Modeling and optimal control

Book reviewModeling,Simulation,and Control of Flexible Manufacturing Systems ±A Petri Net Approach;Meng Chu Zhou;Kurapati Venkatesh;Yushun Fan;World Scienti®c,Singapore,19991.IntroductionA ¯exible manufacturing system (FMS)is an automated,mid-volume,mid-va-riety,central computer-controlled manufacturing system.It can be used to produce a variety of products with virtually no time lost for changeover from one product to the next.FMS is a capital-investment intensive and complex system.In order to get the best economic bene®ts,the design,implementation and operation of FMS should be carefully made.A lot of researches have been done regarding the modeling,simulation,scheduling and control of FMS [1±6].From time to time,Petri net (PN)method has also been used as a tool by di erent researcher in studying the problems regarding the modeling,simulation,scheduling and control of FMS.A lot of papers and books have been published in this area [7±14].``Modeling,Simulation,and Control of Flexible Manufacturing Systems ±A PN Approach''is a new book written by Zhou and Venkatesh which is focused on studying FMS using PN as a systematic method and integrated tool.The book's contents can be classi®ed into four parts.The four parts are introduction part (Chapter 1to Chapter 4),PNs application part (Chapter 5to Chapter 8),new research results part (Chapter 9to Chapter 13),and future development trend part (Chapter 14).In the introduction part,the background,motivation and objectives of the book are described in Chapter 1.The brief history of manufacturing systems and PNs is also presented in Chapter 1.The basic de®nitions and problems in FMS design and implementation are introduced in Chapter 2.The authors divide FMS related problems into two major areas ±managerial and technical.In Chapter 4,basic de®nitions,properties,and analysis techniques of PNs are presented,Chapter 4can be used as the fundamentals of PNs for those who are not familiar with PN method.In Chapter 3,the authors presented their approach to studying FMS related prob-lems,the approach uses PNs as an integrated tool and methodology in FMS design and implementation.In Chapter 3,various applications in modeling,analysis,sim-ulation,performance evaluation,discrete event control,planning and scheduling of FMS using PNs are presented.Through reading the introduction part,the readers can obtain basic concepts and methods about FMS and PNs.The readers can also get a clear picture about the relationshipbetween FMS and PNs.Mechatronics 11(2001)947±9500957-4158/01/$-see front matter Ó2001Elsevier Science Ltd.All rights reserved.PII:S 0957-4158(00)00057-X948Book review/Mechatronics11(2001)947±950The second part of the book is about PNs applications.In this part,various applications of using PNs in solving FMS related problems are introduced.FMS modeling is the basis for simulation,analysis,planning and scheduling.In Chapter5, after introduction of several kinds of PNs,a general modeling method of FMS using PNs is given.The systematic bottom-up and top-down modeling method is pre-sented.The presented method is demonstrated by modeling a real FMS cell in New Jersey Institute of Technology.The application of PNs in FMS performance analysis is introduced in Chapter 6.The stochastic PNs and the time distributions are introduced in this Chapter. The analysis of a¯exible workstation performance using the PN tool called SPNP developed at Duke University is given in Section6.4.In Chapter7,the procedures and steps involved for discrete event simulation using PNs are discussed.The use of various modeling techniques such as queuing network models,state-transition models,high-level PNs,object-oriented models for simulations are brie¯y explained.A software package that is used to simulate PN models is introduced.Several CASE tools for PNs simulations are brie¯y intro-duced.In Chapter8,PNs application in studying the di erent e ects between push and pull paradigms is shown.The presented application method is useful for the selection of suitable management paradigm for manufacturing systems.A manufacturing system is modeled considering both push and pull paradigms in Section8.3which is used as a practical example.The general procedures for performance evaluation of FMS with pull paradigm are given in Section8.4.The third part of the book is mainly the research results of the authors in the area of PNs applications.In Chapter9,an augmented-timed PN is put forward. The proposed method is used to model the manufacturing systems with break-down handling.It is demonstrated using a¯exible assembly system in Section9.3. In Chapter10,a new class of PNs called Real-time PN is proposed.The pro-posed PN method is used to model and control the discrete event control sys-tems.The comparison of the proposed method and ladder logic diagrams is given in Chapter11.Due to the signi®cant advantages of Object-oriented method,it has been used in PNs to de®ne a new kind of PNs.In Chapter12,the authors propose an Object-oriented design methodology for the development of FMS control software.The OMT and PNs are integrated in order to developreusable, modi®able,and extendible control software.The proposed methodology is used in a FMS.The OMT is used to®nd the static relationshipamong di erent objects.The PN models are formulated to study the performance of the FMS.In Chapter12,the scheduling methods of FMS using PNs are introduced.Some examples are presented for automated manufacturing system and semiconductor test facility.In the last Chapter,the future research directions of PNs are pointed out.The contents include CASE tool environment,scheduling of large production system,su-pervisory control,multi-lifecycle engineering and benchmark studies.Book review/Mechatronics11(2001)947±950949 mentsAs a monograph in PNs and its applications in FMS,the book is abundant in contents.Besides the rich knowledge of PNs,the book covers almost every aspects regarding FMS design and analysis,such as modeling,simulation,performance evaluation,planning and scheduling,break down handling,real-time control,con-trol software development,etc.So,the reader can obtain much knowledge in PN, FMS,discrete event system control,system simulation,scheduling,as well as in software development.The book is a very good book in the combinations of PNs theory and prac-tical applications.Throughout the book,the integrated style is demonstrated.It is very well suited for the graduate students and beginners who are interested in using PN methods in studying their speci®c problems.The book is especially suited for the researchers working in the areas of FMS,CIMS,advanced man-ufacturing technologies.The feedback messages from our graduate students show that compared with other books about PNs,this book is more interested and easy to learn.It is easy to get a clear picture about what is PNs method and how it can be used in the FMS design and analysis.So,the book is a very good textbook for the graduate students whose majors are manufacturing systems, industrial engineering,factory automation,enterprise management,and computer applications.Both PNs and FMS are complex and research intensive areas.Due to the deep understanding for PNs,FMS,and the writing skills of the authors,the book has good advantages in describing complex problems and theories in a very easy read and understandable fashion.The easy understanding and abundant contents enable the book to be a good reference book both for the students and researchers. Through reading the book,the readers can also learn the new research results in PNs and its applications in FMS that do not contained in other books.Because the most new results given in the book are the study achievements of the authors,the readers can better know not only the results,but also the background,history,and research methodology of the related areas.This would helpthe researchers who are going to do the study to know the state-of-art of relevant areas,thus the researchers can begin the study in less preparing time and to get new results more earlier.As compared to other books,the organization of the book is very application oriented.The aims are to present new research results in FMS applications using PNs method,the organization of the book is cohesive to the topics.A lot of live examples have reinforced the presented methods.These advantages make the book to be a very good practical guide for the students and beginners to start their re-search in the related areas.The history and reference of related research given in this book provides the reader a good way to better know PNs methods and its applications in FMS.It is especially suited for the Ph.D.candidates who are determined to choose PNs as their thesis topics.950Book review/Mechatronics11(2001)947±9503.ConclusionsDue to the signi®cant importance of PNs and its applications,PNs have become a common background and basic method for the students and researchers to do re-search in modeling,planning and scheduling,performance analysis,discrete event system control,and shop-¯oor control software development.The book under re-view provides us a good approach to learn as well as to begin the research in PNs and its application in manufacturing systems.The integrated and application oriented style of book enables the book to be a very good book both for graduate students and researchers.The easy understanding and step-by-step deeper introduction of the contents makes it to be a good textbook for the graduate students.It is suited to the graduated students whose majors are manufacturing system,industrial engineering, enterprise management,computer application,and automation.References[1]Talavage J,Hannam RG.Flexible manufacturing systems in practice:application,design,andsimulation.New York:Marcel Dekker Inc.;1988.[2]Tetzla UAW.Optimal design of¯exible manufacturing systems.New York:Springer;1990.[3]Jha NK,editor.Handbook of¯exible manufacturing systems.San Diego:Academic Press,1991.[4]Carrie C.Simulation of manufacturing.New York:John Wiley&Sons;1988.[5]Gupta YP,Goyal S.Flexibility of manufacturing systems:concepts and measurements.EuropeanJournal of Operational Research1989;43:119±35.[6]Carter MF.Designing¯exibility into automated manufacturing systems.In:Stecke KE,Suri R,editors.Proceedings of the Second ORSA/TIMS Conference on FMS:Operations Research Models and Applications.New York:Elsevier;1986.p.107±18.[7]David R,Alla H.Petri nets and grafcet.New York:Prentice Hall;1992.[8]Zhou MC,DiCesare F.Petri net synthesis for discrete event control of manufacturing systems.Norwell,MA:Kluwer Academic Publishers;1993.[9]Desrochers AA,Al-Jaar RY.Applications of petri nets in manufacturing systems.New York:IEEEPress;1995.[10]Zhou MC,editor.Petri nets in¯exible and agile automation.Boston:Kluwer Academic Publishers,1995.[11]Lin C.Stochastic petri nets and system performance evaluations.Beijing:Tsinghua University Press;1999.[12]Peterson JL.Petri net theory and the modeling of systems.Englewood Cli s,NJ:Prentice-Hall;1981.[13]Resig W.Petri nets.New York:Springer;1985.[14]Jensen K.Coloured Petri Nets.Berlin:Springer;1992.Yushun FanDepartment of Automation,Tsinghua UniversityBeijing100084,People's Republic of ChinaE-mail address:*****************。

中英文对照资料外文翻译文献FEM Optimization for Robot StructureAbstractIn optimal design for robot structures, design models need to he modified and computed repeatedly. Because modifying usually can not automatically be run, it consumes a lot of time. This paper gives a method that uses APDL language of ANSYS 5.5 software to generate an optimal control program, which mike optimal procedure run automatically and optimal efficiency be improved.1）IntroductionIndustrial robot is a kind of machine, which is controlled by computers. Because efficiency and maneuverability are higher than traditional machines, industrial robot is used extensively in industry. For the sake of efficiency and maneuverability, reducing mass and increasing stiffness is more important than traditional machines, in structure design of industrial robot.A lot of methods are used in optimization design of structure. Finite element method is a much effective method. In general, modeling and modifying are manual, which is feasible when model is simple. When model is complicated, optimization time is longer. In the longer optimization time, calculation time is usually very little, a majority of time is used for modeling and modifying. It is key of improving efficiency of structure optimization how to reduce modeling and modifying time.APDL language is an interactive development tool, which is based on ANSYS and is offered to program users. APDL language has typical function of some large computer languages. For example, parameter definition similar to constant and variable definition, branch and loop control, and macro call similar to function and subroutine call, etc. Besides these, it possesses powerful capability of mathematical calculation. The capability of mathematical calculation includes arithmetic calculation, comparison, rounding, and trigonometric function, exponential function and hyperbola function of standard FORTRAN language, etc. By means of APDL language, the data can be read and then calculated, which is in database of ANSYS program, and running process of ANSYS program can be controlled.Fig. 1 shows the main framework of a parallel robot with three bars. When the length of three bars are changed, conjunct end of three bars can follow a given track, where robot hand is installed. Core of top beam is triangle, owing to three bars used in the design, which is showed in Fig.2. Use of three bars makes top beam nonsymmetrical along the plane that is defined by two columns. According to a qualitative analysis from Fig.1, Stiffness values along z-axis are different at three joint locations on the top beam and stiffness at the location between bar 1 and top beam is lowest, which is confirmed by computing results of finite element, too. According to design goal, stiffness difference at three joint locations must he within a given tolerance. In consistent of stiffness will have influence on the motion accuracy of the manipulator under high load, so it is necessary to find the accurate location of top beam along x-axis.To the questions presented above, the general solution is to change the location of the top beam many times, compare the results and eventually find a proper position, The model will be modified according to the last calculating result each time. It is difficult to avoid mistakes if the iterative process is controlled manually and the iterative time is too long. The outer wall and inner rib shapes of the top beam will be changed after the model is modified. To find the appropriate location of top beam, the model needs to be modified repetitiously.Fig. 1 Solution of Original DesignThis paper gives an optimization solution to the position optimization question of the top beam by APDL language of ANSYS program. After the analysis model first founded, the optimization control program can be formed by means of modeling instruction in the log file. The later iterative optimization process can be finished by the optimization control program and do not need manual control. The time spent in modifying the model can be decreased to the ignorable extent. The efficiency of the optimization process is greatly improved.2）Construction of model for analysisThe structure shown in Fig. 1 consists of three parts: two columns, one beam and three driving bars. The columns and beam are joined by the bolts on the first horizontal rib located on top of the columns as shown in Fig.1. Because the driving bars are substituted by equivalentforces on the joint positions, their structure is ignored in the model.The core of the top beam is three joints and a hole with special purpose, which can not be changed. The other parts of the beam may be changed if needed. For the convenience of modeling, the core of the beam is formed into one component. In the process of optimization, only the core position of beam along x axis is changed, that is to say, shape of beam core is not changed. It should be noticed that, in the rest of beam, only shape is changed but the topology is not changed and which can automatically be performed by the control program.Fig.1, six bolts join the beam and two columns. The joint surface can not bear the pull stress in the non-bolt joint positions, in which it is better to set contact elements. When the model includes contact elements, nonlinear iterative calculation will be needed in the process of solution and the computing time will quickly increase. The trial computing result not including contact element shows that the outside of beam bears pulling stress and the inner of beam bears the press stress. Considering the primary analysis object is the joint position stiffness between the top beam and the three driving bars, contact elements may not used, hut constructs the geometry model of joint surface as Fig.2 showing. The upper surface and the undersurface share one key point in bolt-joint positions and the upper surface and the under surface separately possess own key points in no bolt positions. When meshed, one node will be created at shared key point, where columns and beam are joined, and two nodes will be created at non shared key point, where column and beam are separated. On right surface of left column and left surface of right column, according to trial computing result, the structure bears press stress. Therefore, the columns and beam will share all key points, not but at bolts. This can not only omit contact element but also show the characteristic of bolt joining. The joining between the bottoms of the columns and the base are treated as full constraint. Because the main aim of analysis is the stiffness of the top beam, it can be assumed that the joint positions hear the same as load between beam and the three driving bars. The structure is the thin wall cast and simulated by shell element . The thickness of the outside wall of the structure and the rib are not equal, so two groups of real constant should he set. For the convenience of modeling, the two columns are alsoset into another component. The components can create an assembly. In this way, the joint positions between the beam core and columns could he easily selected, in the modifying the model and modifying process can automatically be performed. Analysis model is showed Fig.1. Because model and load are symmetric, computing model is only half. So the total of elements is decreased to 8927 and the total of nodes is decreased to 4341. All elements are triangle.3.）Optimization solutionThe optimization process is essentially a computing and modifying process. The original design is used as initial condition of the iterative process. The ending condition of the process is that stiffness differences of the joint locations between three driving bars and top beam are less than given tolerance or iterative times exceed expected value. Considering the speciality of the question, it is foreseen that the location is existent where stiffness values are equal. If iterative is not convergent, the cause cannot be otherwise than inappropriate displacement increment or deficient iterative times. In order to make the iterative process convergent quickly and efficiently, this paper uses the bisection searching method changing step length to modify the top beam displacement. This method is a little complex but the requirement on the initial condition is relatively mild.The flow chart of optimization as follows:1. Read the beam model data in initial position from backup file;2. Modify the position of beam;3. Solve;4. Read the deform of nodes where beam and three bars are joined;5. Check whether the convergent conditions are satisfied, if not, then continue to modify the beam displacement and return to 3, otherwise, exit the iteration procedure.6. Save the results and then exit.The program's primary control codes and their function commentaries are given in it, of which the detailed modeling instructions are omitted. For the convenience of comparing with the control flow, the necessary notes are added.the flag of the batch file in ANSYSBATCH RESUME, robbak.db, 0read original data from the backupfile robbak,.db/PREP7 enter preprocessordelete the joint part between beam core and columnsmove the core of the beam by one :step lengthapply load and constraint on the geometry meshing thejoint position between beam core and columns FINISH exit the preprocessorISOLU enter solverSOLVE solveFINISH exit the solverPOST1 enter the postprocessor*GET ,front,NODE,2013,U,Z read the deformation of first joint node on beam*GET,back,NODE, 1441 ,U,Z read the deformation of second joint node on beam intoparameter hacklastdif-1 the absolute of initial difference between front and hacklast timeflag=- 1 the feasibility flag of the optimizationstep=0.05 the initial displacement from initial position to the currentposition*D0,1,1,10,1 the iteration procedure begin, the cycle variable is I andits value range is 1-10 and step length is 1dif=abs(front-back) the absolute of the difference between front and hack inthe current result*IF,dif,LE,l .OE-6,THEN check whether the absolute difference dif satisfies therequest or noflag=l yes, set flag equal to 1*EXIT exit the iterative calculation*ELSEIF,dif,GE,lastdif,THEN check whether the dif value becomes great or not flag=2yes, set flag 2 modify step length by bisection methodperform the next iterative calculation, use the lastposition as the current position and modified last steplength as the current step lengthELSE if the absolute of difference value is not less thanexpected value and become small gradually, continue tomove top beam read the initial condition from back upfile enter the preprocessorMEN, ,P51X, , , step,, , ,1 move the core of the beam by one step length modify thejoint positions between beam core and column applyload and constraint meshingFINISH exit preprocessorISOLU enter solverSOLVE solveFINISH exit the solver/POST1 exit the postprocessor*GET,front,NODE,201 3,U,Z read the deformation of first joint node to parameter front *GET,back,NODE, 144 1,U,Z read the deformation of second joint node to parameter back lastdif-dif update the value of last dif*ENDIF the end of the if-else*ENDDO the end of the DO cycleMost of the control program above is copied from log file, which is long. The total of lines is up to about 1000 lines. Many codes such as modeling and post-process codes are used repeatedly. To make the program construct clear, these instructions can he made into macros, which are called by main program. This can efficiently reduce the length of the main program. In addition, modeling instructions from log file includes lots of special instructions that are only used under graphic mode but useless under hatch mode. Deleting and modifying these instructions when under batch mode in ANSYS can reduce the length of the file, too.In the program above, the deformation at given position is read from node deformation. In meshing, in order to avoid generating had elements, triangle mesh is used. In optimization, the shape of joint position between columns and beam continually is changed. This makes total of elements different after meshing each time and then element numbering different, too. Data read from database according to node numbering might not he data to want. Therefore, beam core first needs to he meshed, then saved. When read next time, its numbering is the same as last time.Evaluating whether the final result is a feasible result or not needs to check the flag value. If only the flag value is I, the result is feasible, otherwise the most proper position is not found. The total displacement of top beam is saved in parameter step. If the result is feasible, the step value is the distance from initial position to the most proper position. The sum of iterative is saved in parameter 1. According to the final value of I, feasibility of analysis result and correctness of initial condition can he evaluated.4）Optimization resultsThe sum of iterative in optimization is seven, and it takes about 2 hour and 37 minutes to find optimal position. Fig.3 shows the deformation contour of the half-construct. In Fig.3, the deformations in three joints between beam and the three driving bars is the same as level, and the corresponding deformation range is between -0.133E-04 and -0.1 15E-O4m, the requirement of the same stiffness is reached. At this time, the position of beam core along x-axis as shown in Fig. 1 has moved -0.71E-01m compared with the original designed positionBecause the speed of computer reading instruction is much faster than modifying model manually, the time modifying model can be ignored. The time necessary foroptimization mostly depends on the time of solution. Compared with the optimization procedure manually modifying model, the efficiency is improved and mistake operating in modeling is avoided.5）ConclusionThe analyzing result reveals that the optimization method given in this paper is effective and reaches the expected goal. The first advantage of this method is that manual mistakes do not easily occur in optimization procedure. Secondly, it is pretty universal and the control codes given in this paper may he transplanted to use in similar structure optimization design without large modification. The disadvantage is that the topology structure of the optimization object can not be changed. The more the workload of modifying the model, the more the advantages of this method are shown. In addition, the topology optimization function provided in ANSYS is usedto solve the optimization problem that needs to change the topology structure.The better optimization results can he achieved if the method in this paper combined with it.中文译文：机器人机构优化设计有限元分析摘要机器人结构最优化设计，设计模型需要反复的修正和计算。

机器人顶刊论文机器人领域内除开science robotics以外，TRO和IJRR是机器人领域的两大顶刊，最近师弟在选择研究方向，因此对两大顶刊的论文做了整理。

TRO的全称IEEE Transactions on Robotics，是IEEE旗下机器人与自动化协会的汇刊，最新的影响因子为6.123。

ISSUE 61 An End-to-End Approach to Self-Folding Origami Structures2 Continuous-Time Visual-Inertial Odometry for Event Cameras3 Multicontact Locomotion of Legged Robots4 On the Combined Inverse-Dynamics/Passivity-Based Control of Elastic-Joint Robots5 Control of Magnetic Microrobot Teams for Temporal Micromanipulation Tasks6 Supervisory Control of Multirotor Vehicles in Challenging Conditions Using Inertial Measurements7 Robust Ballistic Catching: A Hybrid System Stabilization Problem8 Discrete Cosserat Approach for Multisection Soft Manipulator Dynamics9 Anonymous Hedonic Game for Task Allocation in a Large-Scale Multiple Agent System10 Multimodal Sensorimotor Integration for Expert-in-the-Loop Telerobotic Surgical Training11 Fast, Generic, and Reliable Control and Simulation of Soft Robots Using Model Order Reduction12 A Path/Surface Following Control Approach to Generate Virtual Fixtures13 Modeling and Implementation of the McKibben Actuator in Hydraulic Systems14 Information-Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving15 Robust Planar Odometry Based on Symmetric Range Flow and Multiscan Alignment16 Accelerated Sensorimotor Learning of Compliant Movement Primitives17 Clock-Torqued Rolling SLIP Model and Its Application to Variable-Speed Running in aHexapod Robot18 On the Covariance of X in AX=XB19 Safe Testing of Electrical Diathermy Cutting Using a New Generation Soft ManipulatorISSUE 51 Toward Dexterous Manipulation With Augmented Adaptive Synergies: The Pisa/IIT SoftHand 22 Efficient Equilibrium Testing Under Adhesion and Anisotropy Using Empirical Contact Force Models3 Force, Impedance, and Trajectory Learning for Contact Tooling and Haptic Identification4 An Ankle–Foot Prosthesis Emulator With Control of Plantarflexion and Inversion–Eversion Torque5 SLAP: Simultaneous Localization and Planning Under Uncertainty via Dynamic Replanning in Belief Space6 An Analytical Loading Model for n -Tendon Continuum Robots7 A Direct Dense Visual Servoing Approach Using Photometric Moments8 Computational Design of Robotic Devices From High-Level Motion Specifications9 Multicontact Postures Computation on Manifolds10 Stiffness Modulation in an Elastic Articulated-Cable Leg-Orthosis Emulator: Theory and Experiment11 Human–Robot Communications of Probabilistic Beliefs via a Dirichlet Process Mixture of Statements12 Multirobot Reconnection on Graphs: Problem, Complexity, and Algorithms13 Robust Intrinsic and Extrinsic Calibration of RGB-D Cameras14 Reactive Trajectory Generation for Multiple Vehicles in Unknown Environments With Wind Disturbances15 Resource-Aware Large-Scale Cooperative Three-Dimensional Mapping Using Multiple Mobile Devices16 Control of Planar Spring–Mass Running Through Virtual Tuning of Radial Leg Damping17 Gait Design for a Snake Robot by Connecting Curve Segments and ExperimentalDemonstration18 Server-Assisted Distributed Cooperative Localization Over Unreliable Communication Links19 Realization of Smooth Pursuit for a Quantized Compliant Camera Positioning SystemISSUE 41 A Survey on Aerial Swarm Robotics2 Trajectory Planning for Quadrotor Swarms3 A Distributed Control Approach to Formation Balancing and Maneuvering of Multiple Multirotor UAVs4 Joint Coverage, Connectivity, and Charging Strategies for Distributed UAV Networks5 Robotic Herding of a Flock of Birds Using an Unmanned Aerial Vehicle6 Agile Coordination and Assistive Collision Avoidance for Quadrotor Swarms Using Virtual Structures7 Decentralized Trajectory Tracking Control for Soft Robots Interacting With the Environment8 Resilient, Provably-Correct, and High-Level Robot Behaviors9 Humanoid Dynamic Synchronization Through Whole-Body Bilateral Feedback Teleoperation10 Informed Sampling for Asymptotically Optimal Path Planning11 Robust Tactile Descriptors for Discriminating Objects From Textural Properties via Artificial Robotic Skin12 VINS-Mono: A Robust and Versatile Monocular Visual-Inertial State Estimator13 Zero Step Capturability for Legged Robots in Multicontact14 Fast Gait Mode Detection and Assistive Torque Control of an Exoskeletal Robotic Orthosis for Walking Assistance15 Physically Plausible Wrench Decomposition for Multieffector Object Manipulation16 Considering Uncertainty in Optimal Robot Control Through High-Order Cost Statistics17 Multirobot Data Gathering Under Buffer Constraints and Intermittent Communication18 Image-Guided Dual Master–Slave Robotic System for Maxillary Sinus Surgery19 Modeling and Interpolation of the Ambient Magnetic Field by Gaussian Processes20 Periodic Trajectory Planning Beyond the Static Workspace for 6-DOF Cable-Suspended Parallel Robots1 Computationally Efficient Trajectory Generation for Fully Actuated Multirotor Vehicles2 Aural Servo: Sensor-Based Control From Robot Audition3 An Efficient Acyclic Contact Planner for Multiped Robots4 Dimensionality Reduction for Dynamic Movement Primitives and Application to Bimanual Manipulation of Clothes5 Resolving Occlusion in Active Visual Target Search of High-Dimensional Robotic Systems6 Constraint Gaussian Filter With Virtual Measurement for On-Line Camera-Odometry Calibration7 A New Approach to Time-Optimal Path Parameterization Based on Reachability Analysis8 Failure Recovery in Robot–Human Object Handover9 Efficient and Stable Locomotion for Impulse-Actuated Robots Using Strictly Convex Foot Shapes10 Continuous-Phase Control of a Powered Knee–Ankle Prosthesis: Amputee Experiments Across Speeds and Inclines11 Fundamental Actuation Properties of Multirotors: Force–Moment Decoupling and Fail–Safe Robustness12 Symmetric Subspace Motion Generators13 Recovering Stable Scale in Monocular SLAM Using Object-Supplemented Bundle Adjustment14 Toward Controllable Hydraulic Coupling of Joints in a Wearable Robot15 Geometric Construction-Based Realization of Spatial Elastic Behaviors in Parallel and Serial Manipulators16 Dynamic Point-to-Point Trajectory Planning Beyond the Static Workspace for Six-DOF Cable-Suspended Parallel Robots17 Investigation of the Coin Snapping Phenomenon in Linearly Compliant Robot Grasps18 Target Tracking in the Presence of Intermittent Measurements via Motion Model Learning19 Point-Wise Fusion of Distributed Gaussian Process Experts (FuDGE) Using a Fully Decentralized Robot Team Operating in Communication-Devoid Environment20 On the Importance of Uncertainty Representation in Active SLAM1 Robust Visual Localization Across Seasons2 Grasping Without Squeezing: Design and Modeling of Shear-Activated Grippers3 Elastic Structure Preserving (ESP) Control for Compliantly Actuated Robots4 The Boundaries of Walking Stability: Viability and Controllability of Simple Models5 A Novel Robotic Platform for Aerial Manipulation Using Quadrotors as Rotating Thrust Generators6 Dynamic Humanoid Locomotion: A Scalable Formulation for HZD Gait Optimization7 3-D Robust Stability Polyhedron in Multicontact8 Cooperative Collision Avoidance for Nonholonomic Robots9 A Physics-Based Power Model for Skid-Steered Wheeled Mobile Robots10 Formation Control of Nonholonomic Mobile Robots Without Position and Velocity Measurements11 Online Identification of Environment Hunt–Crossley Models Using Polynomial Linearization12 Coordinated Search With Multiple Robots Arranged in Line Formations13 Cable-Based Robotic Crane (CBRC): Design and Implementation of Overhead Traveling Cranes Based on Variable Radius Drums14 Online Approximate Optimal Station Keeping of a Marine Craft in the Presence of an Irrotational Current15 Ultrahigh-Precision Rotational Positioning Under a Microscope: Nanorobotic System, Modeling, Control, and Applications16 Adaptive Gain Control Strategy for Constant Optical Flow Divergence Landing17 Controlling Noncooperative Herds with Robotic Herders18 ε⋆: An Online Coverage Path Planning Algorithm19 Full-Pose Tracking Control for Aerial Robotic Systems With Laterally Bounded Input Force20 Comparative Peg-in-Hole Testing of a Force-Based Manipulation Controlled Robotic HandISSUE 11 Development of the Humanoid Disaster Response Platform DRC-HUBO+2 Active Stiffness Tuning of a Spring-Based Continuum Robot for MRI-Guided Neurosurgery3 Parallel Continuum Robots: Modeling, Analysis, and Actuation-Based Force Sensing4 A Rationale for Acceleration Feedback in Force Control of Series Elastic Actuators5 Real-Time Area Coverage and Target Localization Using Receding-Horizon Ergodic Exploration6 Interaction Between Inertia, Viscosity, and Elasticity in Soft Robotic Actuator With Fluidic Network7 Exploiting Elastic Energy Storage for “Blind”Cyclic Manipulation: Modeling, Stability Analysis, Control, and Experiments for Dribbling8 Enhance In-Hand Dexterous Micromanipulation by Exploiting Adhesion Forces9 Trajectory Deformations From Physical Human–Robot Interaction10 Robotic Manipulation of a Rotating Chain11 Design Methodology for Constructing Multimaterial Origami Robots and Machines12 Dynamically Consistent Online Adaptation of Fast Motions for Robotic Manipulators13 A Controller for Guiding Leg Movement During Overground Walking With a Lower Limb Exoskeleton14 Direct Force-Reflecting Two-Layer Approach for Passive Bilateral Teleoperation With Time Delays15 Steering a Swarm of Particles Using Global Inputs and Swarm Statistics16 Fast Scheduling of Robot Teams Performing Tasks With Temporospatial Constraints17 A Three-Dimensional Magnetic Tweezer System for Intraembryonic Navigation and Measurement18 Adaptive Compensation of Multiple Actuator Faults for Two Physically Linked 2WD Robots19 General Lagrange-Type Jacobian Inverse for Nonholonomic Robotic Systems20 Asymmetric Bimanual Control of Dual-Arm Exoskeletons for Human-Cooperative Manipulations21 Fourier-Based Shape Servoing: A New Feedback Method to Actively Deform Soft Objects into Desired 2-D Image Contours22 Hierarchical Force and Positioning Task Specification for Indirect Force Controlled Robots。

1 预测性维护系统需求分析1.1 工业设备维护管理现状由于传统石化生产设备信息管理程度低、预测性维护水平低，难以满足当下互联网智能化、精准化预测的管理及维护模式需求。

因此，刘凤燕[1]提出设备预测性维修的概念，实现提前维护不同石化设备的目标，减少工企业经济、生产设备的损失。

例如袁烨等[2]提出控制图解释方法，通过构建预测性维护优化模型，分析控制图解释参数，以确定设备预测性维护周期。

沈梦姣等[3-4]提出根据标识解析技术构建多服务单元数据预测智能化系统，当设备出现故障时，数据服务集合能有效标识解析单一服务设备单元，实现设备故障提前监测和预测的功能，从而有效提高设备健康运行效率。

1.2 功能性需求分析基于上述设备预测性维护的关键性作用，该文根据标识解析技术构建科学、可信的工业互联网标识解析二级节点设备预测性维护服务系统[5]，该系统具有以下功能：1） 服务系统需要具备完整的工业设备标识数据管理功能（注册、查询、更新以及监测等），将可信和非可信数据分布式存储在数据库中，实现标识数据可信同步的目标。

2） 工业设备标识数据管理记录公开化、共享化，提高石化行业设备异常数据信息监管和数据溯源能力，以解析标识数据，从而实现设备的预测性维护[6]。

3） 石化工业设备预测性维护系统需要准确、高效获取和反馈标识设备状态数据，提高供应链系统和石化工业设备系统的对接精度，实现设备预测性维护的目标[7]。

2 基于标识解析技术的工业设备预测性维护系统架构2.1 设计原则标识解析是一种信息映射的关键技术，一方面，其具备数据交互、设备获取快捷等优势。

另一方面，该技术能将虚拟对基于标识解析技术的工业设备预测性维护系统设计与实现房 兢（中海油信息科技有限公司，广东 深圳 516068）摘 要：针对石化行业设备因异常振动、断裂而导致的被动高成本维修，该文提出了基于标识解析技术与设备预测性维护相结合的预防维护智能化管理系统，该系统具备标识设备注册管理、可信数据存储、多类型数据监测以及数据校验等功能，有效提高了石化行业标识设备信息互通互联可信安全和设备故障预测维护能力，降低了被动维护的经济成本，实现了标识解析技术下设备实时运行参数采集和算法预测功能，进一步解决了由未及时发现的设备故障所引起的许多问题，为石化行业设备管理及维护人员提供了科学、准确的决策依据。

第40卷第9期2023年9月控制理论与应用Control Theory&ApplicationsV ol.40No.9Sep.2023不对称约束多人非零和博弈的自适应评判控制李梦花,王鼎,乔俊飞†(北京工业大学信息学部,北京100124;计算智能与智能系统北京市重点实验室,北京100124;智慧环保北京实验室,北京100124;北京人工智能研究院,北京100124)摘要:本文针对连续时间非线性系统的不对称约束多人非零和博弈问题,建立了一种基于神经网络的自适应评判控制方法.首先,本文提出了一种新颖的非二次型函数来处理不对称约束问题,并且推导出最优控制律和耦合Hamilton-Jacobi方程.值得注意的是,当系统状态为零时,最优控制策略是不为零的,这与以往不同.然后,通过构建单一评判网络来近似每个玩家的最优代价函数,从而获得相关的近似最优控制策略.同时,在评判学习期间发展了一种新的权值更新规则.此外,通过利用Lyapunov理论证明了评判网络权值近似误差和闭环系统状态的稳定性.最后,仿真结果验证了本文所提方法的有效性.关键词:神经网络;自适应评判控制;自适应动态规划;非线性系统;不对称约束;多人非零和博弈引用格式:李梦花,王鼎,乔俊飞.不对称约束多人非零和博弈的自适应评判控制.控制理论与应用,2023,40(9): 1562–1568DOI:10.7641/CTA.2022.20063Adaptive critic control for multi-player non-zero-sum games withasymmetric constraintsLI Meng-hua,WANG Ding,QIAO Jun-fei†(Faculty of Information Technology,Beijing University of Technology,Beijing100124,China;Beijing Key Laboratory of Computational Intelligence and Intelligent System,Beijing100124,China;Beijing Laboratory of Smart Environmental Protection,Beijing100124,China;Beijing Institute of Artificial Intelligence,Beijing100124,China)Abstract:In this paper,an adaptive critic control method based on the neural networks is established for multi-player non-zero-sum games with asymmetric constraints of continuous-time nonlinear systems.First,a novel nonquadratic func-tion is proposed to deal with asymmetric constraints,and then the optimal control laws and the coupled Hamilton-Jacobi equations are derived.It is worth noting that the optimal control strategies do not stay at zero when the system state is zero, which is different from the past.After that,only a critic network is constructed to approximate the optimal cost function for each player,so as to obtain the associated approximate optimal control strategies.Meanwhile,a new weight updating rule is developed during critic learning.In addition,the stability of the weight estimation errors of critic networks and the closed-loop system state is proved by utilizing the Lyapunov method.Finally,simulation results verify the effectiveness of the method proposed in this paper.Key words:neural networks;adaptive critic control;adaptive dynamic programming;nonlinear systems;asymmetric constraints;multi-player non-zero-sum gamesCitation:LI Menghua,WANG Ding,QIAO Junfei.Adaptive critic control for multi-player non-zero-sum games with asymmetric constraints.Control Theory&Applications,2023,40(9):1562–15681引言自适应动态规划(adaptive dynamic programming, ADP)方法由Werbos[1]首先提出,该方法结合了动态规划、神经网络和强化学习,其核心思想是利用函数近似结构来估计最优代价函数,从而获得被控系统的近似最优解.在ADP方法体系中,动态规划蕴含最优收稿日期:2022−01−21;录用日期:2022−11−10.†通信作者.E-mail:***************.cn.本文责任编委:王龙.科技创新2030–“新一代人工智能”重大项目(2021ZD0112302,2021ZD0112301),国家重点研发计划项目(2018YFC1900800–5),北京市自然科学基金项目(JQ19013),国家自然科学基金项目(62222301,61890930–5,62021003)资助.Supported by the National Key Research and Development Program of China(2021ZD0112302,2021ZD0112301,2018YFC1900800–5),the Beijing Natural Science Foundation(JQ19013)and the National Natural Science Foundation of China(62222301,61890930–5,62021003).第9期李梦花等:不对称约束多人非零和博弈的自适应评判控制1563性原理提供理论基础,神经网络作为函数近似结构提供实现手段,强化学习提供学习机制.值得注意的是, ADP方法具有强大的自学习能力,在处理非线性复杂系统的最优控制问题上具有很大的潜力[2–7].此外, ADP作为一种近似求解最优控制问题的新方法,已经成为智能控制与计算智能领域的研究热点.关于ADP的详细理论研究以及相关应用,读者可以参考文献[8–9].本文将基于ADP的动态系统优化控制统称为自适应评判控制.近年来,微分博弈问题在控制领域受到了越来越多的关注.微分博弈为研究多玩家系统的协作、竞争与控制提供了一个标准的数学框架,包括二人零和博弈、多人零和博弈以及多人非零和博弈等.在零和博弈问题中,控制输入试图最小化代价函数而干扰输入试图最大化代价函数.在非零和博弈问题中,每个玩家都独立地选择一个最优控制策略来最小化自己的代价函数.值得注意的是,零和博弈问题已经被广泛研究.在文献[10]中,作者提出了一种改进的ADP方法来求解多输入非线性连续系统的二人零和博弈问题.An等人[11]提出了两种基于积分强化学习的算法来求解连续时间系统的多人零和博弈问题.Ren等人[12]提出了一种新颖的同步脱策方法来处理多人零和博弈问题.然而,关于非零和博弈[13–14]的研究还很少.此外,控制约束在实际应用中也广泛存在.这些约束通常是由执行器的固有物理特性引起的,如气压、电压和温度.因此,为了确保被控系统的性能,受约束的系统需要被考虑.Zhang等人[15]发展了一种新颖的事件采样ADP方法来求解非线性连续约束系统的鲁棒最优控制问题.Huo等人[16]研究了一类非线性约束互联系统的分散事件触发控制问题.Yang和He[17]研究了一类具有不匹配扰动和输入约束的非线性系统事件触发鲁棒镇定问题.这些文献考虑的都是对称约束,而实际应用中,被控系统受到的约束也可能是不对称的[18–20],例如在污水处理过程中,需要通过氧传递系数和内回流量对溶解氧浓度和硝态氮浓度进行控制,而根据实际的运行条件,这两个控制变量就需要被限制在一个不对称约束范围内[20].因此,在控制器设计过程中,不对称约束问题将是笔者研究的一个方向.到目前为止,关于具有控制约束的微分博弈问题,有一些学者取得了相应的研究成果[12,21–23].但可以发现,具有不对称约束的多人非零和博弈问题还没有学者研究.同时,在多人非零和博弈问题中,相关的耦合Hamilton-Jacobi(HJ)方程是很难求解的.因此,本文针对一类连续时间非线性系统的不对称约束多人非零和博弈问题,提出了一种自适应评判控制方法来近似求解耦合HJ方程,从而获得被控系统的近似最优解.本文的主要贡献如下:1)首次将不对称约束应用到连续时间非线性系统的多人非零和博弈问题中;2)提出了一种新颖的非二次型函数来处理不对称约束问题,并且当系统状态为零时,最优控制策略是不为零的,这与以往不同;3)在学习期间,用单一评判网络结构代替了传统的执行–评判网络结构,并且提出了一种新的权值更新规则;4)利用Lyapunov方法证明了评判网络权值近似误差和系统状态的一致最终有界(uniformly ultimately bounded,UUB)稳定性.2问题描述考虑以下具有不对称约束的N–玩家连续时间非线性系统:˙x(t)=f(x(t))+N∑j=1g j(x(t))u j(t),(1)其中:x(t)∈Ω⊂R n是状态向量且x(0)=x0为初始状态,R n代表由所有n-维实向量组成的欧氏空间,Ω是R n的一个紧集;u j(t)∈T j⊂R m为玩家j在时刻t所选择的策略,且T j为T j={[u j1u j2···u jm]T∈R m:u j min u jl u j max, |u j min|=|u j max|,l=1,2,···,m},(2)其中:u jmin∈R和u j max∈R分别代表控制输入分量的最小界和最大界,R表示所有实数集.假设1非线性系统(1)是可控的,并且x=0是被控系统(1)的一个平衡点.此外,∀j∈N,f(x)和g j(x)是未知的Lipschitz函数且f(0)=0,其中集合N={1,2,···,N},N 2是一个正整数.假设2∀j∈N,g j(0)=0,且存在一个正常数b gj使∥g j(x)∥ b gj,其中∥·∥表示在R n上的向量范数或者在R n×m上的矩阵范数,R n×m代表由所有n×m维实矩阵组成的空间.注1假设1–3是自适应评判领域的常用假设,例如文献[6,13,19],是为了保证系统的稳定性以及方便后文中的稳定性证明,其中假设3出现在后文中的第3.2节.定义与每个玩家相关的效用函数为U i(x,U)=x T Q i x+N∑j=1S j(u j),i∈N,(3)其中U={u1,u2,···,u N}并且Q i是一个对称正定矩阵.此外,为了处理不对称约束问题,令S j(u j)为S j(u j)=2αj m∑l=1ujlβjtanh−1(z−βjαj)d z,(4)其中αj和βj分别为αj=u jmax−u j min2,βj=u jmax+u jmin2.(5)因此,与每个玩家相关的代价函数可以表示为J i(x0,U)=∞U i(x,U)dτ,i∈N,(6)1564控制理论与应用第40卷本文希望构建一个Nash均衡U∗={u∗1,u∗2,···,u∗N},来使以下不等式被满足:J i(u∗1,···,u∗i,···,u∗N)J i(u∗1,···,u i,···,u∗N),(7)其中i∈N.为了方便,将J i(x0,U)简写为J i(x0).于是,每个玩家的最优代价函数为J∗i (x0)=minu iJ i(x0,U),i∈N.(8)在本文中,如果一个控制策略集的所有元素都是可容许的,那么这个集合是可容许的.定义1(容许控制[24])如果控制策略u i(x)是连续的,u i(x)可以镇定系统(1),并且J i(x0)是有限的,那么它是集合Ω上关于代价函数(6)的可容许控制律,即u i(x)∈Ψ(Ω),i∈N,其中,Ψ(Ω)是Ω上所有容许控制律的集合.对于任意一个可容许控制律u i(x)∈Ψ(Ω),如果相关代价函数(6)是连续可微的,那么非线性Lyapu-nov方程为0=U i(x,U)+(∇J i(x))T(f(x)+N∑j=1g j(x)u j),(9)其中:i∈N,J i(0)=0,并且∇(·) ∂(·)∂x.根据最优控制理论,耦合HJ方程为0=minU H i(x,U,∇J∗i(x)),i∈N,(10)其中,Hamiltonian函数H i(x,U,∇J∗i(x))为H i(x,U,∇J∗i(x))=U i(x,U)+(∇J∗i (x))T(f(x)+N∑j=1g j(x)u j),(11)进而,由∂H i(x,U,∇J∗i(x))∂u i=0可得出最优控制律为u∗i (x)=−αi tanh(12αig Ti(x)∇J∗i(x))+¯βi,i∈N,(12)其中¯βi=[βiβi···βi]T∈R m.注2根据式(2)和式(5),能推导出βi=0,即¯βi=0,又根据式(12)可知u∗i(0)=0,i∈N.因此,为了保证x=0是系统(1)的平衡点,在假设2中提出了条件∀j∈N,g j(0)=0.将式(12)代入式(10),耦合HJ方程又能表示为(∇J∗i (x))T f(x)+N∑j=1((∇J∗i(x))T g j(x)¯βj)+x T Q i x−N∑j=1((∇J∗i(x))Tαj g j(x)tanh(A j(x)))+N∑j=1S j(−αj tanh(A j(x))+¯βj)=0,i∈N,(13)其中J∗i(0)=0并且A j(x)=12αjg Tj(x)∇J∗j(x).如果已知每个玩家的最优代价函数值,那么相关的最优状态反馈控制律就可以直接获得,也就是说式(13)是可解的.可是,式(13)这种非线性偏微分方程的求解是十分困难的.同时,随着系统维数的增加,存储量和计算量也随之以指数形式增加,也就是平常所说的“维数灾”问题.因此,为了克服这些弱点,在第3部分提出了一种基于神经网络的自适应评判机制,来近似每个玩家的最优代价函数,从而获得相关的近似最优状态反馈控制策略.3自适应评判控制设计3.1神经网络实现本节的核心是构建并训练评判神经网络,以得到训练后的权值,从而获得每个玩家的近似最优代价函数值.首先,根据神经网络的逼近性质[25],可将每个玩家的最优代价函数J∗i(x)在紧集Ω上表示为J∗i(x)=W Tiσi(x)+ξi(x),i∈N,(14)其中:W i∈Rδ是理想权值向量,σi(x)∈Rδ是激活函数,δ是隐含层神经元个数,ξi(x)∈R是重构误差.同时,可得出每个玩家的最优代价函数梯度为∇J∗i(x)=(∇σi(x))T W i+∇ξi(x),i∈N,(15)将式(15)代入式(12),有u∗i(x)=−αi tanh(B i(x)+C i(x))+¯βi,i∈N,(16)其中:B i(x)=12αig Ti(x)(∇σi(x))T W i∈R m,C i(x)=12αig Ti(x)∇ξi(x)∈R m.然后,将式(15)代入式(13),耦合HJ方程变为W Ti∇σi(x)f(x)+(∇ξi(x))T f(x)+x T Q i x+N∑j=1((W Ti∇σi(x)+(∇ξi(x))T)g j(x)¯βj)−N∑j=1(αj W Ti∇σi(x)g j(x)tanh(B j(x)+C j(x)))−N∑j=1(αj(∇ξi(x))T g j(x)tanh(B j(x)+C j(x)))+N∑j=1S j(−αj tanh(B j(x)+C j(x))+¯βj)=0,i∈N.(17)值得注意的是,式(14)中的理想权值向量W i是未知的,也就是说式(16)中的u∗i(x)是不可解的.因此,第9期李梦花等:不对称约束多人非零和博弈的自适应评判控制1565构建如下的评判神经网络:ˆJ∗i (x)=ˆW Tiσi(x),i∈N,(18)来近似每个玩家的最优代价函数,其中ˆW i∈Rδ是估计的权值向量.同时,其梯度为∇ˆJ∗i(x)=(∇σi(x))TˆW i,i∈N.(19)考虑式(19),近似的最优控制律为ˆu∗i(x)=−αi tanh(D i(x))+¯βi,i∈N,(20)其中D i(x)=12αig Ti(x)(∇σi(x))TˆW i.同理,近似的Hamiltonian可以写为ˆHi(x,ˆW i)=ˆW T i ϕi+x T Q i x+N∑j=1(ˆW Ti∇σi(x)g j(x)¯βj)−N ∑j=1(αjˆW Ti∇σi(x)g j(x)tanh(D j(x)))+N∑j=1S j(−αj tanh(D j(x))+¯βj),i∈N,(21)其中ϕi=∇σi(x)f(x).此外,定义误差量e i=ˆH i(x,ˆW i )−H i(x,U∗,∇J∗i(x))=ˆH i(x,ˆW i).为了使e i足够小,需要训练评判网络来使目标函数E i=12e Tie i最小化.在这里,本文采用的训练准则为˙ˆW i =−γi1(1+ϕTiϕi)2(∂E i∂ˆW i)=−γiϕi(1+ϕTiϕi)2e i,i∈N,(22)其中:γi>0是评判网络的学习率,(1+ϕT iϕi)2用于归一化操作.此外,定义评判网络的权值近似误差为˜Wi=W i−ˆW i.因此,有˙˜W i =γiφi1+ϕTiϕie Hi−γiφiφT i˜W i,i∈N,(23)其中:φi=ϕi(1+ϕTiϕi),e Hi=−(∇ξi(x))T f(x)是残差项.3.2稳定性分析本节的核心是通过利用Lyapunov方法讨论评判网络权值近似误差和闭环系统状态的UUB稳定性.这里,给出以下假设:假设3∥∇ξi(x)∥ b∇ξi ,∥∇σi(x)∥ b∇σi,∥e Hi∥ b e Hi,∥W i∥ b W i,其中:b∇ξi,b∇σi,b e Hi,b W i 都是正常数,i∈N.定理1考虑系统(1),如果假设1–3成立,状态反馈控制律由式(20)给出,且评判网络权值通过式(22)进行训练,则评判网络权值近似误差˜W i是UUB 稳定的.证选取如下的Lyapunov函数:L1(t)=N∑i=1(12˜W Ti˜Wi)=N∑i=1L1i(t),(24)计算L1i(t)沿着式(23)的时间导数,即˙L1i(t)=γi˜W Tiφi1+ϕTiϕie Hi−γi˜W TiφiφTi˜Wi,i∈N,(25)利用不等式¯X T¯Y12∥¯X∥2+12∥¯Y∥2(注:¯X和¯Y都是具有合适维数的向量),并且考虑1+ϕTiϕi 1,能得到˙L1i(t)γi2(∥φTi˜Wi∥2+∥e Hi∥2)−γi˜W TiφiφTi˜Wi=−γi2˜W TiφiφTi˜Wi+γi2∥e Hi∥2,i∈N.(26)根据假设3,有˙L1i(t) −γi2λmin(φiφTi)∥˜W i∥2+γi2b2e Hi,i∈N,(27)其中λmin(·)表示矩阵的最小特征值.因此,当不等式∥˜W i∥>√b2e Hiλmin(φiφTi),i∈N(28)成立时,有˙L1i(t)<0.根据标准的Lyapunov定理[26],可知评判网络权值近似误差˜W i是UUB稳定的.证毕.定理2考虑系统(1),如果假设1–3成立,状态反馈控制律由式(20)给出,且评判网络权值通过式(22)进行训练,则系统状态x(t)是UUB稳定的.证选取如下的Lyapunov函数:L2i(t)=J∗i(x),i∈N.(29)计算L2i(t)沿着系统˙x=f(x)+N∑j=1g j(x)ˆu∗j的时间导数,即˙L2i(t)=(∇J∗i(x))T(f(x)+N∑j=1g j(x)ˆu∗j)=(∇J∗i(x))T(f(x)+N∑j=1g j(x)u∗j)+N∑j=1((∇J∗i(x))T g j(x)(ˆu∗j−u∗j)),i∈N.(30)考虑式(13),有˙L2i(t)=−x T Q i x−N∑j=1S j(u∗j)+N∑j=1((∇J∗i(x))T g j(x)(ˆu∗j−u∗j))Σi,i∈N,(31)1566控制理论与应用第40卷利用不等式¯XT ¯Y 12∥¯X ∥2+12∥¯Y ∥2,并且考虑式(15)–(16)(20),可得Σi 12N ∑j =1∥−αj tanh (D j (x ))+αj tanh (F j (x ))∥2+12N ∑j =1∥g Tj (x )((∇σi (x ))T W i +∇ξi (x ))∥2,i ∈N ,(32)其中F j (x )=B j (x )+C j (x ).然后,利用不等式∥¯X+¯Y∥2 2∥¯X ∥2+2∥¯Y ∥2,有Σi N ∑j =1(∥αj tanh (D j (x ))∥2+∥αj tanh (F j (x ))∥2)+N ∑j =1∥g Tj (x )(∇σi (x ))T W i ∥2+N ∑j =1∥g T j (x )∇ξi (x )∥2,i ∈N ,(33)其中D j (x )∈R m ,F j (x )∈R m 分别被表示为[D j 1(x )D j 2(x )···D jm (x )]T 和[F j 1(x )F j 2(x )···F jm (x )]T .易知,∀θ∈R ,tanh 2θ 1.因此,有∥tanh (D j (x ))∥2=m ∑l =1tanh 2(D jl (x )) m,(34)∥tanh (F j (x ))∥2=m ∑l =1tanh 2(F jl (x )) m.(35)同时,根据假设2–3,有Σi N ∑j =1(2α2j m +b 2g j b 2∇σi b 2W i +b 2g j b 2∇ξi ),i ∈N ,(36)根据式(2)(4)–(5),可知S j (u ∗j ) 0.于是,有˙L2i (t ) −λmin (Q i )∥x ∥2+ϖi ,i ∈N ,(37)其中ϖi =N ∑j =1(2α2j m +b 2g j b 2∇σi b 2W i +b 2g j b 2∇ξi ).因此,根据式(37)可知,当不等式∥x ∥>√ϖiλmin (Q i )成立时,有˙L2i (t )<0.即,如果x (t )满足下列不等式:∥x ∥>max {√ϖ1λmin (Q 1),···,√ϖNλmin (Q N )},(38)则,∀i ∈N ,都有˙L 2i (t )<0.同理,可得闭环系统状态x (t )也是UUB 稳定的.证毕.4仿真结果考虑如下的3–玩家连续时间非线性系统:˙x =[−1.2x 1+1.5x 2sin x 20.5x 1−x 2]+[01.5sin x 1cos x 1]u 1(x )+[1.2sin x 1cos x 2]u 2(x )+[01.1sin x 2]u 3(x ),(39)其中:x (t )=[x 1x 2]T ∈R 2是状态向量,u 1(x )∈T 1={u 1∈R :−1 u 1 2},u 2(x )∈T 2={u 2∈R :−0.2 u 2 1}和u 3(x )∈T 3={u 3∈R :−0.4 u 3 0.8}是控制输入.令Q 1=2I 2,Q 2=1.8I 2,Q 3=0.3I 2,其中I 2代表2×2维单位矩阵.同时,根据式(5)可知,α1=1.5,β1=0.5,α2=0.6,β2=0.4,α3=0.6,β3=0.2.因此,与每个玩家相关的代价函数可以表示为J i (x 0)= ∞0(x TQ i x +3∑j =1S j (u j ))d τ,i =1,2,3,(40)其中S j (u j )=2αju jβj tanh −1(z −βjαj)d z =2αj (u j −βj )tanh −1(u j −βjαj)+α2j ln (1−(u j −βj )2α2j).(41)然后,本文针对系统(39)构建3个评判神经网络,每个玩家的评判神经网络权值分别为ˆW1=[ˆW 11ˆW 12ˆW13]T ,ˆW 2=[ˆW 21ˆW 22ˆW 23]T ,ˆW 3=[ˆW 31ˆW 32ˆW33]T ,激活函数被定义为σ1(x )=σ2(x )=σ3(x )=[x 21x 1x 2x 22]T,且隐含层神经元个数为δ=3.此外,系统初始状态取x 0=[0.5−0.5]T ,每个评判神经网络的学习率分别为γ1=1.5,γ2=0.8,γ3=0.2,且每个评判神经网络的初始权值都在0和2之间选取.最后,引入探测噪声η(t )=sin 2(−1.2t )cos(0.5t )+cos(2.4t )sin 3(2.4t )+sin 5t +sin 2(1.12t )+sin 2t ×cos t +sin 2(2t )cos(0.1t ),使得系统满足持续激励条件.执行学习过程,本文发现每个玩家的评判神经网络权值分别收敛于[6.90912.99046.6961]T ,[4.89012.23475.2062]T ,[1.79450.33212.4583]T .在60个时间步之后去掉探测噪声,每个玩家的评判网络权值收敛过程如图1–3所示.然后,将训练好的权值代入式(20),能得到每个玩家的近似最优控制律,将其应用到系统(39),经过10个时间步之后,得到的状态轨迹和控制轨迹分别如图4–5所示.由图4可知,系统状态最终收敛到了平衡点.由图5可知,每个玩家的控制轨迹都没有超出预定的边界,并且可以观察到u 1,u 2和u 3分别收敛于0.5,0.4和0.2.综上所述,仿真结果验证了所提方法的有效性.第9期李梦花等:不对称约束多人非零和博弈的自适应评判控制1567䇴 㖁㔌U / s图1玩家1的评判网络权值收敛过程Fig.1Convergence process of the critic network weights forplayer1䇴 㖁㔌U / s图2玩家2的评判网络权值收敛过程Fig.2Convergence process of the critic network weights forplayer2﹣䇴 㖁㔌U / s图3玩家3的评判网络权值收敛过程Fig.3Convergence process of the critic network weights forplayer 35结论本文首次将不对称约束应用到连续时间非线性系统的多人非零和博弈问题中.首先,获得了最优状态反馈控制律和耦合HJ 方程,并且为了解决不对称约束问题,建立了一种新的非二次型函数.值得注意的是,当系统状态为零时,最优控制策略是不为零的.其次,由于耦合HJ 方程不易求解,提出了一种基于神经网络的自适应评判算法来近似每个玩家的最优代价函数,从而获得相关的近似最优控制律.在实现过程中,用单一评判网络结构代替了经典的执行–评判结构,并且建立了一种新的权值更新规则.然后,利用Lyap-unov 理论讨论了评判网络权值近似误差和系统状态的UUB 稳定性.最后,仿真结果验证了所提算法的可行性.在未来的工作中,会考虑将事件驱动机制引入到连续时间非线性系统的不对称约束多人非零和博弈问题中,并且将该研究内容应用到污水处理系统中也是笔者的一个重点研究方向.﹣0.5﹣0.4﹣0.3﹣0.2﹣0.10.00.10.20.00.10.20.30.40.5(U )Y 1(U )Y 2图4系统(39)的状态轨迹Fig.4State trajectory of the system (39)0.00.51.01.52.00.00.20.40.60.81.01.200.012345678910﹣0.40.4﹣0.20.2(U )V 3(U )V 2(U )V 1U / s 012345678910U / s 012345678910U / s (c)(b)(a)(U )V 1(U )V 2(U )V 3图5系统(39)的控制轨迹Fig.5Control trajectories of the system (39)1568控制理论与应用第40卷参考文献:[1]WERBOS P J.Beyond regression:New tools for prediction andanalysis in the behavioral sciences.Cambridge:Harvard Universi-ty,1974.[2]HONG Chengwen,FU Yue.Nonlinear robust approximate optimaltracking control based on adaptive dynamic programming.Control Theory&Applications,2018,35(9):1285–1292.(洪成文,富月.基于自适应动态规划的非线性鲁棒近似最优跟踪控制.控制理论与应用,2018,35(9):1285–1292.)[3]CUI Lili,ZHANG Yong,ZHANG Xin.Event-triggered adaptive dy-namic programming algorithm for the nonlinear zero-sum differential games.Control Theory&Applications,2018,35(5):610–618.(崔黎黎,张勇,张欣.非线性零和微分对策的事件触发自适应动态规划算法.控制理论与应用,2018,35(5):610–618.)[4]WANG D,HA M,ZHAO M.The intelligent critic framework foradvanced optimal control.Artificial Intelligence Review,2022,55(1): 1–22.[5]WANG D,QIAO J,CHENG L.An approximate neuro-optimal solu-tion of discounted guaranteed cost control design.IEEE Transactions on Cybernetics,2022,52(1):77–86.[6]YANG X,HE H.Adaptive dynamic programming for decentralizedstabilization of uncertain nonlinear large-scale systems with mis-matched interconnections.IEEE Transactions on Systems,Man,and Cybernetics:Systems,2020,50(8):2870–2882.[7]ZHAO B,LIU D.Event-triggered decentralized tracking control ofmodular reconfigurable robots through adaptive dynamic program-ming.IEEE Transactions on Industrial Electronics,2020,67(4): 3054–3064.[8]WANG Ding.Research progress on learning-based robust adaptivecritic control.Acta Automatica Sinica,2019,45(6):1037–1049.(王鼎.基于学习的鲁棒自适应评判控制研究进展.自动化学报, 2019,45(6):1037–1049.)[9]ZHANG Huaguang,ZHANG Xin,LUO Yanhong,et al.An overviewof research on adaptive dynamic programming.Acta Automatica Sini-ca,2013,39(4):303–311.(张化光,张欣,罗艳红,等.自适应动态规划综述.自动化学报, 2013,39(4):303–311.)[10]L¨U Yongfeng,TIAN Jianyan,JIAN Long,et al.Approximate-dynamic-programming H∞controls for multi-input nonlinear sys-tem.Control Theory&Applications,2021,38(10):1662–1670.(吕永峰,田建艳,菅垄,等.非线性多输入系统的近似动态规划H∞控制.控制理论与应用,2021,38(10):1662–1670.)[11]AN P,LIU M,WAN Y,et al.Multi-player H∞differential gameusing on-policy and off-policy reinforcement learning.The16th In-ternational Conference on Control and Automation.Electr Network: IEEE,2020,10:1137–1142.[12]REN H,ZHANG H,MU Y,et al.Off-policy synchronous iterationIRL method for multi-player zero-sum games with input constraints.Neurocomputing,2020,378:413–421.[13]LIU D,LI H,WANG D.Online synchronous approximate optimallearning algorithm for multiplayer nonzero-sum games with unknown dynamics.IEEE Transactions on Systems,Man,and Cybernetics: Systems,2014,44(8):1015–1027.[14]V AMVOUDAKIS K G,LEWIS F L.Non-zero sum games:Onlinelearning solution of coupled Hamilton-Jacobi and coupled Riccati equations.IEEE International Symposium on Intelligent Control.Denver,CO,USA:IEEE,2011,9:171–178.[15]ZHANG H,ZHANG K,XIAO G,et al.Robust optimal controlscheme for unknown constrained-input nonlinear systems via a plug-n-play event-sampled critic-only algorithm.IEEE Transactions on Systems,Man,and Cybernetics:Systems,2020,50(9):3169–3180.[16]HUO X,KARIMI H R,ZHAO X,et al.Adaptive-critic design fordecentralized event-triggered control of constrained nonlinear inter-connected systems within an identifier-critic framework.IEEE Trans-actions on Cybernetics,2022,52(8):7478–7491.[17]YANG X,HE H.Event-triggered robust stabilization of nonlin-ear input-constrained systems using single network adaptive critic designs.IEEE Transactions on Systems,Man,and Cybernetics:Sys-tems,2020,50(9):3145–3157.[18]WANG L,CHEN C L P.Reduced-order observer-based dynamicevent-triggered adaptive NN control for stochastic nonlinear systems subject to unknown input saturation.IEEE Transactions on Neural Networks and Learning Systems,2021,32(4):1678–1690.[19]YANG X,ZHU Y,DONG N,et al.Decentralized event-driven con-strained control using adaptive critic designs.IEEE Transactions on Neural Networks and Learning Systems,2022,33(10):5830–5844.[20]WANG D,ZHAO M,QIAO J.Intelligent optimal tracking withasymmetric constraints of a nonlinear wastewater treatment system.International Journal of Robust and Nonlinear Control,2021,31(14): 6773–6787.[21]LI M,WANG D,QIAO J,et al.Neural-network-based self-learningdisturbance rejection design for continuous-time nonlinear con-strained systems.Proceedings of the40th Chinese Control Confer-ence.Shanghai,China:IEEE,2021,7:2179–2184.[22]SU H,ZHANG H,JIANG H,et al.Decentralized event-triggeredadaptive control of discrete-time nonzero-sum games over wireless sensor-actuator networks with input constraints.IEEE Transactions on Neural Networks and Learning Systems,2020,31(10):4254–4266.[23]YANG X,HE H.Event-driven H∞-constrained control using adap-tive critic learning.IEEE Transactions on Cybernetics,2021,51(10): 4860–4872.[24]ABU-KHALAF M,LEWIS F L.Nearly optimal control laws for non-linear systems with saturating actuators using a neural network HJB approach.Automatica,2005,41(5):779–791.[25]HORNIK K,STINCHCOMBE M,WHITE H.Universal approxima-tion of an unknown mapping and its derivatives using multilayer feed-forward networks.Neural Networks,1990,3(5):551–560.[26]LEWIS F L,JAGANNATHAN S,YESILDIREK A.Neural NetworkControl of Robot Manipulators and Nonlinear Systems.London:Tay-lor&Francis,1999.作者简介:李梦花博士研究生,目前研究方向为自适应动态规划、智能控制,E-mail:*********************;王鼎教授,博士生导师,目前研究方向为智能控制、强化学习,E-mail:*****************.cn;乔俊飞教授,博士生导师,目前研究方向为智能计算、智能优化控制,E-mail:***************.cn.。

改革开放以来,随着我国工业的迅速发展和科学技术的进步,电气控制技术在工业上的运用也越来越广泛,对于一个国家的科技水平高低来说,电气控制技术水平是一项重要的衡量因素.电气控制技术主要以电动机作为注重的对象,通过一系列的电气控制技术,买现生产或者监控的自动化.下面是搜索整理的电气控制英文参考文献，欢迎借鉴参考。

电气控制英文参考文献一： [1]Laiqing Xie,Yugong Luo,Donghao Zhang,Rui Chen,Keqiang Li. Intelligent energy-saving control strategy for electric vehicle based on preceding vehicle movement[J]. Mechanical Systems andSignal Processing,2019,130. [2]F.N. Tan,Q.Y. Wong,W.L. Gan,S.H. Li,H.X. Liu,F. Poh,W.S. Lew. Electric field control for energy efficient domain wallinjection[J]. Journal of Magnetism and Magnetic Materials,2019,485. [3]N. Nursultanov,W.J.B. Heffernan,M.J.W.M.R. van Herel,J.J. Nijdam. Computational calculation of temperature and electrical resistance to control Joule heating of green Pinus radiata logs[J]. Applied Thermal Engineering,2019,159. [4]Min Cheng,Junhui Zhang,Bing Xu,Ruqi Ding,Geng Yang. Anti-windup scheme of the electronic load sensing pump via switchedflow/power control[J]. Mechatronics,2019,61. [5]Miles L. Morgan,Dan J. Curtis,Davide Deganello. Control of morphological and electrical properties of flexographic printed electronics through tailored ink rheology[J]. OrganicElectronics,2019,73. [6]Maciej ?awryńczuk,Pawe?Oc?oń. Model Predictive Control and energy optimisation in residential building with electric underfloor heating system[J]. Energy,2019,182. [7]Lorenzo Niccolai,Alessandro Anderlini,GiovanniMengali,Alessandro A. Quarta. Electric sail displaced orbit control with solar wind uncertainties[J]. Acta Astronautica,2019,162. [8]Patrik Beňo,Matej Kubi?. Control and stabilization of single-wheeled electric vehicle with BLDC engine[J]. Transportation Research Procedia,2019,40. [9]André Murilo,Rafael Rodrigues,Evandro Leonardo SilvaTeixeira,Max Mauro Dias Santos. Design of a Parameterized Model Predictive Control for Electric Power Assisted Steering[J]. Control Engineering Practice,2019,90. [10]Kazusa Yamamoto,Olivier Sename,Damien Koenig,Pascal Moulaire. Design and experimentation of an LPV extended state feedback control on Electric Power Steering systems[J]. Control EngineeringPractice,2019,90. [11]Pedro de A. Delou,Julia P.A. de Azevedo,Dinesh Krishnamoorthy,Maurício B. de Souza,Argimiro R. Secchi. Model Predictive Control with Adaptive Strategy Applied to an Electric Submersible Pump in a Subsea Environment[J]. IFACPapersOnLine,2019,52(1). [12]Unal Yilmaz,Omer Turksoy,Ahmet Teke. Intelligent control of high energy efficient two-stage battery charger topology forelectric vehicles[J]. Energy,2019,186. [13]Qiuyi Guo,Zhiguo Zhao,Peihong Shen,Xiaowen Zhan,Jingwei Li. Adaptive optimal control based on driving style recognition forplug-in hybrid electric vehicle[J]. Energy,2019,186. [14]Leonid Lobanov,Nikolai Pashсhin. Electrodynamic treatment by electric current pulses as effective method of control of stress-strain states and improvement of life of welded structures[J]. Procedia Structural Integrity,2019,16. [15]Evangelos Pournaras,Seoho Jung,Srivatsan Yadhunathan,Huiting Zhang,Xingliang Fang. Socio-technical smart grid optimization via decentralized charge control of electric vehicles[J]. Applied Soft Computing Journal,2019,82. [16]Guoming Huang,Xiaofang Yuan,Ke Shi,Xiru Wu. A BP-PID controller-based multi-model control system for lateral stability of distributed drive electric vehicle[J]. Journal of the Franklin Institute,2019,356(13). [17]Ioannis Kalogeropoulos,Haralambos Sarimveis. Predictive control algorithms for congestion management in electric power distribution grids[J]. Applied Mathematical Modelling,2020,77. [18]Junjun Zhu,Zhenpo Wang,Lei Zhang,David G. Dorrell.Braking/steering coordination control for in-wheel motor drive electric vehicles based on nonlinear model predictive control[J]. Mechanism and Machine Theory,2019,142. [19]Jiechen Wu,Junjie Hu,Xin Ai,Zhan Zhang,Huanyu Hu. Multi-time scale energy management of electric vehicle model-based prosumers by using virtual battery model[J]. Applied Energy,2019,251. [20]G. Coorey,D. Peiris,T. Usherwood,L. Neubeck,J. Mulley,J. Redfern. An Internet-Based Intervention Integrated with the Primary Care Electronic Health Record to Improve Cardiovascular Disease Risk Factor Control: a Mixed-Methods Evaluation of Acceptability, Usage Trends and Persuasive Design Characteristics[J]. Heart, Lung and Circulation,2019,28. [21]Félice Lê-Scherban,Lance Ballester,Juan C. Castro,Suzanne Cohen,Steven Melly,Kari Moore,James W. Buehler. Identifying neighborhood characteristics associated with diabetes and hypertension control in an urban African-American population usinggeo-linked electronic health records[J]. Preventive Medicine Reports,2019,15. [22]Yuekuan Zhou,Sunliang Cao. Energy flexibility investigation of advanced grid-responsive energy control strategies with thestatic battery and electric vehicles: A case study of a high-rise office building in Hong Kong[J]. Energy Conversion and Management,2019,199. [23]D. Aravindh,R. Sakthivel,B. Kaviarasan,S. MarshalAnthoni,Faris Alzahrani. Design of observer-based non-fragile load frequency control for power systems with electric vehicles[J]. ISA Transactions,2019,91. [24]Augusto Matheus dos Santos Alonso,Danilo IglesiasBrandao,Tommaso Caldognetto,Fernando Pinhabel Maraf?o,Paolo Mattavelli. A selective harmonic compensation and power control approach exploiting distributed electronic converters inmicrogrids[J]. International Journal of Electrical Power and Energy Systems,2020,115. [25]Hay Wong,Derek Neary,Eric Jones,Peter Fox,Chris Sutcliffe. Benchmarking spatial resolution in electronic imaging for potential in-situ Electron Beam Melting monitoring[J]. Additive Manufacturing,2019,29. [26]Yunfei Bai,Hongwen He,Jianwei Li,Shuangqi Li,Ya-xiong Wang,Qingqing Yang. Battery anti-aging control for a plug-in hybrid electric vehicle with a hierarchical optimization energy management strategy[J]. Journal of Cleaner Production,2019,237. [27]N. Samartin-Veiga,A.J. González-Villar,M.T. Carrillo-de-la-Pe?a. Neural correlates of cognitive dysfunction in fibromyalgia patients: Reduced brain electrical activity during the execution ofa cognitive control task[J]. NeuroImage: Clinical,2019,23. [28]Masato Nakaya,Shinta Watanabe,Jun Onoe. Control of electric, optical, thermal properties of C 60 films by electron-beam irradiation[J]. Carbon,2019,152. [29]R. Saadi,M.Y. Hammoudi,O. Kraa,M.Y. Ayad,M. Bahri. A robust control of a 4-leg floating interleaved boost converter for fuel cell electric vehicle application[J]. Mathematics and Computers in Simulation,2019. [30]Frederik Banis,Daniela Guericke,Henrik Madsen,Niels Kj?lstad Poulsen. Supporting power balance in Microgrids with Uncertain Production using Electric Vehicles and Indirect Control ? ? This work has been supported by ENERGINET.DK under the project microgrid positioning - uGrip and the CITIES project.[J]. IFAC PapersOnLine,2019,52(4). 电气控制英文参考文献二： [31]Huijuan Luo,Jinpeng Yu,Chong Lin,Zhanjie Liu,Lin Zhao,Yumei Ma. Finite-time dynamic surface control for induction motors with input saturation in electric vehicle drive systems[J]. Neurocomputing,2019. [32]Peter K. Joseph,D. Elangovan,G. Arunkumar. Linear control of wireless charging for electric bicycles[J]. Applied Energy,2019,255. [33]Yu Congyang,Zhu Dequan,Wang Chaoxian,Zhu Lin,Chu Tingting,Jen Tien-Chien,Liao Juan. Optimizing Electric Adjustment Mechanism Using the Combination of Multi-body Dynamics and Control[J]. Procedia Manufacturing,2019,35. [34]Hussein Termous,Xavier Moreau,Clovis Francis,Hassan Shraim. Effect of fractional order damping control on braking performancefor electric vehicles ? ? This work was supported by the Lebanese research program and the AUF-CNRSL-UL program.[J]. IFAC PapersOnLine,2019,52(5). [35]Manuel Schwartz,Florian Siebenrock,S?ren Hohmann. Model Predictive Control Allocation of an Over-actuated Electric Vehicle with Single Wheel Actuators[J]. IFAC PapersOnLine,2019,52(8). [36]Di Wu,Nikitha Radhakrishnan,Sen Huang. A hierarchical charging control of plug-in electric vehicles with simpleflexibility model[J]. Applied Energy,2019,253. [37]Abhishek Nayak,Rubi Rana,Sukumar Mishra. Frequency Regulation by Electric Vehicle during Grid Restoration using Adaptive Optimal Control[J]. IFAC PapersOnLine,2019,52(4). [38]Nicolò Robuschi,Mauro Salazar,Pol Duhr,FrancescoBraghin,Christopher H. Onder. Minimum-fuel Engine On/Off Control for the Energy Management of a Hybrid Electric Vehicle via Iterative Linear Programming ? ? We thank Ferrari S.p.A. for supporting this project.[J]. IFAC PapersOnLine,2019,52(5). [39]Anas A. Ahmed,M.R. Hashim,Marzaini Rashid. Control of the structural, electrical and optical properties of spin coated NiO films by varying precursor molarity[J]. Thin Solid Films,2019,690. [40]Wilco van Harselaar,Niels Schreuders,Theo Hofman,Stephan Rinderknecht. Improved Implementation of Dynamic Programming on the Example of Hybrid Electric Vehicle Control[J]. IFACPapersOnLine,2019,52(5). [41]Jose A. Matute,Mauricio Marcano,Sergio Diaz,Joshue Perez. Experimental Validation of a Kinematic Bicycle Model Predictive Control with Lateral Acceleration Consideration ? ? This project has received funding from the Electronic Component Systems for European Leadership Joint Undertaking under grant agreement No 737469 (AutoDrive Project). This Joint Undertaking receives support fromthe European Union Horizon 2020 research and innovation programmeand Germany, Austria, Spain, Italy, Latvia, Belgium, Netherlands, Sweden, Finland, Lithuania, Czech Republic, Romania,[J]. IFAC PapersOnLine,2019,52(8). [42]Vladislav S. Gromov,Oleg I. Borisov,Sergey S. Shavetov,AntonA. Pyrkin,FatimatB. Karashaeva. Modeling and Control of Robotic Systems Course: from Fundamentals to Applications ? ? The work was written with the support of the Ministry of Science and Higher Education of the Russian Federation, project unique identifier RFMEFI57818X0271 “Adaptive Sensorless Control for Synchronous Electric Drives in Intelligent Robotics and Transport Systems”.[J]. IFAC PapersOnLine,2019,52(9). [43]H. Mbarak,A.K. Kodeary,S.M. Hamidi,E. Mohajarani,Y. Zaatar. Control of nonlinear refractive index of AuNPs doped with nematic liquid crystal under external electric field[J]. Optik,2019,198. [44]Yanzhao Jia,Rabee Jibrin,Yutaro Itoh,Daniel G?rges. Energy-Optimal Adaptive Cruise Control for Electric Vehicles in Both Time and Space Domain based on Model Predictive Control[J]. IFAC PapersOnLine,2019,52(5). [45]Lukas Engbroks,Daniel G?rke,Stefan Schmiedler,TobiasG?decke,Bastian Beyfuss,Bernhard Geringer. Combined energy and thermal management for plug-in hybrid electric vehicles -analyses based on optimal control theory ? ? This work has been performed within the Daimler AG in Stuttgart, Germany in cooperation with the Institute for Powertrains and Automotive Technology at Vienna University of Technology, Austria.[J]. IFAC PapersOnLine,2019,52(5). [46]Jean Kuchly,Dominique Nelson-Gruel,Alain Charlet,Yann Chamaillard,Cédric Nouillant. Projected Gradient and ModelPredictive Control : Optimal Energy and Pollutants Management for Hybrid Electric Vehicle[J]. IFAC PapersOnLine,2019,52(5). [47]Pier Giuseppe Anselma,Yi Huo,Joel Roeleveld,Giovanni Belingardi,Ali Emadi. From Off-line to On-line Control of a Multimode Power Split Hybrid Electric Vehicle Powertrain[J]. IFAC PapersOnLine,2019,52(5). [48]Xiaoyong Zhu,Deyang Fan,Zixuan Xiang,Li Quan,Wei Hua,Ming Cheng. Systematic multi-level optimization design and dynamiccontrol of less-rare-earth hybrid permanent magnet motor for all-climatic electric vehicles[J]. Applied Energy,2019,253. [49]. Engineering - Industrial Engineering; Findings from Southwest Jiaotong University Provides New Data about Industrial Engineering (Optimal Energy Management and Control In Multimode Equivalent Energy Consumption of Fuel Cell/supercapacitor of Hybrid Electric Tram)[J]. Energy Weekly News,2019. [50]. SK Planet Co. Ltd.; Patent Issued for Electronic Stamp System For Security Intensification, Control Method Thereof, And Non-Transitory Computer Readable Storage Medium Having ComputerProgram Recorded Thereon (USPTO 10,361,857)[J]. Computers, Networks & Communications,2019. [51]. Energy - Electric Power; Study Data from National Institute of Technology Calicut Update Understanding of Electric Power (Modified switching scheme-based explicit torque control of brush-less direct current motor drive)[J]. Energy Weekly News,2019. [52]. Energy; Findings from School of Mechanical Engineering Reveals New Findings on Energy (Deep Reinforcement Learning of Energy Management With Continuous Control Strategy and Traffic Information for a Series-parallel Plug-in Hybrid Electric Bus)[J]. Energy Weekly News,2019. [53]. Energy - Electric Power; Reports Outline Electric Power Study Results from Dalian Maritime University (Direct VoltageControl of Stand-alone Dfig Under Asymmetric Loads Based On Non-singular Terminal Sliding Mode Control and Improved Extended State Observer)[J]. Energy Weekly News,2019. [54]. Energy - Electric Power; Studies from Xi'an Jiao Tong University Add New Findings in the Area of Electric Power (A model predictive control approach for matching uncertain wind generation with PEV charging demand in a microgrid)[J]. Energy WeeklyNews,2019. [55]. Energy - Electric Power; Researchers from Northwestern Polytechnical University Discuss Findings in Electric Power (Decoupling Start Control Method for Aircraft Wound-rotor Synchronous Starter-generator Based On Main Field Current Estimation)[J]. Energy Weekly News,2019. [56]. Energy - Electric Power; Wuhan University Reports Findings in Electric Power (Adjustable virtual inertia control of supercapacitors in PV-based AC microgrid cluster)[J]. Energy Weekly News,2019. [57]. Lg Electronic Inc.; Researchers Submit Patent Application, "Method And Apparatus For Monitoring Control Channel In Unlicensed Band", for Approval (USPTO 20190229825)[J]. Computers, Networks & Communications,2019. [58]. Special Conditions: Pilatus Aircraft Ltd., Model PC-12/47E Airplanes; Electronic Engine Control System Installation[J]. The Federal Register / FIND,2019,84(158). [59]. Apple Inc.; Patent Issued for Offset Control For Assembling An Electronic Device Housing (USPTO 10,368,457)[J]. Computers, Networks & Communications,2019. [60]. Mitsubishi Electric Corporation; Researchers Submit Patent Application, "Synchronization Control System And Control Device",for Approval (USPTO 20190238071)[J]. Computers, Networks & Communications,2019. 电气控制英文参考文献三： [61]. Technology - Cybernetics; Findings from North ChinaElectric Power University Provides New Data about Cybernetics (Hierarchical Distributed Model Predictive Control of Standalone Wind/solar/battery Power System)[J]. Energy Weekly News,2019. [62]. Nidec Corporation; "Motor Control System And Electric Power Steering System" in Patent Application Approval Process (USPTO 20190233002)[J]. Energy Weekly News,2019. [63]. Mobvoi Information Technology Co. LTD.; Researchers Submit Patent Application, "Display Device, Electronic Device And Display Control Method For Screen", for Approval (USPTO 20190235540)[J]. Computers, Networks & Communications,2019. [64]. Engineering - Power Delivery; Studies from North China Electric Power University Have Provided New Data on Power Delivery (Fault Tripping Criteria In Stability Control Device Adapting ToHalf-wavelength Ac Transmission Line)[J]. Energy Weekly News,2019. [65]. Samsung Electronics Co. Ltd.; "Electronic Device For Sensing Biometric Information And Control Method Thereof" in Patent Application Approval Process (USPTO 20190231235)[J]. Medical Patent Business Week,2019. [66]Asiabar Aria Noori,Kazemi Reza. A direct yaw momentcontroller for a four in-wheel motor drive electric vehicle using adaptive sliding mode control[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(3). [67]. Energy - Electrical Energy Systems; New Electrical Energy Systems Findings Has Been Reported by Investigators at University of Sfax (Constrained design and control of trapezoidal waves-forms hybrid excitation synchronous motor increasing energy accumulator lifetime)[J]. Energy Weekly News,2019. [68]. Energy; Findings from School of Mechanical Engineering Has Provided New Data on Energy (Considering Well-to-Wheels Analysis in Control Design: Regenerative Suspension Helps to Reduce Greenhouse Gas Emissions from Battery Electric Vehicles)[J]. Energy Weekly News,2019. [69]. Mitsubishi Electric Corporation; Patent Application Titled "Electric-Power Control Device, Electric Motor, Air-Conditioning Apparatus, And Method For Manufacturing Electric Motor" Published Online (USPTO 20190242594)[J]. Energy Weekly News,2019. [70]. Energy; Reports Summarize Energy Study Results from Warsaw University of Technology (Model Predictive Control and energy optimisation in residential building with electric underfloorheating system)[J]. Energy Weekly News,2019. [71]. Energy - Nuclear Power; Researchers from Korea Electric Power Corporation Report New Studies and Findings in the Area of Nuclear Power (Development of Anti-windup Pi Control and Bumpless Control Transfer Methodology for Feedwater Control System)[J]. Energy Weekly News,2019. [72]. Energy - Electric Power; Data on Electric Power Discussed by Researchers at School of Electrical and Electronics Engineering (Analysis of the Performance Characteristics and Arm Current Control for Modular Multilevel Converter With Asymmetric Arm Parameters)[J]. Energy Weekly News,2019. [73]. Energy - Electric Power; Study Findings on Electric Power Are Outlined in Reports from University of Technology (Direct power control for VSC-HVDC systems: An application of the global tracking passivity-based PI approach)[J]. Energy Weekly News,2019. [74]Allous Manel,Mrabet Kais,Zanzouri Nadia. Fast fault-tolerant control of electric power steering systems in the presence of actuator fault[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(12). [75]. Energy - Electric Power; Researchers from College of Engineering Detail New Studies and Findings in the Area of Electric Power (Power Control Strategy of Photovoltaic Plants for Frequency Regulation In a Hybrid Power System)[J]. Energy Weekly News,2019. [76]. Energy - Electric Power; Researchers at Shiv Nadar University Report New Data on Electric Power (Methods for overcoming misalignment effects and charging control of a dynamic wireless electric vehicle charging system)[J]. Energy Weekly News,2019. [77]Zhang Bing,Zong Changfu,Chen Guoying,Li Guiyuan. An adaptive-prediction-horizon model prediction control for path tracking in a four-wheel independent control electric vehicle[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(12). [78]Ren Yue,Zheng Ling,Yang Wei,Li Yinong. Potential field–based hierarchical adaptive cruise control for semi-autonomous electric vehicle[J]. Proceedings of the Institution of MechanicalEngineers,2019,233(10). [79]. Energy - Electric Power; Data from University of the Basque Country Advance Knowledge in Electric Power (Sliding Mode Control of an Active Power Filter With Photovoltaic Maximum Power Tracking)[J]. Energy Weekly News,2019. [80]Izadbakhsh Alireza,Kheirkhahan Payam. Adaptive fractional-order control of electrical flexible-joint robots: Theory and experiment[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(9). [81]Yang Weiwei,Liang Jiejunyi,Yang Jue,Zhang Nong. Optimal control of a novel uninterrupted multi-speed transmission for hybrid electric mining trucks[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(12). [82]Guercioni Guido Ricardo,Vigliani Alessandro. Gearshiftcontrol strategies for hybrid electric vehicles: A comparison of powertrains equipped with automated manual transmissions and dual-clutch transmissions[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(11). [83]. Energy - Electric Power; Findings from PontificalUniversity Provides New Data on Electric Power (A Communication-free Reactive-power Control Strategy In Vsc-hvdc Multi-terminal Systems To Improve Transient Stability)[J]. Energy Weekly News,2019. [84]. Energy - Electric Power; Findings from Yazd University in the Area of Electric Power Reported (An adaptive time-graded control method for VSC-MTDC networks)[J]. Energy Weekly News,2019. [85]Liu Hui,Li Xunming,Wang Weida,Han Lijin,Xin Huibin,Xiang Changle. Adaptive equivalent consumption minimisation strategy and dynamic control allocation-based optimal power management strategy for four-wheel drive hybrid electric vehicles[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(12). [86]. Networks - Neural Networks; Findings on Neural Networks Reported by Investigators at School of Electrical Engineering and Automation (Stability Analysis of Fractional Order Hopfield Neural Networks With Optimal Discontinuous Control)[J]. Computers, Networks & Communications,2019. [87]. Energy - Electric Power; Researchers from NanjingUniversity of Aeronautics and Astronautics Describe Findings in Electric Power (Synchronous Vibration Control for a Class of Cross-coupled Antisymmetric Msr Systems)[J]. Energy Weekly News,2019. [88]. Energy - Electric Power; Investigators at Chung Ang University Detail Findings in Electric Power (Flexible Risk Control Strategy Based On Multi-stage Corrective Action With Energy Storage System)[J]. Energy Weekly News,2019. [89]. Energy - Electric Power; Findings in Electric Power Reported from National Institute of Technology (An adaptive PI control scheme to balance the neutral-point voltage in a solar PV fed grid connected neutral point clamped inverter)[J]. Energy Weekly News,2019. [90]Najjari Behrouz,Mirzaei Mehdi,Tahouni Amin. Constrained stability control with optimal power management strategy for in-wheel electric vehicles[J]. Proceedings of the Institution of Mechanical Engineers,2019,233(4). 电气控制英文参考文献四： [91]. Energy - Wind Farms; Investigators at School of Electrical Power Detail Findings in Wind Farms (Theoretical Study On Control Strategy of Grid-connected High Voltage Ride Through In Doubly-fed Wind Farm)[J]. Energy Weekly News,2019. [92]. Kia Motors Corporation; Patent Issued for Wireless Charging Control Apparatus And Method For Optimal Charging By Adjusting The Inclination Of The Electric Vehicle Being Charged (USPTO10,399,449)[J]. Computers, Networks & Communications,2019. [93]. Energy; New Data from Institute of Electrical Engineering Illuminate Findings in Energy (Charging-Discharging Control Strategy for a Flywheel Array Energy Storage System Based on the Equal Incremental Principle)[J]. Energy Weekly News,2019. [94]. Science - Applied Sciences; Findings from North China Electric Power University Broaden Understanding of Applied Sciences (Coordinated Frequency Control Strategy with the Virtual Battery Model of Inverter Air Conditionings)[J]. Science Letter,2019. [95]. Science - Materials Science; Studies from Tsinghua University in the Area of Materials Science Described (ElectricField Control of Neel Spin-orbit Torque In an Antiferromagnet)[J]. Science Letter,2019. [96]. Electronics - Power Electronics; Studies from Nanjing University of Aeronautics and Astronautics Have Provided New Data on Power Electronics (Wireless battery charging control for electric vehicles: a user-involved approach)[J]. Computers, Networks & Communications,2019. [97]Kivanc,Ustun. Dynamic control of electronic differential in the field weakening region[J]. International Journal ofElectronics,2019,106(10). [98]Mohit Batra,John McPhee,Nasser L. Azad. Real-time model predictive control of connected electric vehicles[J]. Vehicle System Dynamics,2019,57(11). [99]Kim Daihyun,Echelmeier Austin,Cruz Villarreal Jorvani,Gandhi Sahir,Quintana Sebastian,Egatz-Gomez Ana,Ros Alexandra. Electric Triggering for Enhanced Control of Droplet Generation.[J].Analytical chemistry,2019,91(15). [100]Kurien Caneon,Srivastava Ajay Kumar. Impact of Electric Vehicles on Indirect Carbon Emissions and Role of Engine Post-Treatment Emission Control Strategies.[J]. Integrated environmental assessment and management,2019. [101]Aravindh D,Sakthivel R,Kaviarasan B,Anthoni SMarshal,Alzahrani Faris. Design of observer-based non-fragile loadfrequency control for power systems with electric vehicles.[J]. ISA transactions,2019,91. [102]Chen Xianzhe,Zhou Xiaofeng,Cheng Ran,Song Cheng,Zhang Jia,Wu Yichuan,Ba You,Li Haobo,Sun Yiming,You Yunfeng,Zhao Yonggang,Pan Feng. Electric field control of Néel spin-orbit torque in an antiferromagnet.[J]. Nature materials,2019,18(9). [103]Lê-Scherban Félice,Ballester Lance,Castro Juan C,Cohen Suzanne,Melly Steven,Moore Kari,Buehler James W. Identifying neighborhood characteristics associated with diabetes and hypertension control in an urban African-American population using geo-linked electronic health records.[J]. Preventive medicine reports,2019,15. [104]Samartin-Veiga N,González-Villar A J,Carrillo-de-la-Pe?a M T. Neural correlates of cognitive dysfunction in fibromyalgia patients: Reduced brain electrical activity during the execution of a cognitive control task.[J]. NeuroImage. Clinical,2019,23. [105]Leibel Sydney,Weber Rachel. Utilizing a PhysicianNotification System in the EPIC Electronic Medical Record to Improve Pediatric Asthma Control: A Quality Improvement Project.[J].Clinical pediatrics,2019,58(11-12). [106]Bernacka-Wojcik Iwona,Huerta Miriam,Tybrandt Klas,Karady Michal,Mulla Mohammad Yusuf,Poxson David J,Gabrielsson Erik O,Ljung Karin,Simon Daniel T,Berggren Magnus,Stavrinidou Eleni. Implantable Organic Electronic Ion Pump Enables ABA Hormone Delivery for Control of Stomata in an Intact Tobacco Plant.[J]. Small (Weinheim an der Bergstrasse, Germany),2019. [107]Stoynova Nevena,Laske Christoph,Plewnia Christian. Combining electrical stimulation and cognitive control training to reduce concerns about subjective cognitive decline.[J]. Brainstimulation,2019,12(4). [108]Bettano Amy,Land Thomas,Byrd Alice,Svencer Susan,Nasuti Laura. Using Electronic Referrals to Address Health Disparities and Improve Blood Pressure Control.[J]. Preventing chronicdisease,2019,16. [109]Xu Meng,Yan Jian-Min,Guo Lei,Wang Hui,Xu Zhi-Xue,Yan Ming-Yuan,Lu Yun-Long,Gao Guan-Yin,Li Xiao-Guang,Luo Hao-Su,ChaiYang,Zheng Ren-Kui. Nonvolatile Control of the Electronic Properties of In_{2- x}Cr_xO₃ Semiconductor Films by Ferroelectric Polarization Charge.[J]. ACS appliedmaterials & interfaces,2019,11(35). [110]Gao Tao,Mirzadeh Mohammad,Bai Peng,Conforti Kameron M,Bazant Martin Z. Active control of viscous fingering using electricfields.[J]. Nature communications,2019,10(1). [111]Chaux Robin,Treussier Isabelle,Audeh Bissan,Pereira Suzanne,Hengoat Thierry,Paviot Béatrice Trombert,Bousquet Cedric. Automated Control of Codes Accuracy in Case-Mix Databases by Evaluating Coherence with Available Information in the Electronic Health Record.[J]. Studies in health technology andinformatics,2019,264. [112]Bolat Mustafa Suat,Cinar Onder,Asci Ramazan,Buyukalpelli Recep. A novel method for pain control: infiltration free local anesthesia technique (INFLATE) for transrectal prostatic biopsy using transcutaneous electrical nerve stimulation (TENS).[J]. International urology and nephrology,2019. [113]Cruz Chad D,Yuan Jennifer,Climent Clàudia,Tierce NathanT,Christensen Peter R,Chronister Eric L,Casanova David,Wolf Michael O,Bardeen Christopher J. Using sulfur bridge oxidation to control electronic coupling and photochemistry in covalent anthracene dimers.[J]. Chemical science,2019,10(32). [114]Zhou Canliang,Sun Linfeng,Zhang Fengquan,Gu Chenjie,Zeng Shuwen,Jiang Tao,Shen Xiang,Ang Diing Shenp,Zhou Jun. Electrical Tuning of the SERS Enhancement by Precise Defect DensityControl.[J]. ACS applied materials & interfaces,2019,11(37). [115]Taeho Park,Hyeongcheol Lee. Optimal Supervisory Control Strategy for a Transmission-Mounted Electric Drive Hybrid Electric Vehicle[J]. International Journal of AutomotiveTechnology,2019,20(4). [116]Zoé Magalh?es,André Murilo,Renato V. Lopes. Development and evaluation with MIL and HIL simulations of a LQR-based upper-level electronic stability control[J]. Journal of the Brazilian Society of Mechanical Sciences and Engineering,2019,41(8). [117]Justin Roger Mboupda Pone,Victor Kamdoum Tamba,Guillaume Honore Kom,Mathieu Jean Pierre Pesdjock,Alain Tiedeu,Martin Kom. Numerical, electronic simulations and experimental analysis of a no-equilibrium point chaotic circuit with offset boosting and partial amplitude control[J]. SN Applied Sciences,2019,1(8). [118]Alberto Cavallo,Antonio Russo,Giacomo Canciello.Hierarchical control for generator and battery in the more electric aircraft[J]. Science China Information Sciences,2019,62(9). [119]Ying Liu,Kai Cao,Jingjun Liu,Zhengping Zhang,Jing Ji,Feng Wang,Zhilin Li. Electrodeposition of copper-doped SnS thin films and their electric transmission properties control for thermoelectric enhancement[J]. Journal of Materials Science: Materials in Electronics,2019,30(17). [120]Feng Tian,Liqi Sui,Yuanfan Zeng,Bo Li,Xingyue Zhou,Lijun Wang,Hongxu Chen. Hardware Design and Test of a Gear-ShiftingControl System of a Multi-gear Transmission for ElectricVehicles[J]. Automotive Innovation,2019,2(3).。

Introduction to Model Predictive ControlLars ImslandAbstractThis note gives a brief introduction to Model Predictive Control(MPC),assuming the reader has been exposed to the LQR controller,and QP optimization algorithms including KKT conditions.Basic knowledge of linear algebra is also assumed.The main focus is linear MPC,while MPC using nonlinear process models is mentioned briefly(assuming some exposure to SQP algorithms for nonlinear programming). Contents1Introduction1 2MPC principle—traditional view2 3Linear MPC—Linear Quadratic Constrained Regulation33.1Model class and infinite horizon opti-mal control problem (3)3.2The unconstrained solution:The linearquadratic regulator(LQR) (3)3.3From LQR,via constrained open-loopcontrol,to MPC (4)3.4The constrained LQR:MPC (5)3.5Stability (8)4Some practical aspects104.1MPC tuning (10)4.2MPC tracking (11)4.3State estimation and integral control..124.4Other issues (13)5Explicit solution of the linear MPC controller14 6Brief introduction to nonlinear MPC(NMPC)166.1Solving the NMPC optimization problem186.2Stability of NMPC (19)7Bibliographic notes20 A Some proofs22A.1Proof of Lemma1 (22)B Exercises24C Some Matlab functions24C.1blkdiag2.m (24)1IntroductionThe term Model Predictive Control(MPC)has come to refer to a controller whicha)uses a(multivariable)process model to predict future behavior,b)uses mathematical programming for optimizing predicted future performance(typically quadratic pro-gramming(QP)in the case of linear process models,or sequential quadratic programming(SQP)for non-linear process models,but also linear-and semi-definite programming have been used),andc)handles constraints on inputs(manipulated variables)and states/controlled variables.This type of controller has become the advanced process control technology in chemical process industry, and usage is spreading to other application areas.Most widespread is the type of MPC usually termed linear MPC–using linear process models for prediction and convex QP for optimization of a quadratic performance objective.Linear MPC will be the main focus for this note.However,over the last10-15years,MPC using nonlinear process models(usually termed nonlinear MPC) has gained ing nonlinear models will typically give a non-convex nonlinear MPC optimization prob-lem.Since nonlinear optimization is much more difficult(time-consuming,error-prone)than convex opti-mization,typically QP,for linear MPC,it is important to justify the need for nonlinear MPC before it is ap-plied.Typical reasons for using nonlinear MPC are that the process operates in several steady states(“grade change”)with significantly different dynamics,that there are large disturbances that excite nonlinearities,orthat the process is a batch process.In the last few years,there have been a significant research focus on efficient implementation of nonlinear optimization techniques tailored for nonlinear MPC.Let us return to linear MPC,which in our case will refer to a MPC using linear state-space models.The ob-jective function is a quadratic weighting of future manipulated manipulated variables and setpoint errors.Al-though thefirst MPC applications optimized performance on afinite horizon,theory developed in the nineties has shown us that(for linear process models)it is just as easy to optimize performance on the infinite horizon, and due to better theoretical properties,this should be the preferred choice.In the unconstrained case,the infinite horizon LQ controller(the LQR)delivers the optimal performance on the infinite horizon.Therefore,the main reason for the widespread use of MPC in place of the LQR is that it offers a straightforward and transparent approach to handling constraints,both on manipulated variables (inputs)and controlled variables(outputs).The linear MPC formulation presented in this note emphasizes this approach to MPC,namely that linear MPC is(should be)an integration of the LQ controller with constraint handling:•The LQ controller is used for stabilization and to obtain good control performance on the infinite horizon.•The MPC QP formulation handles the constraints in an optimal manner.Before we continue,some brief notes on notation.We consider in this note discrete-time dynamic systems, and variables that change with time have a subscript that denote time instant.For example,x k denotes vector x at time instant k.We will set up optimization problems,where we will calculate future variables(“predict”)to optimize future behavior.The notation we use will not reflect the difference between these predicted variables and the ’real’variables.In reality,predicted and real variables will invariably be different,due to noise and model error.2MPC principle—traditional viewOriginally,the MPC controller was based on step-response or impulse-response models,and in most industrial implementations,this is still the case.To illustrate the MPC principle,we assume the process model is an single-input single-output(SISO)finite impulse response(FIR)model1,y k+j=N∑i=1h i u k+j−i.This model predicts that the output at a given time depends on a linear combination of past input values;the summation weights h i are the impulse response coefficients.The sum is truncated at the point where past inputs no longer influence the output.Therefore,this representation is only possible for stable plants.The MPC optimization problem is,at time instant k,to optimize future performance(as measured by a quadratic function)over a horizon N:minN∑i=0q(y k+i+1−y d k+i+1)2+r(u k+i−u d k+i)2(1a)subject to y≤y k+i+1≤y,u≤u k+i≤u,i=0,...,N.(1b) Superscript d signifies“desired”values,or“reference”.This objective function can easily be shown to be a QP. The scalars q and r are weights that can be used as tuning parameters(in addition to the horizon-length N). The optimization problem is illustrated in Figure1.The MPC principle is to repeat this optimization at every time instant,using only thefirst of the calculated inputs as input to the process:1.At time instant k,solve the QP to obtain an optimal,feasible input sequence.2.Apply thefirst input in the input sequence(u k,u k+1,...,u k+N)to process.3.Set k=k+1(next time instant),go to step1.1Actually,this is more or less the models used in thefirst reported industrial MPC applications,[24,25,7].k +NPast Present Future kFigure 1:MPC principleA small remark is in place here:The reader will note that measurements are not used in this procedure,while the whole idea of MPC is to re-optimize when new measurements are available at each sample instant.In a real implementation of the approach used here,feedback from measurements are obtained by updating of biases on the measured outputs (see Section 4.3),which are then used in the prediction.The MPC controller we will study closer in this text,starting from next section,is based on the same principle as in the procedure above,but differs in two important aspects:It uses state-space models,and it optimizes performance on an infinite horizon (while using a finite number of degrees of freedom).3Linear MPC —Linear Quadratic Constrained Regulation 3.1Model class and infinite horizon optimal control problemThe model (2)is typically obtained ei-ther by linearization of a nonlinear processmodel,or by system identification methodsbased on measured data.The state x andinput u are deviation variables,expressingthe difference between the ’real’state (in-put)and the value of the state (input)in thedesired process equilibrium.For instance,if a nonlinear process model is found frommass and energy balances to be˙ξ=f (ξ,ν)where ξis the state and νthe input,thenwe could define x =ξ−ξd and u =ν−νdwhere ξd and νd are the desired equilib-rium,satisfying f (ξd ,νd )=0.The matri-ces A and B could be found,using forwardEuler discretization with discretization in-terval h ,from A =I +h ∂f ∂ξ (ξd ,νd ),B =h ∂f ∂ν (ξd ,νd ).We will assume that the process to be controlled is described by a discrete state space model x k +1=Ax k +Bu k ,(2)where dim x =n x and dim u =n u .The process could be stable or unstable (there are no restrictions on the eigenvalues of A ),but we assume that the pair (A ,B )is stabilizable.We will most of the time assume that the state x is measured.The objective we want to obtain by control,is to optimize perfor-mance by minimizing the infinite horizon cost J (x 0,{u k }k =0,...,∞)=∞∑k =0x T k Qx k +u T k Ru k ,(3)where future x k are given by the model (2)based on x 0and {u k }k =0,...,∞.The matrices Q and R are tuning parameters,but must fulfill that Q ≥0and R >0,and the state must be detectable through Q (that is,the pair (A ,√Q )must be detectable,where √Q is a matrix such that Q =(√Q )2).It is implicit in the cost function (3)that we treat the regulation problem,that is,we want to control x k →0.The problem of tracking time-varying references are treated in Section 4.2.3.2The unconstrained solution:The linear quadratic regulator (LQR)When there are no constraints,the optimal future controls that minimizes the infinite horizon objective func-tion (3),is given as a fixed (time-invariant)state feedbacku k =−Kx k .(4)This controller is often denoted the(discrete-time)infinite horizon LQ controller,or the Linear Quadratic Reg-ulator(LQR).The state feedback matrix K=(R+B T PB)−1B T PA is given by the positive semi-definite solution of the discrete-time algebraic Riccati equation(DARE),A T PA−P−A T PB(R+B T PB)−1B T PA+Q=0.(5)A fact that we will use in the sequel,is that when u k=−Kx k,∞∑k=0x T k Qx k+u T k Ru k=∞∑k=0x T kQ+K T RKx k=x T0Px0.(6)We do not prove this here,but remark that it is rather easy to show by noting that(5)can be written(A−BK)T P(A−BK)−P+Q+K T RK=0,and using the same trick as in the beginning of Section A.1.3.3From LQR,via constrained open-loop control,to MPCThe LQR has some important advantages;+it is inherently multivariable(it takes care of couplings in the process),+it is optimal on the infinite horizon,and+it can be shown that it has good robustness properties(due to feedback2).However,it has one important disadvantage:-it does not handle constraints on states and inputs.The optimization problem of minimizing(3)with constraints on states and inputs,is a QP with an infinite number of optimization variables.This is in general an intractable optimization problem.In the special case when there are no constraints,the solution of this QP is given by the LQR state feedback,as we learned in the previous section.Including constraints,the QP can,however,be approximated by a QP with afinite number of optimization variables.As afirst approach,consider truncating the sum in the objective function at a given’horizon length’L:L−1∑k=0x T k+1Qx k+1+u T k Ru k,(7)and using u0,u1,...,u L−1(and x1,x2,...,x L)as optimization variables.This corresponds to what thefirst MPC controllers did,and is still common.The main problem with this approach is that thisfinite horizon objective function is not what we really want to minimize,since in regulation we generally want to optimize all(infinite) future behavior.Therefore,in practice,one often chooses large values for L,for which the approximation can be close.A rule of thumb is’larger than the longest time-constant in the process’,however,this can lead to excessively large optimization problems,and one is still not guaranteed stability.However,in the nineties,it was discovered[20,6,28]that it is possible to optimize performance on the infinite horizon,with afinite number of optimization variables.(It turned out this was also the key to proving closed-loop stability for MPC approaches.More on this in Section3.5.)The simple trick is to divide the objective function into two parts;∞∑k=0x T k Qx k+u T k Ru k=L−1∑k=0x T k Qx k+u T k Ru k+∞∑k=Lx T k Qx k+u T k Ru k,and let the control moves in thefirst part be free optimization variables,while the control moves in the second part is given by the LQR controller u k=−Kx k,k≥L.The rationale is that after some time,we have’resolved’the constraints such that the LQR controller is optimal on the rest of the horizon.If we do this,we can use(6) to obtain∞∑k=Lx T k Qx k+u T k Ru k=x T L Px L,2However,implemented in combination with state estimation,e.g.the Kalmanfilter(that is,the LQG controller)it can be non-robust[11].where P is the Riccati matrix.This means that we can write∞∑k=0x T k Qx k+u T k Ru k=L−1∑k=0x T k Qx k+u T k Ru k+x T L Px L.(8)Note that this is exact,and not an approximation.However,since we now have put structure on the allowed control moves,minimizing this objective function is not necessarily the same as minimizing(3),unless the horizon is’long enough’(more on this later).In the next section we will use this objective function,but we do a change of optimization variables by letting the optimization variables on thefirst part of the horizon be perturbations to u k=−Kx k.It will then turn out that(8)takes a particularly simple form(Lemma1).Once we have such a QP with afinite number of optimization variables,we can calculate the optimizing future control at a given point,and apply them for all future(or at least for an horizon L).However,this approach is not robust,since it does not incorporate feedback.The simple principle of MPC is that feedback can be achieved by performing the open-loop optimization over again at each sample instant,when new measurements are available,and applying only thefirst part of the optimal inputs.3.4The constrained LQR:MPCIn many practical control problems,constraints on inputs and states(and/or outputs)are important.We assume here that these constraints are written compactly asD x x≤d x,D u u≤d u,(9)and that these polytopes contains the origin in the interior.Note that this constraint formulation includes the most usual’box’constraints on inputs and outputs(or states),u≤u≤¯u,y≤y=Cx≤¯y.Constraints that are’mixed’(depending on both x and u)can also easily be added,but we avoid them here as they require a slight redefinition of the MPC problem to come.The control law that will be used in optimization of future performance subject to these constraints,is:u k=−Kx k+c k,k=0,...,L−1,−Kx k,k≥L.(10)It is important to note that this is not the’real’control law,but the one that is used when predicting future performance.The real control law will be defined later in this section(Algorithm1).The c k are degrees of freedom available for constraint handling,entering as’perturbations’to a linear state feedback law.This formulation allows active constraint handling during transients,on the control horizon k=0,...,L−1.Furthermore,thefixed state feedback K affects the asymptotic behavior.Algorithms us-ing this separation of future predicted inputs is often denoted as’dual mode’.We will assume that K is the unconstrained optimal on the infinite horizon,that is,the LQR controller.In this case(3)takes the form of a quadratic function:Lemma1The objective function(3)with future x k and u k given by(2)and(10),respectively,is given byJ(x0,c)=c T W c c+x T0Σx0,(11) where c=[c T0,...,c T L−1]T,and the symmetric matrix3W c=diag(W,...,W),where W=B TΣB+R,andΣis the solution of the Lyapunov equation4Σ−(A−BK)TΣ(A−BK)=Q+K T RK.The proof of this lemma can be found in the Appendix(Section A.1).If we accept that this function must be quadratic,it is easy to see that there must be no cross-terms(terms involving both x k and c):Close to x k=0, there are no constraints active,and hence c=0must be the minimizer of J.If there were cross-terms,the minimizer would not be c=0.3The operator diag gives a block-diagonal matrix with the arguments as the blocks along the diagonal,cf.the Matlab-function blkdiag.4This Lyapunov equation can be solved in Matlab with the function dlyap.However,you might already have calculated it sinceΣ=P, where P is the solution of(5).Note that the last term of J(x0,c)is not dependent on the degrees of freedom c and can therefore be omitted when optimizing J.If there are no constraints in the problem formulation,the minimum to J(x0,c)is clearly given by c=0, giving the LQR controller,as expected.Thus,the interesting case is when there are constraints on inputs u and states x(and/or outputs).For these constraints to be added to our MPC optimization problem,we must calculate future u k and x k as functions of x0and c:x1=Ax0+B(−Kx0+c0)=(A−BK)x0+Bc0x2=Ax1+B(−Kx1+c1)=(A−BK)x1+Bc1=(A−BK)2x0+(A−BK)Bc0+Bc1...u0=−Kx0+c0u1=−Kx1+c1=−K(A−BK)x0−KBc0+c1....As we see,these functions are linear,and straightforward but somewhat tedious to define,hence we omit the detailed definitions here(an example is given in the code listing below).To ensure that constraints are fulfilled for all future time,we must calculate all future u k and x k.As optimization problems with an infinite number of constraints are undesirable,we usually stop at afinite number,and here we choose the constraint horizon to be equal to the control horizon.This is discussed closer below.Writing the linear mappings asx T1x T2···x T L T=P x1x0+P c1c,(12a)u T0u T1···u T L−1 T=P x2x0+P c2c,(12b)we can combine this with(9)to put together constraints asMx0+Nc≤b.(13) We are now ready to state the MPC algorithm:Algorithm1(LTI dual mode MPC)At each sampling instant k,perform the optimization(the QP)mincc T W c c subject to Mx k+Nc≤b.(14) Use thefirst block element of the optimal c,c0,to calculate u k=−Kx k+c0.An important observation is that if,for a given x k,c=0is feasible for Mx k+Nc≤b,then c=0is the solution to the QP problem in the algorithm above(the region of x k where this happen is defined by the polytope{x:Mx≤b}).An interpretation of this is that if u k=−Kx k is feasible,then c=0,and c=0only if u k=−Kx k is not feasible(in the sense that implementing it will lead to a constraint violation on the control horizon).We thus see that we have separated stabilization and optimal control on the infinite horizon(−Kx k) from optimized constraint handling(c).An implementation example forfinding the matrices W c and M,N and b is given in the form of a Matlab function below.The constraints are only enforced on the control horizon.It makes use of the function blkdiag2, which can be found in Appendix C.Listing1:Function for generating matrices for solving MPC problem function[Wc,M,N,b,K,Sigma]=genMPCprob(A,B,Q,R,Dx,dx,Du,du,L,A_s,b_s)%Generate matrices for MPC QP problem%Inputs:%A,B System matrices in discrete time system:x+=A x+B u%Q,R Weights in infinite horizon cost:J=sum(x’Q x+u’R u)%L Control horizon length(degrees of freedom)%Dx,dx State constraints on the horizon:Dx x<=dx%Du,du Input constraints on the horizon:Du x<=du%A_S,b_S Stability constraints on end of control horizon:%A_s x(L)<=b_S(may be omitted)%%Outputs:%Wc Resulting cost function:J=c’Wc c+const%M,N,b Constraints on horizon:M x0+N c<=b%K Unconstrained optimal state feedback%Sigma Riccati equation solution%3/3/2007Lars Imsland[K,SS]=dlqr(A,B,Q,R);Phi=A-B*K;nx=size(A,1);nu=size(B,2);%Define costfunction,c’*W_c*cSigma=dlyap(Phi’,Q+K’*R*K);%Note that Sigma=SS,calculated earlierW=B’*Sigma*B+R;Wc=blkdiag2(W,L);%Define predictions:%[x_1x_2...x_L]=Px1*x_0+Pc1*[c_0c_1...c_L-1]%[u_0u_1...u_L-1]=Px2*x_0+Pc2*[c_0c_1...c_L-1]Px1=[eye(nx);Phi];Pc1=zeros(L*nx,L*nu);Pc1(1:nx,1:nu)=B;for i=2:L,Px1=[Px1;Px1((i-1)*nx+1:i*nx,:)*Phi];for j=1:i,Pc1((i-1)*nx+1:i*nx,(j-1)*nu+1:j*nu)=...Px1((i-j)*nx+1:(i-j+1)*nx,1:nx)*B;endendPx2=blkdiag2(-K,L)*Px1(1:L*nx,:);Pc2=blkdiag2(-K,L)*([zeros(nx,L*nu);Pc1(1:(L-1)*nx,...1:(L-1)*nu),zeros((L-1)*nx,nu)])+blkdiag2(eye(nu),L);Px1=Px1(nx+1:end,:);%remove first I%Define constraints,N*c<=b-M*x0N=[blkdiag2(Dx,L)*Pc1;blkdiag2(Du,L)*Pc2];M=[blkdiag2(Dx,L)*Px1;blkdiag2(Du,L)*Px2];b=[repmat(dx,L,1);repmat(du,L,1)];if(exist(’A_S’,’var’)),%If you have"stability constraints"A_S x<=b_SPxlast=Px1(nx*(L-1)+1:nx*L,:);Pclast=Pc1(nx*(L-1)+1:nx*L,:);N=[N;A_S*Pclast];M=[M;A_S*Pxlast];b=[b;b_S];endThe following example uses this function to control a process consisting of a double integrator with state-and input constraints.Example1(Double integrator)The double integrator,two integrators in series,discretized with sample in-terval T s,can be written in state-space form(2)withA=1T s01,B=T2sT s.We will control this system using MPC,that is,minimize the infinite horizon cost(3)withQ=1000,R=1,using L=5degrees of freedom(horizon)for constraint handling.The constraints are−0.5≤x2≤0.5,and −1≤u≤1.Matlab code using the genMPCprob-function in Listing1for specifying and simulating the MPC controller is given below:Listing2:Code for simulating QP MPC example%Control problem:Double integratorTs=0.05;%Sampling timeA=[1Ts;01];B=[Tsˆ2;Ts];Q=diag([10]);R=1;L=5;nu=1;%Constraints:%-0.5<=x_2<=0.5%-1<=u<=1Dx=[01;0-1];dx=[.5;.5];Du=[1;-1];du=[1;1];[Wc,M,N,b,K]=genMPCprob(A,B,Q,R,Dx,dx,Du,du,L);i=1;x(:,i)=[-2;0];time=1:Ts:10;opt=optimset;opt.Display=’notify’;opt.Diagnostics=’off’;rgeScale=’off’;for i=1:length(time),%Find control for state x(:,i)c(:,i)=quadprog(2*Wc,zeros(1,L*nu),N,b-M*x(:,i),[],[],[],[],[],opt);u(:,i)=-K*x(:,i)+c(1:nu,i);%Simulatex(:,i+1)=A*x(:,i)+B*u(:,i);end%Phaseplotfigure(1);plot(x(1,:),x(2,:));xlabel(’x_1(position)’);ylabel(’x_2(velocity)’);title(’Phase plot double integrator’);%Input and perturbationfigure(2);subplot(211);plot(time,c(1:nu,:));xlabel(’Time[s]’);ylabel(’c’);subplot(212);plot(time,u);xlabel(’Time[s]’);ylabel(’u’);The calculated control u k and’perturbation’c k are plotted in Figure2,for a case where we start in x0= (−2,0)T.As we can see,in the beginning c k=0for handling constraints.First,the input constraint is active, but after a while,the state constraint is active(see Figure3).Eventually,we get closer to the origin and the LQ controller is optimal(c k=0).The states are plotted in a phase plot(the state-space)in Figure3.The colored region is the largest polytope (even though it might not look like a polytope)where the LQ-controller does not violate constraints for this example.The red ring is plotted at the time instant when c k=0,and as we can see,in this case5this happens when the state enters the polytope.3.5StabilityWefirst note that with an infinite number of degrees of freedom(that is,L=∞),the MPC controller in Algorithm1is nominally6stabilizing in the sense that x k→0as k→∞,if•the full state is detectable through the objective function as mentioned in Section3.1,5This will typically be the case,but not always.6Not taking into account model errors and noise.Figure 2:Upper plot:The perturbation c .Lower plot:The implemented input u .Figure 3:Phase plot.The polytope is the maximal region were u =−Kx can be applied without violating constraints.The circle marks when the state enters this polytope (that is,when c k becomes 0).•and the optimization problem is feasible for the initial state.This is straightforward to prove,but a detailed treatment of stability is outside the scope of this rmally,due to the infinite horizon and since we assumed no noise/model error,the implemented control calculated by Algorithm 1for all future k will be given by the optimal solution at the first step,and since the optimization problem is assumed to be feasible,the objective function must be finite,implying that x k →0as k →∞.Since the objective function in (11)is the same as (3),Algorithm 1will also be stabilizing under the same assumptions using the same arguments,unless constraints will be violated after the end of the control horizon (or the constraint horizon,if this is different from the control horizon).Thus,guaranteeing stability amounts to ensuring that the constraints will hold on the infinite horizon,and not only on the horizon where we enforce them.With the formulation above,we cannot ensure this,since we only check constraints on the control horizon,that is,for 1,...,L .Since one often has to choose rather short horizon L to limit computational complexity,stability can in many cases be a practical problem (and not only theoretically).It is straightforward to increase the constraint horizon beyond L by changing the specification of (12),and one can actually guarantee that constraints will not be broken by choosing this constraint horizon long enough.This was recognized in [28,6],where an upper bound on the constraint horizon was also found.The problem with this approach is that this upper bound is rather conservative,such that a large number of extra constraints might have to be added.Another way to guarantee that constraints are not broken,without increasing the computational burden excessively,is via what is often called ’quasi-infinite horizon’.This idea was originally proposed for nonlinear MPC [5],but applies equally well (if not better)for linear MPC.For linear MPC this approach amounts to:1.Calculate the (maximum)region where the LQ control law u k =−Kx k is unconstrained and stabiliz-ing.This set is often called the maximal admissible set,or maximal feasible set.This set is rather straightforward to calculate,and one can show that it will always be a polytope,that is,on the form {x :A S x ≤b S }[13].2.Add A S x L ≤b S as an additional (linear)constraint to the constraints in Algorithm 1(that is,include them in (13)).This is often called a terminal state constraint .The last state on the control horizon,x L ,is found as a (linear)function of x 0and c as in (12).The function shown in Listing 1has A S and b S as optional inputs.This method can guarantee stability for even small L ,but a possible drawback is that the feasibility region (see below)might become small.We summarize in a theorem:Theorem 1If we apply the MPC algorithm (Algorithm 1)to the system (2),we are guaranteed that x k →0as k →∞7if•the MPC optimization problem is feasible at k =0,•(A ,√Q )is detectable,•and we can guarantee that the constraints holds on the infinite horizon,either by–choosing L long enough,or–adding a terminal state constraint to ensure constraint satisfaction on the infinite horizon.We do not prove this here,but refer to e.g.[19].The region of the state space where the MPC optimization problem is feasible,is called the feasibility region ,or the feasible set,and is defined by S ={x :∃c s.t.Mx +Nc ≤b }.4Some practical aspects 4.1MPC tuningIt is seldom that we know exactly which Q and R that correspond to the desired closed-loop behavior.Tuning of the MPC controller is the procedure where we change the MPC parameters to obtain acceptable closed-loop performance,but also,importantly,the region where the MPC controller is feasible (the feasibility region).7Wecan say more:The closed loop is asymptotically stable in the sense of Lyapunov,which implies x k →0as k →∞.Since the MPC control law is nonlinear (we will see that it is piecewise affine)we cannot use a definition of stability in terms of eigenvalues.The main parameters for tuning are the weighting matrices Q and R,and the horizon length L8.The tuning of Q and R is similar to the LQ case,that is,it is customary to choose them as diagonal matrices,and increase elements in Q(relative to elements in R)to get more aggressive closed loop behavior.Similarly,detuning is obtained by increasing the elements in R.Beyond these simple and obvious remarks,the best way to gain insight in tuning,is by trial-and-error(closed-loop simulation).More can be said about the choice of horizon length L.In addition to affecting closed-loop performance,this parameter also affects computational complexity.Generally,shorter L gives lower complexity optimization problem and hence lower computational load for the online optimization problem,while longer L gives better performance,at the cost of higher computational load.In the nominal case,one could argue that one should choose L as large as computational limitations permit, as this will make the closed-loop behave closer to the infinite horizon controller9,which is our measure of good control.Importantly,the use of an infinite prediction horizon allows to weigh also states beyond the control horizon length,thus one could choose control horizon lengths shorter than the longest time constants.In the real world,the issue of model-plant mismatch(due to nonlinearities,modeling errors and simplifi-cations,and process noise)is always important,and will quite often imply that one should not choose too long horizon lengths.One reason for this is that model uncertainties may tend to be amplified as one predicts far into the future.Lastly,forfixed Q and R,increasing L will generally(nominally)increase the feasibility region.This is especially the case if a terminal state constraint is added to the MPC optimization problem.4.2MPC trackingThe theory presented in Section3concerns regulation,that is controlling the system to a constant setpoint. In this section we look at how we can expand the controller to tracking,including,for example,changing set points.Assume that the variables y k that we want to track some given time-varying reference are given as a linear mapping of the states,y k=Cx k.Note that generally,these variables does not need to be the same as the measurements,but to keep notation simple we assume this here.Let the time-varying reference be denoted with y d k.For this reference,generate matching reference states x d k and inputs u d k such that10x d k+1=Ax d k+Bu d k,y d k=Cx d k,(15) where A and B are the same as the process model.Example2The simplest example of tracking is tracking a constant reference,that is,y dk =r where r is aconstant.Then also x d k and u d k must be constant.Then,y d k=Cx d k=C(I−A)−1Bu d k,meaning we canfindu d k=C(I−A)−1B−1r,x d k=(I−A)−1BC(I−A)−1B−1r.Note,however,that when changing between(constant)references,it is often a good idea to let the reference change smoothly to the new value.The objective function we want to minimize in tracking is∞∑k=0(y k−y d k)T˜Q(y k−y d k)+(u k−u d k)T R(u k−u d k)=∞∑k=0(x k−x d k)T Q(x k−x d k)+(u k−u d k)T R(u k−u d k)(16)8One could use different Q and R in calculation of W c and K(for example,allowing shorter L(for lower computational complexity)or larger feasibility region by having a more detuned K),but we will not consider this further.9Surprisingly,this is not always the case[9].10This can be interpreted as a setpointfilter,which is generally always a good idea.。

Control ReferencesRecent Review ArticlesBequette, B.W. “Practical Approaches to Nonlinear Control: A Review of Process Applications,”in Nonlinear Model-based Process Control, NATO ASI Series, Ser. E, Vol. 353, pp. 3-32, R.Berber and C. Kravaris (eds.), Kluwer, Dordrecht (1998).Bequette, B.W., “Nonlinear Control of Chemical Processes: A Review,” Ind. Eng. Chem. Res. 30, 1391-1413 (1991).Garcia, C.E., D.M. Prett and M. Morari, “Model Predictive Control: Theory and Practice-A Survey,” Automatica, 25, 335-348 (1989).Henson, M.A. and D.E. Seborg, “Critique of Exact Linearization Strategies for Process Control,”J. Process Control, 1(3), 122-139 (1991).Kravaris, C. and J.C. Kantor, “Geometric Methods for Nonlinear Process Control. I.Background,” Ind. Eng. Chem. Res., 29, 2295-2310 (1990a)Kravaris, C. and J.C. Kantor, “Geometric Methods for Nonlinear Process Control. II. Controller Synthesis,” Ind. Eng. Chem. Res., 29, 2310-2324 (1990b)McLellan, P.J., T.J. Harris and D.W. Bacon, “Error Trajectory Descriptions of Nonlinear Controller Designs,” Chem. Eng. Sci., 45, 3017-3034 (1990).Morari, M., “Robust Process Control,” Chem. Eng. Res. Des., 65, 462-479 (1987). Rawlings, J.B., E.S. Meadows and K.R. Muske, “Nonlinear Model Predictive Control: A Tutorial and Survey,” Preprints ADCHEM ‘94, IFAC Symposium on the Advanced Control ofChemical Processes, Kyoto, Japan, May 25-27, 203-214 (1994).Seborg, D.E., T.F. Edgar and S.L. Shah, “Adaptive Control Strategies for Process Control: A Survey,” AIChE J., 32, 881-913 (1986)."Classic" Review ArticlesFoss, A.S., “Critique of Chemical Process Control Theory,” AIChE J., 19, 209-214 (1973). Lee, W. and V.W. Weekman, “Advanced Control Practice in the Chemical Process Industry: A View from Industry,” AIChE J., 22, 27-38 (1976).Morari, M., “Flexibility and Resiliency of Process Systems,” Comp. Chem. Engng., 7, 423-437 (1983).Ray, W.H., “Multivariable Process Control - A Survey,” Comp. Chem. Engng., 7, 367 (1983).Basic Process ControlBequette, B.W. An Introduction to Model-Based Control, Prentice Hall, Upper Saddle River, NJ (to31 August 1999 - B.W. BequetteCoughanowr, D.R. Process Systems Analysis and Control, 2nd ed., McGraw-Hill, New York (1991).Luyben, W.L., Process Modeling Simulation and Control for Chemical Engineers, 2nd ed., McGraw-Hill, New York (1990).Luyben, W.L. and M.L. Luyben, Essentials of Process Control, McGraw Hill (1997).Marlin, T.E., Process Control: Designing Processes and Control Systems for Dynamic Performance, McGraw Hill, New York (1995).Ogunnaike, B.A. and W.H. Ray, Process Dynamics, Modeling and Control, Oxford, New York (1994).Riggs, J.B., Chemical Process Control, Ferret Publishing, Lubbock, TX (1999).Seborg, D.E., T.F. Edgar and D.A. Mellichamp, Process Dynamics and Control, Wiley, New York (1989).Smith, C.A. and A.B. Corripio, Principles and Practice of Automatic Process Control, 2nd ed., Wiley, New York (1997).Stephanopoulos, G., Chemical Process Control, Prentice Hall, Englewood Cliffs (1984).Graduate Process ControlLevine, W.S. (ed.) The Control Handbook, CRC Press (1996).Morari, M. and E. Zafiriou, Robust Process Control, Prentice Hall, Englewood Cliffs (1989).Newell, R.B. and P.L. Lee, Applied Process Control - A Case Study, Prentice Hall, Englewood Cliffs (1989).Prett., D. and C. Garcia, Fundamental Process Control, Butterworths, Boston (1988).Ray, W.H., Advanced Process Control, Butterworths, Boston (1989). This is a paperback reprint of the original version published by McGraw-Hill in 1981.Skogestad, S. and I. Postlethwaite Multivariable Feedback Control. Analysis and Design, Wiley, New York (1996).Specialty Process ControlBuckley, P.S., W.L. Luyben and J.P. Shunta, Design of Distillation Control Systems, ISA, Research Triangle Park (1985).Henson, M.A. and D.E. Seborg (eds.), Nonlinear Process Control, Prentice Hall, Upper Saddle River, NJ (1997).Liptak, B.G. and K.Venczel (eds.) Instrument Engineers Handbook, Process Control Volume, Chilton Book Company, Radnor, PA (1985).Luyben, W.L., B.D Tyreus and M.L. Luyben, Plantwide Process Control, McGraw Hill (1999).Schork, F.J., Deshpande, P.B. and K.W. Leffew, Control of Polymerization Reactors, Marcel Dekker, New York (1993).Shinskey, F.G. Distillation Control, McGraw Hill, New York (1977).Shunta, J.P. Achieving World Class Manufacturing Through Process Control, Prentice Hall, Upper Saddle River, NJ (1995).Nonlinear Systems and ControlIsidori, A. Nonlinear Control Systems, 2nd ed., Springer-Verlag, New York (1989).Khalil, K. Nonlinear Systems, Macmillan, New York (1992).Mohler, R.R., Nonlinear Systems. Volume 1. Dynamics and Control, Prentice Hall, Englewood Cliffs (1991).Mohler, R.R., Nonlinear Systems. Volume 2. Applications to Bilinear Control, Prentice Hall, Englewood Cliffs (1991).Slotine, J.-J. E. and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs (1991).Adaptive, Digital and Predictive ControlAstrom, K.J. and B. Wittenmark, Computer Controlled Systems. Theory and Design, Prentice Hall, Englewood Cliffs (1984).Astrom, K.J. and B. Wittenmark, Adaptive Control, Addison Wesley, Reading, MA (1989). Bitmead, R.R., M. Gevers and V. Wertz, Adaptive Optimal Control - The Thinking Man's GPC, Prentice Hall, Englewood Cliffs (1990).Goodwin, G.C. and K.S. Sin, Adaptive Filtering Prediction and Control, Prentice Hall, Englewood Cliffs (1985).Roffel, B., P.J. Vermeer and P.A. Chin, Simulation and Implementation of Self-Tuning Controllers, Prentice Hall, Englewood Cliffs (1989).Soeterboek, R. Predictive Control. A Unified Approach, Prentice Hall, New York (1992).Chemical Process Control ConferencesArkun, Y. and W.H. Ray, Chemical Process Control - CPC IV, Elsevier, Amsterdam (1991). Edgar, T.F. and D.E. Seborg (eds.), Chemical Process Control II, Engineering Foundation, New York (1982).Foss, A.S. and M.M. Denn (eds.), Chemical Process Control, AIChE Symp. Ser. No. 159, Vol. 72 (1976).Kantor, J.C., C.E. Garcia and B. Carnahan, Chemical Process Control - CPC V, AIChE Symp. Ser.No. 316, Vol. 93 (1997).McAvoy, T.J., Y. Arkun and E. Zafiriou, Model Based Process Control, Pergamon Press, New York (1989).Morari, M. and T.J. McAvoy (eds.), Chemical Process Control - CPC III, Elsevier, Amsterdam (1986).Prett, D.M., C.E. Garcia and B.L. Ramaker, The Second Shell Process Control Workshop, Butterworths, Boston (1990).Prett, D.M. and M. Morari (eds.), Shell Process Control Workshop, Butterworths, Boston (1986).Linear and Optimal Control and EstimationChen, C.T. Linear System Theory and Design, Holt Rinehart & Winston, New York (1984).Gelb, A. (ed.) Applied Optimal Estimation, MIT Press, Cambridge, MA (1974).Kailath, T. Linear Systems, Prentice Hall, Englewood Cliffs (1980).Kirk, D.E. Optimal Control Theory, Prentice Hall, Englewood Cliffs (1970).Ljung, L. System Identification: Theory for the User, Prentice Hall , Upper Saddle River, NJ (1987).Stengel, R.F. Optimal Control and Estimation, Dover, NY (1994). This is a paperback reprint of the original version published by McGraw-Hill in 1986.Modeling, Simulation, Design and OptimizationAris, R. Mathematical Modeling Techniques, Pitman, San Francisco (1978).Bequette, B.W. Process Dynamics. Modeling, Analysis and Simulation, Prentice Hall, Upper Saddle River, NJ (1998).Biegler, L.T., I.E. Grossmann and A.W. Westerberg, Systematic Methods of Chemical Process Design, Prentice Hall, Upper Saddle River, NJ (1997).Close, C.M. and D.K. Frederick Modeling and Analysis of Dynamic Systems, 2nd ed., Houghton Mifflin, Boston (1993).Davis, M.E. Numerical Methods and Modeling for Chemical Engineers, Wiley, New York (1984). Denn, M.M. Process Modeling, Longman, New York (1986).Edgar, T.F. and D.M. Himmelblau, Optimization of Chemical Processes, McGraw-Hill, New York (1988).Finlayson, B.A. Nonlinear Analysis in Chemical Engineering, McGraw-Hill, New York (1980).Friedly, J.C. Dynamic Behavior of Processes, Prentice Hall, Englewood Cliffs (1972).Holland, C.D. Fundamentals and Modeling of Separation Processes, Prentice Hall, Englewood Cliffs (1975).Holland, C.D. and A.I. Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill (1983).Jenson, V.G. and G.V. Jeffreys Mathematical Methods in Chemical Engineering, Academic Press, New York (1977).Ramirez, W.F. Computational Methods for Process Simulation, Butterworths, Boston (1989).Riggs, J.B. An Introduction to Numerical Methods for Chemical Engineers, 2nd ed., Texas Tech University Press, Lubbock (1994).Seider, W.D., J.D. Seader and D.R. Lewin, Process Design Principles, Wiley (1999).Walas, S.M. Modeling with Differential Equations in Chemical Engineering, Butterworths (1989).Nonlinear Dynamics, Stability and Bifurcation TheoryAbraham, R.H. and C.D. Shaw Dynamics - The Geometry of Behavior. Part 1.Periodic Behavior (1984), Part 2. Chaotic Behavior (1984), Part 3. Global Behavior (1985), Part 4. Bifurcation Behavior (1988), Aerial Press, Santa Cruz.Arnold, V.I., Ordinary Differential Equations, MIT Press, Cambridge (1973).Boyce, W.E. and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 5th ed., Wiley, New York (1992).Golubitsky, M. and D.G. Schaeffer, Singularities and Groups in Bifurcation Theory, Springer-Verlag, New York (1985).Guckenheimer, J. and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York (1983).Hale J.K. and H. Kocak, Dynamics and Bifurcations, Springer-Verlag, New York (1991).Jackson, E.A. Perspectives of Nonlinear Dynamics, Volumes 1 and 2, Cambridge University Press, Cambridge (1991).Marek, M. and I. Schreiber Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press, Cambridge (1991).Moon, F.C. Chaotic and Fractal Dynamics, Wiley, New York (1992).Scheinerman, E.A. Invitation to Dynamical Systems, Prentice Hall, Upper Saddle River, NJ (1996). Strogatz, S.H. Nonlinear Dynamics and Chaos, Addison Wesley, Reading, MA (1994).Thompson, J.M.T. and H.B. Stewart, Nonlinear Dynamics and Chaos, Wiley, New York (1986). Tufillaro, N.B., T. Abbott and J. Reilly An Experimental Approach to Nonlinear Dynamics and Chaos, Addison-Wesley, Redwood City (1992).MATLAB-Based Control BooksKuo, B.C. and D.C. Hanselman MATLAB Tools for Control System Analysis and Design, Prentice Hall, Englewood Cliffs (1994).Ogata, K. Designing Linear Control Systems with MATLAB, Prentice Hall, Englewood Cliffs (1994).Journals - ControlAIChE JournalAmerican Control Conference (ACC) Proceedings (yearly)ASME Journal of Dynamic Systems, Measurement and ControlAutomatica (IFAC)Canadian Journal of Chemical EngineeringChemical Engineering CommunicationsChemical Engineering Research and DesignChemical Engineering ScienceComputers and Chemical EngineeringConference on Decision and Control (CDC) Proceedings (yearly)Control Engineering Practice (IFAC)Control Systems Magazine (IEEE)Industrial and Engineering Chemistry Research (I&EC Research)IEEE Transactions on Automatic ControlIEEE Transactions on Control System TechnologyInternational Federation of Automatic Control (IFAC) ProceedingsInternational Journal of ControlInternational Journal of Systems SciencesJournal of Process ControlProceedings of the IEESIAM Journal on Control and OptimizationThe following conference proceedings and magazines often provide interesting applied process control problems and solutions.Advances in Instrumentation and Control (ISA Annual Conference)Chemical Engineering Magazine (McGraw Hill)Chemical Engineering ProgressControl Systems Magazine (IEEE)Hydrocarbon Processing (petroleum refining and bulk organic chemicals)Instrumentation Technology (InTech, an instrumentation industry magazine)ISA (Instrument Society of America) TransactionsTAPPI Journal (pulp and paper industry)。

A Model for Dynamic Airline Seat Inventory Control with Multiple Seat BookingsTak C. Lee andMarvin HershNational University of Malaysia, Selangor D.E., MalaysiaFlorida Atlantic University, Boca Raton, Florida 33431AbstractIt is common for the airlines to sell a pool of identical seats atdifferent prices according to different booking classes to improverevenues in a very competitive market. Under this practice, a complexyet very crucial problem is to determine whether a booking requestfor seats in a certain booking class occurring at some point in timeduring the booking period should be accepted or denied. This paper develops a discrete-time dynamic programming model for finding an optimal booking policy, which can be reduced to a set of criticalvalues. Unlike many existing models, this model does not requireany assumptions about the arrival pattern for the various booking classes. Furthermore, multiple seat bookings, which are a practicalissue in airline seat inventory control, are also incorporated intothe model. In this paper, the basic properties of the model are studied. Numerical examples are presented. Computational issues, including the computational efficiency of the model, are also discussed.A Taxonomy and Research Overview of Perishable-Asset Revenue Management: Yield Management, Overbooking, and PricingLawrence R. Weatherford andSamuel E. BodilyUniversity of Wyoming, Laramie, WyomingUniversity of Virginia, Charlottesville, VirginiaThis paper proposes the term perishable-asset revenue management to denote the field that combines the areas of yield management, overbooking, and pricing for perishable assets. After summarizing the characteristics common to problems in this field, the paper discusses the objectives and constraints faced by decision makers. Then it offers a comprehensive taxonomy with 14 different elements and reviews the research that has been done related to each element. Finally, it suggests some important areas of future research that can help bridge the gap between theory and application. Airline network seat inventory control : methodologies and revenue impacts Williamson, Elizabeth LouiseIn the airline industry, it is customary for carriers to offer a widerange of fares for any given seat in the same cabin on the same flight.In order to control the number of seats made available in each fare class, airlines practice what is called seat inventory control. Traditionally,airline seat inventory control has been the process of allocating seats among varies fare classes on a flight leg in order to maximize expected revenues. Reservations for travel on a flight leg are accepted based onthe availability of a particular fare class on that flight leg. Apassenger's ultimate destination, overall itinerary, or total revenue contribution to the airline is not taken into account. The typical route structure of a large airline, however, is built around a complex networkof connecting flights. Maximizing revenues on individual flight legs isnot the same as maximizing total network revenues. The objective of this dissertation is to address the seat inventory control problem at thenetwork level, taking into account the interaction of flight legs and theflow of traffic across a network. BeginninAllocation of Airline Seats between Stochastically Dependent DemandsS. L. Brumelle,J. I. McGill,T. H. Oum,K. Sawaki andM. W. TrethewayAbstractThis paper examines the problem of allocating airline seats between two nested fare classes when the demands for the classes are stochastically dependent. The well known simple seat allotment formula of Littlewood which requires the assumption of statistical independence between demands is generalized to a formula which requires only a much weaker monotonic association assumption. The model employed here is also used to examine the problems of full fare passenger spillage and passenger upgrades fromthe discount class.An Airline Seat Management Model for a Single Leg Route When Lower Fare Classes Book FirstRichard D. WollmerAbstractIt is common practice for airlines to charge several different faresfor a common pool of seats. This paper presents a model that has beenused to address the problem of when to refuse booking requests for agiven fare level to save the seat for a potential request at a higherfare level. This model assumes that lower fare passengers book before higher fare passengers book. This occurs quite frequently in practice,where lower fare passengers are usually vacationers and higher fare passengers are usually business travelers. In fact, many airlinesmandate this practice by imposing advance booking requirements onlower fare classes to prevent business travelers from using them.We present an algorithm that finds a seat management policy that maximizes mean revenue by establishing a critical value for eachfare class. Booking requests for a particular fare level are acceptedif the number of empty seats is strictly greater than its criticalvalue and rejected otherwise. This critical value is a decreasingfunction of the fare price and is equal to zero for the class withthe highest fare.Dynamic and static yield management modelsWen Zhao, University of PennsylvaniaAbstractThis dissertation studies yield management, i.e., how to sell a finitestock of perishable assets for maximum revenues. Here the assets can be goods, seats on a scheduled flight and hotel rooms, or capacity of a manufacturing facility. We consider two types of models: dynamic pricing models, in which only a single price is offered at any time, and multi-class models, in which the same assets can be sold at different prices.For a general dynamic pricing model, we show that the optimal price decreases in the number of items left, but does not necessarily decreaseover time. The optimal price may go up when the reservation price distribution shifts up. We identify a sufficient condition under whichthe optimal price decreases over time. Numerical studies are used to show the revenue impact of price changes in compensating for demand statistical fluctuations and in exploiting shifts of customer reservation price.We study a multi-class dynamic model with non-homogeneous demand, whichhas useful applications in both service and manufacturing firms. The model assumes that a class can be reopened after being closed. We prove that the optimal policy is a monotone threshold policy. This structural property enables us to identify an analytical solution. Numerical examples showthat the optimal dynamic policy outperforms the nave first-come-first-serve policies that are used by many manufacturers. ^ However, the policies suggestedby the above model are quite different from those suggested by the static modelsthat are widely accepted by the airline industry. Therefore, we further proposea new dynamic multi-class model for airline seat allocation that does not allowreopening a class. This model incorporates passenger diversion and no-shows.We show that the optimal policy is a threshold policy, which may not be monotone.The optimal policy of this dynamic model is closely related to those fromstatic models.Finally, we study a static multi-class model for airline seat allocation with overbooking. Under some mild conditions, we prove that a book-up-to policy is optimal. For the special case where no-show probability is homogeneous across all demand classes, a nested class policy is optimal.Network models for seat allocation on flightsMoshe Dror?,Pierre TrudeauCentre de recherche sur les transports, Universite de Montreal, Canada Shaul P. LadanyAbstractThis paper examines the problem of proper (optimal) control over the seat allocation on flights. Given a heterogeneous fleet of aircraft types,multi-leg flights, a number of different passenger categories, and cancelations, an airline's objective is to devise an effective systemwhich aids in setting the seat allocation targets for each category of passengers on each flight. This issue is analyzed by a number of authorsin the context of economic, simulation based, probabilistic, and mathematical programming studies. We present an attempt to addressthis problem from the systems prospective emphasizing characteristicssuch as: passenger cancelations, multi-leg flights, and rolling tactical planning time horizon. Starting from a very simple network flow modelsfor a single flight with a number of intermediate stops, a number of progressively complex models are presented. The airline flights andthe seat allocation system are represented as a generalized networkflow model (with gains/losses on arcs) with the objective of flow maximization (profit maximization). This modelling approach does not claim to replace the seat allocation approaches presented in Alstrup et al. (1985), Mayer (1976), Richter (1982), Simpson (1985a), and Wang (1983), but rather construct seat allocations utilizing some of those referenced schemes in a parameter setting mode for a large network model. The objective of this paper is not to report on computational experiments, but topresent a modeling approach which seems to be promising, if somewhat speculative.Optimal Airline Seat Allocation with Fare Classes Nested by Origins and DestinationsRenwick E. CurryAeronomics Incorporated, Palo Alto, California 94301AbstractPrevious research in the optimal allocation of airline seats has followedone of two themes: marginal seat revenue or mathematical programming. Bothapproaches capture important elements of the revenue management problem. The marginal seat revenue approach accounts for the “nesting” of fare classes in computer reservation systems, but can only control seatinventory by bookings on legs. The mathematical programming approach willhandle realistically large problems and will account for multiple origin–destination itineraries and side constraints, but it does not account forfare class nesting in the reservation systems. This paper combines both approaches by developing equations to find the optimal allocation of seats when fare classes are nested on an origin–destination itinerary and the inventory is not shared among origin–destinations. These results are applicable to seat allocation in certain reservations systems,point-of-sale control, and acceptance of groups over their entireitinerary. A special case of the analysis produces the optimal bookinglimits for leg-based seat allocation with nested fare classes.Optimal and Approximate Control Policies for Airline Booking with Sequential Nonmonotonic Fare ClassesLawrence W. RobinsonAbstractThis paper addresses the question of when to refuse discount bookingsfrom airline passengers to reserve seats for potential future passengerswho are willing to pay a higher fare. When passengers arrive in sequential fare classes, the optimal policy will be to accept reservation requests aslong as the cumulative seats booked does not exceed a given booking limit. This paper relates the probability of filling the plane, under theoptimal policy, with the ratios of the current to the highest remainingfare classes. In addition, it extends the solution from monotonically increasing fares to fares occurring in arbitrary order. Finally,it demonstrates how Monte Carlo integration is easy to use to getarbitrarily close approximations to the optimal policy.Revenue Management Without Forecasting or Optimization: An Adaptive Algorithm for Determining Airline Seat Protection LevelsWe investigate a simple adaptive approach to optimizing seatprotection levels in airline revenue management systems.The approach uses only historical observations of the relativefrequencies of certain seat-filling events to guide directadjustments of the seat protection levels in accordance withthe optimality conditions of Brumelle and McGill (1993).Stochastic approximation theory is used to prove the convergenceof this adaptive algorithm to the optimal protection levels.In a simulation study, we compare the revenue performance ofthis adaptive approach to a more traditional method that combinesa censored forecasting method with a common seat allocationheuristic (EMSR-b).The Passenger-Mix Problem in the Scheduled AirlinesFred Glover,Randy Glover,Joe Lorenzo andClaude McMillanAbstractDeregulation has opened up many opportunities and challenges in the transportation industry—opportunities to increase profits and challenges to keep from being outflanked by competition. A goal of particular interest to the scheduled airlines is to set prices more adaptivelyand to change them more rapidly. A difficult problem arises when many passengers with different itineraries compete for a limited number of seats on a single-flight segment. The problem is complicated by the existence of different fare classes, many flight segments, and different demands across time. For any given set of prices, flight-segment capacities, and passenger-carrying demand, there is some number of passengers at each fare class on each flight segment that will optimize revenue. Knowledge of such an optimum can be used not only in pricing analysis but also in setting policies to influence the passengerfare-class mix so that the optimum will be more nearly achieved in actual practice.We describe a method for identifying the optimum fare-class mix and the design of a system for that purpose which we built and implemented for Frontier Airlines. The recognition and formulation of the problem has become even more important as the number of aircraft in the sky has been reduced and the competition for a limited number of seats has become more intense.。