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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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Adapting to the stream:an instance-attention GNN method for irregular multivariate time series data
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作者 Kun HAN Abigail M Y KOAY +2 位作者 Ryan K L KO Weitong CHEN Miao XU 《Frontiers of Computer Science》 2025年第8期23-35,共13页
Multivariate time series(MTS)data are vital for various applications,particularly in machine learning tasks.However,challenges such as sensor failures can result in irregular and misaligned data with missing values,th... Multivariate time series(MTS)data are vital for various applications,particularly in machine learning tasks.However,challenges such as sensor failures can result in irregular and misaligned data with missing values,thereby complicating their analysis.While recent advancements use graph neural networks(GNNs)to manage these Irregular Multivariate Time Series(IMTS)data,they generally require a reliable graph structure,either pre-existing or inferred from adequate data to properly capture node correlations.This poses a challenge in applications where IMTS data are often streamed and waiting for future data to estimate a suitable graph structure becomes impractical.To overcome this,we introduce a dynamic GNN model suited for streaming characteristics of IMTS data,incorporating an instance-attention mechanism that dynamically learns and updates graph edge weights for real-time analysis.We also tailor strategies for high-frequency and low-frequency data to enhance prediction accuracy.Empirical results on real-world datasets demonstrate the superiority of our proposed model in both classification and imputation tasks. 展开更多
关键词 multivariate time series irregular multivariate time series graph neural networks
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Model distance-based approach for global sensitivity analysis in engineering systems with multivariate outputs
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作者 Kumar VIDHYA Greegar GEORGE 《Frontiers of Structural and Civil Engineering》 2025年第9期1493-1511,共19页
In engineering systems,uncertainties in input parameters can significantly influence the output responses.This paper proposes a model distance-based approach to perform global sensitivity analysis for quantifying the ... In engineering systems,uncertainties in input parameters can significantly influence the output responses.This paper proposes a model distance-based approach to perform global sensitivity analysis for quantifying the influence of input uncertainties on multiple responses in an engineering system.The sensitivity indices are determined by comparing a reference model that incorporates all system uncertainties,with an altered model,where specific uncertainties are constrained.The proposed framework employs probability distance measures such as Hellinger distance,Kullback-Leibler divergence,and I2 norm which are based on joint probability density functions.The study also demonstrates the equivalence between the l2 norm-based approach and Sobol's analysis in multivariate sensitivity context.The proposed methodology effectively manages correlated random variables,accommodates both Gaussian and non-Gaussian distributions,and allows for the grouping of input variables.Ilustrative examples consist of static analysis of a truss system and dynamic analysis of a frame subjected to seismic excitation.The sensitivity indices are estimated using brute-force Monte Carlo simulations.The relative ranking of these sensitivity indices can be utilized to identify the most and least significant variables contributing to the response uncertainty.The numerical results show a consistent ranking of input variables across different probability measures,indicating the robustness of proposed framework. 展开更多
关键词 multivariate outputs moment-independent sensitivity analysis probability distance measures multivariate Sobol's analysis Monte Carlo simulation uncertainty quantification
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On the Zero Coprime Equivalence of Multivariate Polynomial Matrices
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作者 CHEN Zuo LI Dongmei GUO Xu 《Wuhan University Journal of Natural Sciences》 2025年第1期32-42,共11页
The zero coprime system equivalence is one of important research in the theory of multidimensional system equivalence,and is closely related to zero coprime equivalence of multivariate polynomial matrices.We first dis... The zero coprime system equivalence is one of important research in the theory of multidimensional system equivalence,and is closely related to zero coprime equivalence of multivariate polynomial matrices.We first discuss the relation between zero coprime equivalence and unimodular equivalence for polynomial matrices.Then,we investigate the zero coprime equivalence problem for several classes of polynomial matrices,some novel findings and criteria on reducing these matrices to their Smith normal forms are obtained.Finally,an example is provided to illustrate the main results. 展开更多
关键词 multidimensional system multivariate polynomial matrix zero coprime equivalence unimodular equivalence Smith normal form
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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 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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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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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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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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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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Intelligent Multivariable Modeling of Blast Furnace Molten Iron Quality Based on Dynamic AGA-ANN and PCA 被引量:4
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作者 Meng YUAN Ping ZHOU +3 位作者 Ming-liang LI Rui-feng LI Hong WANG Tian-you CHAI 《Journal of Iron and Steel Research International》 SCIE EI CAS CSCD 2015年第6期487-495,共9页
Blast furnace (BF) ironmaking process has complex and nonlinear dynamic characteristics. The molten iron temperature (MIT) as well as Si, P and S contents of molten iron is difficult to be directly measured online... Blast furnace (BF) ironmaking process has complex and nonlinear dynamic characteristics. The molten iron temperature (MIT) as well as Si, P and S contents of molten iron is difficult to be directly measured online, and large-time delay exists in offline analysis through laboratory sampling. A nonlinear multivariate intelligent modeling method was proposed for molten iron quality (MIQ) based on principal component analysis (PCA) and dynamic ge- netic neural network. The modeling method used the practical data processed by PCA dimension reduction as inputs of the dynamic artificial neural network (ANN). A dynamic feedback link was introduced to produce a dynamic neu- ral network on the basis of traditional back propagation ANN. The proposed model improved the dynamic adaptabili- ty of networks and solved the strong fluctuation and resistance problem in a nonlinear dynamic system. Moreover, a new hybrid training method was presented where adaptive genetic algorithms (AGA) and ANN were integrated, which could improve network convergence speed and avoid network into local minima. The proposed method made it easier for operators to understand the inside status of blast furnace and offered real-time and reliable feedback infor- mation for realizing close-loop control for MIQ. Industrial experiments were made through the proposed model based on data collected from a practical steel company. The accuracy could meet the requirements of actual operation. 展开更多
关键词 molten iron quality blast furnace nonlinear multivariate modeling dynamic neural network principalcomponent analysis adaptive genetic algorithm
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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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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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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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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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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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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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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 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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PSEUDO-DIVISION ALGORITHM FOR MATRIX MULTIVARIABLE POLYNOMIAL AND ITS APPLICATION 被引量:1
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作者 阿拉坦仓 张鸿庆 钟万勰 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2000年第7期733-740,共8页
Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equation... Pseudo-division algorithm for matrix multivariable polynomial are given, thereby with the view of differential algebra, the sufficient and necessary conditions for transforming a class of partial differential equations into infinite dimensional Hamiltonianian system and its concrete form are obtained. Then by combining this method with Wu's method, a new method of constructing general solution of a class of mechanical equations is got, which several examples show very effective. 展开更多
关键词 matrix multivariable polynomial infinite dimensional Hamiltonianian system Wu's method general solution
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