLI'S POLYNOMIAL
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作者 刘国栋 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1998年第9期895-906,共12页
In this paper, the definitons of both higher-order multivariable Euler's numbersand polynomial. higher-order multivariable Bernoulli's numbers and polynomial aregiven and some of their important properties... In this paper, the definitons of both higher-order multivariable Euler's numbersand polynomial. higher-order multivariable Bernoulli's numbers and polynomial aregiven and some of their important properties are expounded. As a result, themathematical relationship between higher-order multivariable Euler's polynomial(numbers) and higher-order higher -order Bernoulli's polynomial (numbers) are thusobtained. 展开更多
关键词 higher-order multivariable Euler's numbers higher-ordermultivariable Euler's polynomial higher-order multivariableBernoulli's numbers higher-order multivariable Bernoulli'spolynomial
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Multivariable Fuzzy Predictive Control Based on the Modified CPN Model
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作者 郑怀林 陈维南 《Journal of Southeast University(English Edition)》 EI CAS 1998年第1期108-113,共6页
Through modifying the CPN model, a kind of multivariable fuzzy model is put forward, and the matching fuzzy multistep predictive control algorithm is deduced based on the model. The modified model works in a competiti... Through modifying the CPN model, a kind of multivariable fuzzy model is put forward, and the matching fuzzy multistep predictive control algorithm is deduced based on the model. The modified model works in a competitive output manner which results in its local representation property. While studying on line, only a few parameters need to be regulated. So the model has the merits of fast learning and on line self organizing modeling. The control algorithm is simple, adaptive and useful in multivariable and time delay systems. Applying the algorithm in a paper making system, simulation shows its good effect. 展开更多
关键词 modified CPN model fuzzy predictive control multivariable time delay systems
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A New Method for Constructing the Inversion of Multivariable Linear Systems
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作者 张新建 童丽 《Chinese Quarterly Journal of Mathematics》 CSCD 2001年第4期18-24,共7页
A new algorithm for constructing an inverse of a multivariable linear system is presented. This algorithm makes the constructing an inverse of the higher order matrices into searching for the equivalent normal form o... A new algorithm for constructing an inverse of a multivariable linear system is presented. This algorithm makes the constructing an inverse of the higher order matrices into searching for the equivalent normal form of the lower order matrices. Consequently, the calculation is more simple efficient and programmed than previous methods. Another result of the paper is that the lower reduced inverse system is obtained, by selecting special bases of the observable space of the original systems, it reveals the effect of the observability of the original systems on the order of the inverse systems. 展开更多
关键词 multivariable linear system inverse system orthogonal decomposition
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Multivariable Decoupling Predictive Control with Input Constraints and Its Application on Chemical Process 被引量:13
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作者 苏佰丽 陈增强 袁著祉 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2006年第2期216-222,共7页
A constrained decoupling (generalized predictive control) GPC algorithm is proposed for MIMO (malti-input multi-output) system. This algorithm takes account of all constraints of inputs and their increments. By solvin... A constrained decoupling (generalized predictive control) GPC algorithm is proposed for MIMO (malti-input multi-output) system. This algorithm takes account of all constraints of inputs and their increments. By solving matrix equations, the multi-step predictive decoupling controllers are realized. This algorithm need not solve Diophantine functions, and weakens the cross-coupling of the variables. At last the simulation results demon- strate the effectiveness of this proposed strategy. 展开更多
关键词 chemical process control multivariable system OPTIMIZATION predictive control input constraint
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Multivariable sales prediction for filling stations via GA improved BiLSTM 被引量:8
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作者 Shi-Yuan Pan Qi Liao Yong-Tu Liang 《Petroleum Science》 SCIE CAS CSCD 2022年第5期2483-2496,共14页
Accurate sales prediction in filling stations is the basis to fill in the refined oil in time and avoid the outof-stock as much as possible.Considering the defect of great“lag”in the general time series model,this p... Accurate sales prediction in filling stations is the basis to fill in the refined oil in time and avoid the outof-stock as much as possible.Considering the defect of great“lag”in the general time series model,this paper summarizes the multiple factors that influence the oil sales and develops a multivariable long short-term memory(LSTM)neural network,with the hyper-parameters being improved by the genetic algorithm(GA).To further improve the prediction accuracy,the proposed LSTM neural network is generalized to bidirectional LSTM(Bi LSTM),in which the impact of future factors on present sales can be taken into account by backward training.Finally,different LSTM structures and genetic algorithm parameters are tested to discuss their impact on prediction accuracy.Results demonstrated that genetic algorithm improved Bi LSTM model is superior to extreme gradient boosting,ARIMA,and artificial neural network,having the highest accuracy of 89%. 展开更多
关键词 Refined oil multivariable prediction BiLSTM Genetic algorithm Future influence
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Observer-based multivariable fixed-time formation control of mobile robots 被引量:5
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作者 LI Yandong ZHU Ling and GUO Yuan 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2020年第2期403-414,共12页
This paper proposes a multivariable fixed-time leaderfollower formation control method for a group of nonholonomic mobile robots, which has the ability to estimate multiple uncertainties. Firstly, based on the state s... This paper proposes a multivariable fixed-time leaderfollower formation control method for a group of nonholonomic mobile robots, which has the ability to estimate multiple uncertainties. Firstly, based on the state space model of the leader-follower formation, a multivariable fixed-time formation kinematics controller is designed. Secondly, to overcome uncertainties existing in the nonholonomic mobile robot system, such as load change,friction, external disturbance, a multivariable fixed-time torque controller based on the fixed-time disturbance observer at the dynamic level is designed. The designed torque controller is cascaded with the formation controller and finally realizes accurate estimation of the uncertain part of the system, the follower tracking of reference velocity and the desired formation of the leader and the follower in a fixed-time. The fixed-time upper bound is completely determined by the controller parameters, which is independent of the initial state of the system. The multivariable fixed-time control theory and the Lyapunov method are adopted to ensure the system stability.Finally, the effectiveness of the proposed algorithm is verified by the experimental simulation. 展开更多
关键词 multivariable fixed-time CONTROL formation CONTROL uncertainty fixed time OBSERVER NONHOLONOMIC mobile robot
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A method for real power transfer allocation using multivariable regression analysis 被引量:6
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作者 Hussain Shareef Azah Mohamed +1 位作者 Saifunizam Abd.Khalid Mohd Wazir Mustafa 《Journal of Central South University》 SCIE EI CAS 2012年第1期179-186,共8页
A multivariable regression(MVR) approach is proposed to identify the real power transfer between generators and loads.Based on solved load flow results,it first uses modified nodal equation method(MNE) to determine re... A multivariable regression(MVR) approach is proposed to identify the real power transfer between generators and loads.Based on solved load flow results,it first uses modified nodal equation method(MNE) to determine real power contribution from each generator to loads.Then,the results of MNE method and load flow information are utilized to determine suitable regression coefficients using MVR model to estimate the power transfer.The 25-bus equivalent system of south Malaysia is utilized as a test system to illustrate the effectiveness of the MVR output compared to that of the MNE method.The error of the estimate of MVR method ranges from 0.001 4 to 0.007 9.Furthermore,when compared to MNE method,MVR method computes generator contribution to loads within 26.40 ms whereas the MNE method takes 360 ms for the calculation of same real power transfer allocation.Therefore,MVR method is more suitable for real time power transfer allocation. 展开更多
关键词 power tracing multivariable regression power systems DEREGULATION
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Multivariable Decoupling Predictive Control Based on QFT Theory and Application in CSTR Chemical Process 被引量:4
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作者 王增会 陈增强 +1 位作者 孙青林 袁著祉 《Chinese Journal of Chemical Engineering》 SCIE EI CAS CSCD 2006年第6期765-769,共5页
A novel method of incorporating generalized predictive control (GPC) algorithms based on quantitative feedback theory (QFT) principles is proposed for solving the feedback control problem of the highly uncertain and c... A novel method of incorporating generalized predictive control (GPC) algorithms based on quantitative feedback theory (QFT) principles is proposed for solving the feedback control problem of the highly uncertain and cross-coupling plants. The quantitative feedback theory decouples the multi-input and multi-output (MIMO) plant and is also used to reduce the uncertainties of the system, stabilize the system, and achieve tracking performance of the system to a certain extent. Single-input and single-output (SISO) generalized predictive control is used to achieve performance with higher performance. In GPC, the model is identified on-line, which is based on the QFT input and the plant output signals. The simulation results show that the performance of the system is superior to the performance when only QFT is used for highly uncertain MIMO plants. 展开更多
关键词 quantitative feedback theory generalized predictive control decouple multivariable UNCERTAIN system frequency domain design
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PID Controller Tuning for a Multivariable Glass Furnace Process by Genetic Algorithm 被引量:6
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作者 Kumaran Rajarathinam James Barry Gomm +1 位作者 Ding-Li Yu Ahmed Saad Abdelhadi 《International Journal of Automation and computing》 EI CSCD 2016年第1期64-72,共9页
Standard genetic algorithms (SGAs) are investigated to optimise discrete-time proportional-integral-derivative (PID) con- troller parameters, by three tuning approaches, for a multivariable glass furnace process w... Standard genetic algorithms (SGAs) are investigated to optimise discrete-time proportional-integral-derivative (PID) con- troller parameters, by three tuning approaches, for a multivariable glass furnace process with loop interaction. Initially, standard genetic algorithms (SGAs) are used to identify control oriented models of the plant which are subsequently used for controller optimisa- tion. An individual tuning approach without loop interaction is considered first to categorise the genetic operators, cost functions and improve searching boundaries to attain the desired performance criteria. The second tuning approach considers controller parameters optimisation with loop interaction and individual cost functions. While, the third tuning approach utilises a modified cost function which includes the total effect of both controlled variables, glass temperature and excess oxygen. This modified cost function is shown to exhibit improved control robustness and disturbance rejection under loop interaction. 展开更多
关键词 Genetic algorithms control optimisation decentralised control proportional-integral-derivative (PID) control modifiedcost function multivariable process loop interaction.
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Multiobjective optimization and multivariable control of the beer fermentation process with the use of evolutionary algorithms 被引量:7
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作者 ANDRES-TOROB. GIRON-SIERRAJ.M. FERNANDEZ-BLANCOP. LOPEZ-OROZCOJ.A. BESADA-PORTASE. 《Journal of Zhejiang University Science》 CSCD 2004年第4期378-389,共12页
This paper describes empirical research on the model, optimization and supervisory control of beer fermentation.Conditions in the laboratory were made as similar as possible to brewery industry conditions. Since mathe... This paper describes empirical research on the model, optimization and supervisory control of beer fermentation.Conditions in the laboratory were made as similar as possible to brewery industry conditions. Since mathematical models that consider realistic industrial conditions were not available, a new mathematical model design involving industrial conditions was first developed. Batch fermentations are multiobjective dynamic processes that must be guided along optimal paths to obtain good results.The paper describes a direct way to apply a Pareto set approach with multiobjective evolutionary algorithms (MOEAs).Successful finding of optimal ways to drive these processes were reported.Once obtained, the mathematical fermentation model was used to optimize the fermentation process by using an intelligent control based on certain rules. 展开更多
关键词 Multiobjective optimization Genetic algorithms Industrial control multivariable control systems Fermenta- tion processes
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MULTIVARIABLE MODEL REFERENCE ADAPTIVE CONTROL FOR A TURBOFAN ENGINE 被引量:6
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作者 Fan Jun Huang Jinquan +1 位作者 Sun Jianguo Feng Zhengping(Dep. Of Power Engineering, Nanjing University of Aeronauticsand Astronautics, Nanjing,China, 210026) 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 1996年第4期300-304,共5页
A decentralized model reference adaptive control (MRAC) scheme is proposed and applied to design a multivariable control system of a dual-spool turbofan engine.Simulation studies show good static and dynamic performan... A decentralized model reference adaptive control (MRAC) scheme is proposed and applied to design a multivariable control system of a dual-spool turbofan engine.Simulation studies show good static and dynamic performance of the system over the fullflight envelope. Simulation results also show the good effectiveness of reducing interactionin the multivariable system with significant coupling. The control system developed has awide frequency band to satisfy the strict engineering requirement and is practical for engineering applications. 展开更多
关键词 turbofan engines model reference adaptive control flight envelopes multivariable control
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Recent developments on applications of sequential loop closing and diagonal dominance control schemes to industrial multivariable system 被引量:4
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作者 A.H.Mazinan M.F.Kazemi 《Journal of Central South University》 SCIE EI CAS 2013年第12期3401-3420,共20页
With a focus on an industrial multivariable system, two subsystems including the flow and the level outputs are analysed and controlled, which have applicability in both real and academic environments. In such a case,... With a focus on an industrial multivariable system, two subsystems including the flow and the level outputs are analysed and controlled, which have applicability in both real and academic environments. In such a case, at first, each subsystem is distinctively represented by its model, since the outcomes point out that the chosen models have the same behavior as corresponding ones. Then, the industrial multivariable system and its presentation are achieved in line with the integration of these subsystems, since the interaction between them can not actually be ignored. To analyze the interaction presented, the Gershgorin bands need to be acquired, where the results are used to modify the system parameters to appropriate values. Subsequently, in the view of modeling results, the control concept in two different techniques including sequential loop closing control(SLCC) scheme and diagonal dominance control(DDC) schemes is proposed to implement on the system through the Profibus network, as long as the OPC(OLE for process control) server is utilized to communicate between the control schemes presented and the multivariable system. The real test scenarios are carried out and the corresponding outcomes in their present forms are acquired. In the same way, the proposed control schemes results are compared with each other, where the real consequences verify the validity of them in the field of the presented industrial multivariable system control. 展开更多
关键词 multivariable system diagonal dominance control sequential loop closing control Profibus network OPC server orifice relation interaction analysis Gershogerin bands non-minimum phase system
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Multivariable Dynamic Modeling for Molten Iron Quality Using Incremental Random Vector Functional-link Networks 被引量:4
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作者 Li ZHANG Ping ZHOU +2 位作者 He-da SONG Meng YUAN Tian-you CHAI 《Journal of Iron and Steel Research International》 SCIE EI CAS CSCD 2016年第11期1151-1159,共9页
Molten iron temperature as well as Si, P, and S contents is the most essential molten iron quality (MIQ) indices in the blast furnace (BF) ironmaking, which requires strict monitoring during the whole ironmaking p... Molten iron temperature as well as Si, P, and S contents is the most essential molten iron quality (MIQ) indices in the blast furnace (BF) ironmaking, which requires strict monitoring during the whole ironmaking production. However, these MIQ parameters are difficult to be directly measured online, and large-time delay exists in off-line analysis through laboratory sampling. Focusing on the practical challenge, a data-driven modeling method was presented for the prediction of MIQ using the improved muhivariable incremental random vector functional-link net- works (M-I-RVFLNs). Compared with the conventional random vector functional-link networks (RVFLNs) and the online sequential RVFLNs, the M-I-RVFLNs have solved the problem of deciding the optimal number of hidden nodes and overcome the overfitting problems. Moreover, the proposed M I RVFLNs model has exhibited the potential for multivariable prediction of the MIQ and improved the terminal condition for the multiple-input multiple-out- put (MIMO) dynamic system, which is suitable for the BF ironmaking process in practice. Ultimately, industrial experiments and contrastive researches have been conducted on the BF No. 2 in Liuzhou Iron and Steel Group Co. Ltd. of China using the proposed method, and the results demonstrate that the established model produces better estima ting accuracy than other MIQ modeling methods. 展开更多
关键词 molten iron quality multivariable incremental random vector functional-link network blast furnace iron-making data-driven modeling principal component analysis
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Multivariable PI Type Generalized Predictive Control 被引量:4
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作者 Chen, Zengqiang Zhao, Tianhang Yuan, Zhuzhi 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 1998年第2期8-13,共6页
This paper presents a multivariable generalized predictive controller with proportion and integration structure by modifying the quadratic criterion of the usual MGPC. The control performance has been improved greatl... This paper presents a multivariable generalized predictive controller with proportion and integration structure by modifying the quadratic criterion of the usual MGPC. The control performance has been improved greatly. The effectiveness of the controller is demonstrated by the simulation result. 展开更多
关键词 Predictive control Self-tuning control multivariable control PI control
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A novel model-based multivariable framework for aircraft gas turbine engine limit protection control 被引量:3
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作者 Shuwei PANG Soheil JAFARI +1 位作者 Theoklis NIKOLAIDIS Qiuhong LI 《Chinese Journal of Aeronautics》 SCIE EI CAS CSCD 2021年第12期57-72,共16页
Control technologies are innovated to satisfy increasingly complicated control demands of gas turbine engines.In terms of limit protection control,a novel model-based multivariable limit protection control method,whic... Control technologies are innovated to satisfy increasingly complicated control demands of gas turbine engines.In terms of limit protection control,a novel model-based multivariable limit protection control method,which is achieved by adaptive command reconstruction and multiplecontrol loop selection and switch logic,is proposed in this paper to address the problem of balancing smaller thrust loss and safe operations by comparing with widely-used Min-Max logic.Five different combination modes of control loops,which represent the online control loop of last time instant and that of current time instant,is analyzed.Different command reconstructions are designed for these modes,which is based on static gain conversion of amplitude beyond limits by using an onboard model.The double-prediction based control loop selection and switch logic is developed to choose a control loop appropriately by comparing converted amplitude beyond limits regardless of one or more parameters tending to exceed limits.The proposed method is implemented in a twin-spool turbofan engine to achieve limit protection with direct thrust control,and the loss of thrust is improved by about 30% in comparison with the loss of thrust caused by Min-Max logic when limit protection control is activated,which demonstrates the effectiveness of the proposed method. 展开更多
关键词 Amplitude conversion Command reconstruction Gas turbine engine Limit protection control multivariable control Onboard prediction
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Incremental multivariable predictive functional control and its application in a gas fractionation unit 被引量:3
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作者 施惠元 苏成利 +3 位作者 曹江涛 李平 宋英莉 李宁波 《Journal of Central South University》 SCIE EI CAS CSCD 2015年第12期4653-4668,共16页
The control of gas fractionation unit(GFU) in petroleum industry is very difficult due to multivariable characteristics and a large time delay.PID controllers are still applied in most industry processes.However,the t... The control of gas fractionation unit(GFU) in petroleum industry is very difficult due to multivariable characteristics and a large time delay.PID controllers are still applied in most industry processes.However,the traditional PID control has been proven not sufficient and capable for this particular petro-chemical process.In this work,an incremental multivariable predictive functional control(IMPFC) algorithm was proposed with less online computation,great precision and fast response.An incremental transfer function matrix model was set up through the step-response data,and predictive outputs were deduced with the theory of single-value optimization.The results show that the method can optimize the incremental control variable and reject the constraint of the incremental control variable with the positional predictive functional control algorithm,and thereby making the control variable smoother.The predictive output error and future set-point were approximated by a polynomial,which can overcome the problem under the model mismatch and make the predictive outputs track the reference trajectory.Then,the design of incremental multivariable predictive functional control was studied.Simulation and application results show that the proposed control strategy is effective and feasible to improve control performance and robustness of process. 展开更多
关键词 gas fractionation unit multivariable process incremental predictive functional control
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Fractional derivative multivariable grey model for nonstationary sequence and its application 被引量:4
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作者 KANG Yuxiao MAO Shuhua +1 位作者 ZHANG Yonghong ZHU Huimin 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2020年第5期1009-1018,共10页
Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problem... Most of the existing multivariable grey models are based on the 1-order derivative and 1-order accumulation, which makes the parameters unable to be adjusted according to the data characteristics of the actual problems. The results about fractional derivative multivariable grey models are very few at present. In this paper, a multivariable Caputo fractional derivative grey model with convolution integral CFGMC(q, N) is proposed. First, the Caputo fractional difference is used to discretize the model, and the least square method is used to solve the parameters. The orders of accumulations and differential equations are determined by using particle swarm optimization(PSO). Then, the analytical solution of the model is obtained by using the Laplace transform, and the convergence and divergence of series in analytical solutions are also discussed. Finally, the CFGMC(q, N) model is used to predict the municipal solid waste(MSW). Compared with other competition models, the model has the best prediction effect. This study enriches the model form of the multivariable grey model, expands the scope of application, and provides a new idea for the development of fractional derivative grey model. 展开更多
关键词 fractional derivative of Caputo type fractional accumulation generating operation(FAGO) Laplace transform multivariable grey prediction model particle swarm optimization(PSO)
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A closed-loop particle swarm optimizer for multivariable process controller design 被引量:2
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作者 Kai HAN Jun ZHAO +1 位作者 Zu-hua XU Ji-xin QIAN 《Journal of Zhejiang University-Science A(Applied Physics & Engineering)》 SCIE EI CAS CSCD 2008年第8期1050-1060,共11页
Design of general multivariable process controllers is an attractive and practical alternative to optimizing design by evolutionary algorithms (EAs) since it can be formulated as an optimization problem. A closed-loop... Design of general multivariable process controllers is an attractive and practical alternative to optimizing design by evolutionary algorithms (EAs) since it can be formulated as an optimization problem. A closed-loop particle swarm optimization (CLPSO) algorithm is proposed by mapping PSO elements into the closed-loop system based on control theories. At each time step, a proportional integral (PI) controller is used to calculate an updated inertia weight for each particle in swarms from its last fitness. With this modification, limitations caused by a uniform inertia weight for the whole population are avoided, and the particles have enough diversity. After the effectiveness, efficiency and robustness are tested by benchmark functions, CLPSO is applied to design a multivariable proportional-integral-derivative (PID) controller for a solvent dehydration tower in a chemical plant and has improved its performances. 展开更多
关键词 multivariable process control Proportional-integral-derivative (PID) control Model predictive control (MPC) Particle swarm optimization (PSO) Closed-loop system
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THE ANALYSIS OF SHALLOW SHELLS BASED ON MULTIVARIABLE WAVELET FINITE ELEMENT METHOD 被引量:2
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作者 Xingwu Zhang Xuefeng Chen Zhengjia He Zhibo Yang 《Acta Mechanica Solida Sinica》 SCIE EI 2011年第5期450-460,共11页
Based on the generalized variational principle and B-spline wavelet on the interval (BSWI), the multivariable BSWI elements with two kinds of variables (TBSWI) for hyperboloidal shell and open cylindrical shell ar... Based on the generalized variational principle and B-spline wavelet on the interval (BSWI), the multivariable BSWI elements with two kinds of variables (TBSWI) for hyperboloidal shell and open cylindrical shell are constructed in this paper. Different from the traditional method, the present one treats the generalized displacement and stress as independent variables. So differentiation and integration are avoided in calculating generalized stress and thus the precision is improved. Furthermore, compared with commonly used Daubechies wavelet, BSWI has explicit expression and excellent approximation property and thus further guarantee satisfactory results. Finally, the efficiency of the constructed multivariable shell elements is validated through several numerical examples. 展开更多
关键词 multivariable B-spline wavelet on the interval hyperboloidal shell open cylindrical shell
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