2004-A-Input-to-state stability of networked control systems

Chapter8Open ProblemsIn this chapter,we would like to give a list of open and unanswered problems in Mathematical Control Theory.The solutions of these open problems will be very important for the development of modern nonlinear control theory.Expectedly novel mathematical analysis and synthesis tools need to be developed to address these challenging problems.The interested reader should also consult the book[3]for other significant and important open problems in Mathematical Control Theory.Open Problem#1Under what conditions WIOS implies IOS?A qualitative characterization of the IOS property for abstract control systems as discussed in this book has not been available yet.For systems described by ODEs, many qualitative characterizations of the ISS and IOS properties are provided in [21–23].Moreover,Theorem4.1in Chap.4gives a complete qualitative character-ization of the WIOS property:“0-GAOS”+“RFC”+“the continuity with respect to initial conditions and external inputs”implies WIOSA similar qualitative characterization for the IOS property in a general context of abstract dynamical systems as discussed in this book will be very important for control designs and applications.Open Problem#2Development of small-gain techniques for dynamical systems described by Partial Differential Equations(PDEs).Small-gain results have been well studied forfinite-dimensional nonlinear sys-tems described by ordinary differential,or difference,equations(see,e.g.,[8–10] and references therein).However,as of today,there is little research devoted to the development of small-gain techniques for nonlinear systems described by Partial Differential Equations(PDEs).We believe that the small-gain results provided in the present book(Theorems5.1and5.2in Chap.5)will pave the road for the appli-cation of small-gain results to systems described by PDEs.I.Karafyllis,Z.-P.Jiang,Stability and Stabilization of Nonlinear Systems,381 Communications and Control Engineering,DOI10.1007/978-0-85729-513-2_8,©Springer-Verlag London Limited2011Open Problem#3Formulas for the Coron–Rosier methodology.Theorem6.1in Chap.6is an existence-type result.Although its proof is con-structive,it cannot be easily applied for feedback design purposes.The creation of formulas for the Coron–Rosier approach will be very significant for control pur-poses,since the Coron–Rosier approach can allow nonconvex control sets and does not require additional properties for the Control Lyapunov Function.The signifi-cance of the solution of this open problem is also noted in[5].Open Problem#4When is a nonlinear,time-varying,time-delay system stabiliz-able?We have recently provided a positive answer to the above question when the sys-tem only involves state-delay[13].A complete answer to the question of when the nonlinear time-varying system with both state and input delays is stabilizable re-mains open and requires deeper investigation.Nonetheless,it should be mentioned that sufficient,but not necessary,conditions for the solution of the stabilization prob-lem with input delays are proposed in the recent work of Krsti´c[14–16](also see [11]).To our knowledge,a necessary and sufficient condition for stabilizability is missing even for linear time-varying systems with input delays.Open Problem#5Application of small-gain results for distributed feedback design of large-scale nonlinear systems.Large-scale systems are abundant in variousfields of science and engineering and have gained increasing attention due to emerging engineering and biomedical applications.Examples of these applications are from smart grids with green and re-newable energy sources,modern transportation networks,and biological networks. There has been some success with the use of decentralized control strategy for both linear and nonlinear large-scale systems;see[7,19]and many references therein. Clearly more remains to be accomplished in this excitingfield.We feel that small-gain is a very appropriate tool for addressing some of these modern-day challenges. The small-gain results of the present book(Theorems5.1and5.2in Chap.5)make a preliminary step forward toward studying some complex large-scale systems be-yond the past literature of decentralized systems and control.Open Problem#6Extension of the discretization approach for autonomous sys-tems.The discretization approach for Lyapunov functionals was described in Chap.2 (Propositions2.4and2.5).However,as remarked in Chap.2,the discretization ap-proach requires good knowledge of some approximation of the solution map,and its use has been restricted to time-varying systems with special structure(see[1,17, 18]).An extension of the discretization approach for autonomous systems wouldbe an important contribution in stability theory because such a result would al-low the use of positive definite functions with non sign-definite derivative.The re-quired extension of the discretization approach must utilize appropriate differential inequalities in the same spirit as the classical Lyapunov’s approach(without requir-ing knowledge of the solution map or a system with special structure).The recent work in[12]is an attempt in this research direction(see also references therein). However,the problem is still completely“untouched.”Open Problem#7Application of feedback design methodologies to other mathe-matical problems.In this book,we have seen the applications of certain tools of modern nonlinear control theory to problems arising from mathematics and economics.Particularly, we have seen•applications of small-gain results to game theory(see Sect.5.5in Chap.5),•applications to numerical analysis(see Sect.7.3).We believe that feedback design methodologies can be applied with success to other areas of mathematical sciences.Fixed Point Theory(see[6])and Optimization Theory can be benefited by the application of certain tools of modern nonlinear con-trol theory.Corollary5.4in Chap.5already shows that small-gain results can have serious consequences in Fixed Point Theory.Further connections between Fixed Point Theory and Stability Theory are provided by the work of Burton(see[4]and references therein)but are in the opposite direction from what we propose,that is, the work of Burton applies results from Fixed Point Theory to Stability Theory.The efforts for the solution of problems in Game Theory,Numerical Analysis, Fixed Point Theory,and Optimization Theory will necessarily demand the creation of novel results in stability theory and feedback stabilization theory.Therefore,the application of modern nonlinear control theory to other areas of applied mathe-matics will result to a“knowledge feedback mechanism”between Mathematical Control Theory and other areas in mathematics!Open Problem#8Integral input-to-state stability(for short,iISS)in complex dy-namical systems.The external stability results of this book are exclusively targeted at extensions of Sontag’s ISS property and its variants to a very general context of complex dynamic systems.That is,we want to address a wide class of dynamical systems which may not satisfy the semigroup property,motivated by important examples of hybrid sys-tems,switched systems,and time-delay systems.It remains an open and important, but interesting,question to know how much we could do with the iISS property introduced in[2,20].References1.Aeyels,D.,Peuteman,J.:A new asymptotic stability criterion for nonlinear time-variant dif-ferential equations.IEEE Transactions on Automatic Control43(7),968–971(1998)2.Angeli,D.,Sontag,E.D.,Wang,Y.:A characterization of integral input-to-state stability.IEEETransactions on Automatic Control45(6),1082–1097(2000)3.Blondel,V.D.,Megretski,A.(eds.):Unsolved Problems in Mathematical Systems and ControlTheory.Princeton University Press,Princeton(2004)4.Burton,T.A.:Stability by Fixed Point Theory for Functional Differential Equations.Dover,Mineola(2006)5.Coron,J.-M.:Control and Nonlinearity.Mathematical Surveys and Monographs,vol.136.AMS,Providence(2007)6.Granas,A.,Dugundji,J.:Fixed Point Theory.Springer Monographs in Mathematics.Springer,New York(2003)7.Jiang,Z.P.:Decentralized control for large-scale nonlinear systems:A review of recent results.Dynamics of Continuous,Discrete and Impulsive Systems11,537–552(2004).Special Issue in honor of Prof.Siljak’s70th birthday8.Jiang,Z.P.:Control of interconnected nonlinear systems:a small-gain viewpoint.In:deQueiroz,M.,Malisoff,M.,Wolenski,P.(eds.)Optimal Control,Stabilization,and Nonsmooth Analysis.Lecture Notes in Control and Information Sciences,vol.301,pp.183–195.Springer, Heidelberg(2004)9.Jiang,Z.P.,Mareels,I.M.Y.:A small-gain control method for nonlinear cascaded systems withdynamic uncertainties.IEEE Transactions on Automatic Control42,292–308(1997)10.Jiang,Z.P.,Teel,A.,Praly,L.:Small-gain theorems for ISS systems and applications.Mathe-matics of Control,Signals,and Systems7,95–120(1994)11.Karafyllis,I.:Stabilization by means of approximate predictors for systems with delayed in-put.To appear in SIAM Journal on Control and Optimization12.Karafyllis,I.:Can we prove stability by using a positive definite function with non sign-definite derivative?Submitted to Nonlinear Analysis Theory,Methods and Applications 13.Karafyllis,I.,Jiang,Z.P.:Necessary and sufficient Lyapunov-like conditions for robustnonlinear stabilization.ESAIM:Control,Optimization and Calculus of Variations(2009).doi:10.1051/cocv/2009029,pp.1–42,August200914.Krsti´c,M.:Delay Compensation for Nonlinear,Adaptive,and PDE Systems.Systems&Con-trol:Foundations&Applications.Birkhäuser,Boston(2009)15.Krsti´c,M.:Input delay compensation for forward complete and feedforward nonlinear sys-tems.IEEE Transactions on Automatic Control55,287–303(2010)16.Krsti´c,M.:Lyapunov stability of linear predictor feedback for time-varying input delay.IEEETransactions on Automatic Control55,554–559(2010)17.Peuteman,J.,Aeyels,D.:Exponential stability of slowly time-varying nonlinear systems.Mathematics of Control,Signals and Systems15,42–70(2002)18.Peuteman,J.,Aeyels,D.:Exponential stability of nonlinear time-varying differential equationsand partial averaging.Mathematics of Control,Signals and Systems15,202–228(2002)19.Siljak,D.:Decentralized Control of Complex Systems.Academic Press,New York(1991)20.Sontag,E.D.:Comments on integral variants of ISS.Systems Control Letters3(1–2),93–100(1998)21.Sontag,E.D.,Wang,Y.:On characterizations of the input-to-state stability property.Systemsand Control Letters24,351–359(1995)22.Sontag,E.D.,Wang,Y.:New characterizations of the input-to-state stability.IEEE Transac-tions on Automatic Control41,1283–1294(1996)23.Sontag,E.D.,Wang,Y.:Lyapunov characterizations of input to output stability.SIAM Journalon Control and Optimization39,226–249(2001)。

PWM Buck ControllerFeatures- PWM Buck Control Circuitry- Operating voltage can be up to 27V- Under voltage Lockout (UVLO) Protection - Short Circuit Protection (SCP) - Soft-start circuit- Variable Oscillator Frequency -- 300Khz Max - 1.25V voltage reference Output - 8-pin PDIP and SOP packagesApplications- Backlight inverter - LCD Monitor- XDROM, XDSL Product- DC/DC converters in computers, etc.General DescriptionThe AP2004 integrates Pulse-Width-Modulation (PWM) control circuit into a single chip, mainly designs for power-supply regulator. All the functions include an on-chip 1.25V reference output, an error amplifier, an adjustable oscillator, a soft-start, UVLO, SCP circuitry, and a push-pull output circuit. Switching frequency is adjustable by trimming CT. During low VCC situation, the UVLO makes sure that the outputs are off until the internal circuit operates normally.Pin AssignmentOUT GND SCPSS FBVCC CT COMP( Top View )PDIP/SOPPin DescriptionsNameDescriptionCT TimingCapacitor FB Voltage Feedback SS Soft-Start. COMPFeedback Loop CompensationOUT PWM OutputGND Ground VCC Supply Voltage SCP Short Circuit ProtectionOrdering InformationS: SOP-8LA : TapingL : Lead Free PackagePWM Buck ControllerBlock DiagramOUTFBVCCSCP CTCOMPGNDSSAbsolute Maximum RatingsSymbol ParameterRating Unit V CC Supply voltage 28 V V I Amplifier input voltage 20 V V O Collector output voltage V CC -1.0VVI SOURCE Source current 200 mA I SINK Sink current200 mA T OP Operating temperature range -20 to +85 o C T ST Storage temperature range-65 to +150o C T LEAD Lead temperature 1.6 mm(1/16 inch) from case for 10 seconds260oCPWM Buck ControllerRecommended Operating ConditionsSymbolParameterMin. Max. Unit V CC Supply voltage 3.6 27V V I Amplifier input voltage 1.05 1.45V V O Collector output voltage Vcc-1.5 VI FBCurrent into feedback terminal45 µA R F Feedback resistor 100 k Ω C T Timing capacitor 100 6800pF F OSC Oscillator frequency 10 300 KHz T OPOperating free-air temperature-2085ºCElectrical Characteristics (T A=25ºC, VCC=6V, f=200 Khz)Reference (REF) Symbol Parameter Conditions Min.Typ. Max.UnitComp connect to FB 1.225 1.25 1.275VT A = -20ºC ~ 25ºC -0.1 ±1 % V REF Output voltage change withtemperature T A = 25ºC ~ 85ºC -0.2 ±1 %Under voltage lockout (UVLO) Symbol Parameter Conditions Min.Typ. Max.Unit V UT Upper threshold voltage (V CC ) 2.9 VV LWT Lower threshold voltage (V CC ) 2.4 VV HT Hysteresis (V CC ) I O(REF) = 0.1mAT A = 25ºC500 mV Short-circuit protection (SCP) control Symbol Parameter Conditions Min.Typ. Max.Unit V IT Input threshold voltage T A = 25ºC 0.600.67 0.75V V STB Standby voltage No pull up 100 130 160 mV V LT Latched input voltage No pull up 50 100 mV I SCP Input (source) current V I = 0.7V, T A = 25ºC -10 -15 -20 µAV CT Comparator threshold voltage(COMP)1.5 VOscillator (OSC) Symbol Parameter Conditions Min.Typ. Max.Unit F OSC Frequency C T =270 pF 200 KHzStandard deviation of frequency C T =270 pF 10∆F OSC Frequency change with voltage V CC =3.6V ~ 20V 1%PWM Buck ControllerElectrical Characteristics (Continued) (T A=25ºC, VCC=6V, f=200 Khz)Error-amplifierSymbol Parameter ConditionsMin.Typ. Max. Unit V IO Input offset voltage V O (FB)=1.25V ±6 mV I IO Input offset current V O (FB)=1.25V ±100 nA I IB Input bias currentV O (FB)=1.25V 160 500 nA V CMCommon-mode input voltagerangeV CC =3.6V ~ 20V1.051.45VAVOpen-loop voltageamplificationR F =200 k Ω 70 80 dBGBW Unity-gain bandwidth1.5 MHz CMRR Common-mode rejection ratio 60 80 dB V OH Max. output voltage V ref -0.1 VV OL Min. output voltage1 V I OI Output (sink) current (COMP)V ID = -0.1V, V O = 1.25V 0.5 1.6 mA I OOOutput (source) current(COMP)V ID = 0.1V, V O = 1.25V-45 -70 µA Output sectionSymbol ParameterConditions Min.Typ. Max. Unit I LEAK Leakage current V O = 25V 10 µA Sink current V IN = 20V 200 mA I DRV Source currentV IN = 20V 200 mA V SATOutput saturation voltageI O = 10 mA 1.0 1.5 V I SC Short-circuit output currentV O = 6V120 mA PWM comparatorSymbol ParameterConditionsMin.Typ.Max.UnitV T0 CT 0.6 0.7 V V T100 Input threshold voltage at f =10 KHz (COMP) Maximum duty cycle 1.2 1.3 V Total deviceSymbolParameterConditionsMin.Typ. Max. Unit I CCA Average supply current C T = 270pF6 10 mA Soft StartSymbol Parameter ConditionsMin.Typ. Max. Unit V SS Soft-start Voltage 2.3 VI SSConstant Charge Current20µAPWM Buck ControllerTypical Application CircuitC3Step-Down DC/DC converterTypical CharacteristicsPWM Buck ControllerTypical Characteristics (Continued)Marking Information(Top View)PDIP/SOPLogo ID codeYear: "01" =2001 "02" =2002Xth week: 01~52~PWM Buck ControllerPackage Information(1) PDIP-8L (Plastic Dual-in-line Package )E-PIN O0.118 inchDimensions in millimeters Dimensions in inchesSymbolMin. Nom. Max. Min. Nom. Max.A - - 5.33 - - 0.210A1 0.38 - - 0.015 - - A2 3.1 3.30 3.5 0.122 0.130 0.138B 0.36 0.46 0.56 0.014 0.018 0.022B1 1.4 1.52 1.65 0.055 0.060 0.065 B2 0.81 0.99 1.14 0.032 0.039 0.045C 0.20 0.25 0.36 0.008 0.010 0.014D 9.02 9.27 9.53 0.355 0.365 0.375E 7.62 7.94 8.26 0.300 0.313 0.325E1 6.15 6.35 6.55 0.242 0.250 0.258e - 2.54 - - 0.100 -L 2.92 3.3 3.81 0.115 0.130 0.150 eB 8.38 8.89 9.40 0.330 0.350 0.370 S 0.71 0.84 0.97 0.028 0.033 0.038PWM Buck ControllerPackage Information (Continued)(2) SOP- 8L(JEDEC Small Outline Package)Dimensions In Millimeters Dimensions In InchesSymbolMin. Nom. Max. Min. Nom. Max.A 1.40 1.60 1.75 0.055 0.063 0.0690.040 - 0.100A1 0.10 - 0.25A2 1.30 1.45 1.50 0.051 0.057 0.059B 0.33 0.41 0.51 0.013 0.016 0.020C 0.19 0.20 0.25 0.0075 0.008 0.010D 4.80 5.05 5.30 0.189 0.199 0.209E 3.70 3.90 4.10 0.146 0.154 0.161e - 1.27 - - 0.050 -H 5.79 5.99 6.20 0.228 0.236 0.244L 0.38 0.71 1.27 0.015 0.028 0.050y - - 0.10 - - 0.004θ0O - 8O0O - 8O。

三亚2024年08版小学4年级下册英语第一单元期中试卷考试时间：90分钟（总分:140）A卷一、综合题(共计100题共100分)1. 填空题：A _____ is a natural formation that rises prominently.2. 选择题：What do you call a place where you can borrow books?A. LibraryB. BookstoreC. SchoolD. Office3. 填空题：The __________ (历史的展望未来) influences policies.4. 填空题：I enjoy playing ______ with my siblings.5. 填空题：The hedgehog curls into a _______ (球).6. se invented ________ around 100 AD. 填空题：The Civi7. 听力题：A chemical reaction may produce ______.8. 填空题：I like to build a __________ when it snows. (雪人)9. 填空题：Planting trees can help combat ______ (气候变化).10. 选择题：What do you call the practice of growing crops?A. AgricultureB. HorticultureC. FarmingD. Gardening11. 选择题：Which planet has the longest day?A. EarthB. VenusC. MarsD. Mercury12. 填空题：My dad is really _______ (形容词) when it comes to fixing things. 他总是 _______ (动词).13. 选择题：How do you say "good night" in French?A. Bonne nuitB. Buenas nochesC. BuonanotteD. Dobranoc14. 听力题：My aunt is a ______. She helps organize events.15. 填空题：________ (植物保护活动) raise awareness.16. 选择题：What do we call a traditional story that explains something in nature?A. MythB. FableC. LegendD. Tale17. 听力题：A compound that has both acidic and basic properties is called an ______.18. 填空题：My dog loves to play with his ________.19. 填空题：She is a _____ (科学家) who studies the ocean.20. 小猫) likes to play with balls of yarn. 填空题：The ___21. 填空题：The butterfly emerges from its _________ (蛹).22. 选择题：What do we call the act of promoting teamwork?A. CollaborationB. CooperationC. PartnershipD. All of the Above答案：D23. 听力题：She is ___ her homework now. (doing, done, do)24. 选择题：What is the color of bananas?A. RedB. YellowC. GreenD. Blue答案: B25. 填空题：The _______ (猴子) eats bananas and berries.26. 听力题：In space, there is no air or ______.27. 选择题：What is the capital of Grenada?a. St. George'sb. Gouyavec. Grenvilled. Carriacou答案:a28. 听力题：I like to ___ puzzles. (solve)29. 听力题：A _______ can measure the pressure of liquids in a container.30. 选择题：What do we call the process of learning through experience?A. EducationB. TrainingC. PracticeD. Apprenticeship答案:A31. 听力题：The _______ of a wave can be visualized with a diagram.32. 听力题：Every planet in our solar system orbits the ______.33. 听力题：My aunt lives in a _____ (city/country).34. 选择题：What is the season after winter?A. FallB. SummerC. SpringD. Autumn答案:C35. 填空题：The _____ (青蛙) has a unique way of communicating.36. 选择题：What is the opposite of 'old'?A. YoungB. MatureC. AgedD. Elderly答案:A37. 听力题：There are _____ states of matter: solid, liquid, and gas.38. 填空题：My __________ (玩具名) is really __________ (形容词) to play with.39. 选择题：How many players are on a baseball team?A. NineB. TenC. ElevenD. Twelve40. 选择题：What is the term for a young female horse?A. ColtB. FillyC. FoalD. Mare答案:B41. 填空题：The __________ (古代文明的遗迹) are found all over the world.42. 填空题：We observed a ________ growing.43. 听力题：We will go _____ (shopping/working) tomorrow.44. 听力题：The clock ticks _____ (slowly/quickly).45. 填空题：My favorite board game is _______ (大富翁).46. 选择题：What is the color of a typical blueberry?A. GreenB. BlueC. RedD. Yellow答案:B47. 选择题：What is the name of the famous volcano in Italy?A. Mount EtnaB. Mount VesuviusC. Mount St. HelensD. Mount Fuji48. 填空题：I have a ________ that helps me learn.49. 选择题：What is the capital of the Republic of the Congo?A. BrazzavilleB. KinshasaC. Pointe-NoireD. Ouesso答案:A. Brazzaville50. 听力题：My mom loves to do ____ (yarn crafts).51. 填空题：The sun is _______ in the sky.52. 填空题：A ______ (蜗牛) carries its home with it wherever it goes.53. 选择题：What do we call a baby dog?A. KittenB. PuppyC. CalfD. Chick54. 听力题：My dad loves to go fishing at the ____ (lake).55. 填空题：I dream of becoming a ______ (艺术家) one day. I want to create beautiful pieces that inspire others.56. 听力题：The Earth's crust contains many valuable ______ resources.57. 填空题：My brother has a knack for __________ (解决问题).58. 听力题：The _______ can be used for decoration.59. 选择题：Which shape is round?A. SquareB. TriangleC. CircleD. Rectangle答案:CA rabbit's foot is considered a ______ (好运) charm.61. 选择题：What is the capital of France?A. BerlinB. LondonC. ParisD. Madrid答案:C62. 选择题：Which planet is known for its blue color?A. EarthB. NeptuneC. UranusD. Both B and C答案: D63. 填空题：A duck's quack can be quite ________________ (响亮).64. 填空题：Many fruits grow from _____ (树) or bushes.65. 选择题：What is the opposite of dark?A. BrightB. LightC. DullD. Shadow66. 选择题：What do we call the first month of the year?A. FebruaryB. MarchC. AprilD. January答案:D67. 填空题：A ladybug is often seen on ______ (绿叶).68. 填空题：The __________ is a major river that flows through Nigeria. (尼日尔河)Many _______ have beautiful flowers.70. 听力题：The Industrial Revolution started in the ________.71. 选择题：What is the term for a young shark?a. Pupb. Kitc. Calfd. Chick答案:a72. 填空题：__________ (化学制剂) can enhance the effectiveness of medications and treatments.73. 选择题：What do you wear on your feet?A. HatB. GlovesC. ShoesD. Scarf答案:C74. 选择题：What is the name of the sweet treat made from sugar and gelatin?A. Gummy BearsB. Candy CornC. Jelly BeansD. Marshmallows答案: D75. 听力题：The ______ helps us see light.76. 选择题：What do bees produce?A. MilkB. HoneyC. SilkD. Wool答案: B77. 填空题：The penguin waddles _______ (走路) on ice.Many cultures use _____ (植物药) for healing.79. 听力题：The cake is ________ and sweet.80. 填空题：The _______ (昆虫) crawls on the ground.81. 填空题：Planting _____ (本地树种) contributes to ecological stability.82. 选择题：What is the main language spoken in the USA?A. SpanishB. FrenchC. EnglishD. German答案:C83. 听力题：The chemical formula for potassium hydrogen phthalate is _______.84. 选择题：What is the opposite of loud?A. QuietB. SoftC. SilentD. Mute答案:A85. 选择题：What is the capital of India?A. DelhiB. MumbaiC. KolkataD. Bangalore86. 听力题：The cat is very ___. (playful)87. 选择题：What do you call the person who teaches students?A. DoctorB. TeacherC. EngineerD. Artist88. 听力题：A reaction that requires energy input is called an ______ reaction.89. 选择题：What is the tallest mountain in the world?A. K2B. Mount EverestC. KilimanjaroD. Denali答案：B90. 听力题：The capital of Papua New Guinea is __________.91. 听力题：The pH scale measures how _______ or basic a solution is.92. 填空题：The ancient city of __________ (雅典) is known for its democracy.93. 填空题：The ancient Egyptians used _____ for mummification.94. 选择题：What do we call a young fish?A. FryB. FingerlingC. LarvaD. Pup95. 选择题：What is the capital of the USA?A. New YorkB. Los AngelesC. Washington,D.C.D. Chicago答案:C96. 选择题：What is the capital of Ecuador?A. QuitoB. GuayaquilC. CuencaD. Loja答案: A97. 选择题：What is the term for a young snake?A. HatchlingB. PupC. KitD. Calf答案:A. Hatchling98. 听力题：My uncle is a fantastic ____ (chef).99. 选择题：Which animal can fly?A. FishB. BirdC. DogD. Cat100. 填空题：We have a ______ (精彩的) event planned for next week.。

FORMATION INPUT-TO-STATE STABILITYHerbert G.Tanner and George J.PappasDepartment of Electrical EngineeringUniversity of PennsylvaniaPhiladelphia,PA19102tanner@,pappasg@Abstract:This paper introduces the notion of formation input-to-state stability in order to characterize the internal stability of leader-follower formations,with respect to inputs received by the formation leader.Formation ISS is a weaker form of stability than string stability since it does not require inter-agent communication.It relates group input to internal state of the group through the formation graph adjacency matrix.In this framework,different formation structures can be analyzed and compared in terms of their stability properties and their robustness.Keywords:Formations,graphs,interconnected systems,input-to-state stability.1.INTRODUCTIONFormation control problems have attracted increased attention following the advances on communication and computation technologies that enabled the de-velopment of distributed,multi-agent systems.Direct fields of application include automated highway sys-tems(Varaiya,1993;Swaroop and Hedrick,1996; Yanakiev and Kanellakopoulos,1996),reconnais-sance using wheeled robots(Balch and Arkin,1998), formationflight control(Mesbahi and Hadaegh,2001; Beard et al.,2000)and sattelite clustering(McInnes, 1995).For coordinating the motion of a group of agents,three different interconnection architectures have been con-sidered,namely behavior-based,virtual structure and leader-follower.In behavior based approach(Balch and Arkin,1998;Lager et al.,1994;Yun et al.,1997), several motion premitives are defined for each agent and then the group behavior is generated as a weighted sum of these primary behaviors.Behavior based con-trol schemes are usually hard to analyze formally, although some attempts have been made(Egerstedt, 2000).In leader-follower approaches(Beard et al., 2000;Desai and Kumar,1997;Tabuada et al.,2001;Fierro et al.,2001),one agent is the leader of the formation and all other agents are required to fol-low the leader,directly or indirectly.Virtual structure type formations(Tan and Lewis,1997;Egerstedt and Hu,2001),on the other hand,usually require a cen-tralized control architecture.Balch and Arkin(1998)implement behavior-based schemes on formations of unmanned ground vehicles and test different formation types.Yun et al.(1997) develop elementary behavior strategies for maintain-ing a circular formation using potentialfield meth-ods.Egerstedt and Hu(2001)adopt a virtual struc-ture architecture in which the agents follow a vir-tual leader using a centralized potential-field control scheme.Fierro et al.(2001)develop feedback lineariz-ing controllers for the control of mobile robot forma-tions in which each agent is required to follow one or two leaders.Tabuada et al.(2001)investigate the conditions under which a set of formation constraints can be satisfied given the dynamics of the agents and consider the problem of obtaining a consistent group abstraction for the whole formation.This paper focuses on a different problem:given a leader-follower formation,investigate how the leader input affects the internal stability of the overall for-mation.Stability properties of interconnected systems have been studied within the framework of string stability(Swaroop and Hedrick,1996;Yanakiev and Kanellakopoulos,1996).String stability actually re-quires the attenuation of errors as they propagate in the formation.However,sting stability conditions are generally restrictive and generally require inter-agent communication.It is known,for instance(Yanakiev and Kanellakopoulos,1996)that string stability in au-tonomous operation of an AHS with constant interve-hicle spacing,where each vehicle receives information only with respect to the preceding vehicle,is impos-sible.We therefore believe that a weaker notion of stability of interconnected system that relates group objectives with internal stability would be useful. Our approach is based on the notion of input-to-state stability(Sontag and Wang,1995)and exploits the fact that the cascade interconnection of two input-to-state stable systems is itself input-to-state stable(Khalil, 1996;Krsti´c et al.,1995).This property allows the propagation of input-to-state gains through the for-mation structure and facilitates the calculation of the total group gains that characterize the formation per-formance in terms of stability.Formation ISS is a weaker form of stability than string stability,in the sense that it does not require inter-agent communica-tion and relies entirely on position feedback only(as opposed to both position and velocity feedback)from each leader to its follower.We represent the formation by means of a formation graph(Tabuada et al.,2001). Graphs are especially suited to capture the intercon-nections(Tabuada et al.,2001;Fierro et al.,2001) and informationflow(Fax and Murray,2001)within a formation.The proposed approach provides a means to link the formation leader’s motion or the external input to the internal state and the adjacency matrix of the formation.It establishes a method for comparing stability properties of different formation schemes. The rest of the paper is organized as follows:in sec-tion2the definitions for formation graphs and for-mation input-to-state stability(ISS)are given.Section 3establishes the ISS properties of an leader-follower interconnection and in section4it is shown how these properties can be propagated from one formation graph edge to another to cover the whole formation. Section5provides examples of two stucturally differ-ent basic formation configurations and indicates how interconnection differences affect stability properties. In section6results are summarized and future re-search directions are highlighted.2.FORMATION GRAPHSA formation is being modeled by means of a formation graph.The graph representation of a formation allows a unified way of capturing both the dynamics of each agent and the inter-agent formation specifications.All agent dynamics are supposed to be expressed by lin-ear,time invariant controllable systems.Formation specifications take the form of reference relative posi-tions between the agents,that describe the shape of the formation and assign roles to each agent in terms of the responcibility to preserve the specifications.Such an assignment imposes a leader-follower relationship that leads to a decentralized control architecture.The assignment is expressed as a directed edge on the formation graph(Figure1).Fig.1.An example of a formation graphDefinition2.1.(Formation Graph).A formation graph F=(V,E,D)is a directed graph that consists of:•Afinite set V={v1,...,v l}of l vertices and amapping v i→T R n that assignes to each verticean LTI control system describing the dynamicsof a particular agent:˙x i=A i x i+B i u iwhere x i∈R n is the state of the agent accociatedwith vectice v i,u i∈R m is the agent control inputand A i∈R n×n,B i∈R m×m is a controllable pairof matrices.•A binary relation E⊂V×V representing aleader-follower link between agents,with(v i,v j)∈E whenever the agent associated with vectice v iis to follow the agent of v j.•Afinite set of formation specifications D indexedby the set E,D={d i j}(vi,v j)∈E.For each edge (v i,v j),d i j∈R n,denotes the desired relativedistance that the agent associated with vectice v ihas to maintain from the agent associated withagent v j.Our discussion specializes in acyclic formation graphs. This implies that there can be at least one agent v L that can play the role of a leader(i.e.a vectice with no outgoing arrow).The input of the leader can be used to control the evolution of the whole formation.Thegraph is ordered starting from the leader and following a breadth-first numbering of its vertices.For every edge (v i ,v j )we associate an error vector that expresses the deviation from the specification prescribed for that edge:z i j x j −x i −d i j ∈R ni jThe formation error z is defined as the augmented vector formed by concatenating the error vectors for all edges (v i ,v j )∈E :z z e e ∈E A natural way to represent the connectivity of the graph is by means of the adjacency matrix,A .We will therefore consider the mapping E →R l ×l that assigns to the set E of ordered vertice pairs (v i ,v j )the adjacency matrix A E ∈R l ×l .Our aim is to investigate the stability properties of the formation with respect to the input u L of the formation leader.We thus need to define the kind of stability in terms of which the formation will be analyzed:Definition 2.2.(Formation Input-to-State Stability).A formation is called input-to-state stable iff there isa classfunction βand a class function γsuch that for any initial formation error z (0)and for any bounded inputs of the formation leader u L (·)the evolution of the formation error satisfies:z (t ) ≤β( z (0) ,t )+γsup τ≤tu L(1)By investigating the formation input-to-state stabilitywe establish a relationship between the amplitude of the input of the formation leader and the evolution of the formation errors.This will provide upper bounds for the leaders input in order for the formation shape to be maintained inside some desired specifications.Further,it will allow to characterize and compare formations according to their stability properties.3.EDGE INPUT-TO-STATE STABILITY In the leader-follower configuration,one agent is re-quired to follow another by maintaining a constant distance,x j −x i =d i j .If agent i is required to follow agent j ,then this objective is naturally pursued by applying a follower feedback control law that depends on the relative distance between the agents.For x i =x j −d i j to be an equilibrium of the closed loop control system:˙x i =A i x i +B i u iit should hold that A i (x j −d i j )∈(B i );otherwise the follower cannot be stabilized at that distance from its leader.Suppose that there exists an e i j such that B i e i j =−A i (x j −d i j ).Then the following feedback law can be used for the follower:u i =K i (x j −x i −d i j )+e i jleading to the closed loop dynamics:˙x i =(A i −B i K i )(x i −x j +d i j )Then the error dynamics of the i -j pair of leader-follower becomes:˙z i j =(A i −B i K i )z i j +˙xj which can be written,assuming that agent j followsagent k :˙z i j =(A i −B i K i )z i j +g i j (2)where g i j −(A j −B j K j )z jk .The stability of the follower is thus directly dependenton the matrix (A i −B i K i ),the eigenvalues of which can be arbitrarily chosen,and the interconnection term g i j .The interconnection term can be bounded as follows:g i j ≤λM (A j −B j K j ) z jkwhere λM (·)is the maximum eigenvalue of a given matrix.If K i is chosen so that A i −B i K i is Hurwitz,then the solution of the Lyapunov equation:P i (A i −B i K i )+(A i −B i K i )T P i =−Iprovides a symmetric and positive definite matrix P i and a natural Lyapunov function candidate V i =x T i P i x i for the interconnection dynamics (2)that satisfies:λm (P i ) x i ≤V i ≤λM (P i ) x iwhere λm (·)and λM (·)denote the minimum and max-imum eigenvalue of a given matrix,respectively.Forthe derivative of V i :˙V i ≤− x i 2+2λM (P i )λM (A j −B j K j ) x i z jk≤−(1−θ) x i 2≤0for all x i ≥2λM (P i )λM (A j −B j K j )λm (P i )12λM (P i )t+2(λM (P i ))3(λm (P i ))12λM (P i )t(3)γi (r )=¯γi r (4)where¯βi =λM (P i )2γi =2(λM (P i ))3(λm (P i ))14.FROM EDGE STABILITY TO FORMATIONSTABILITYAn important property of input-to-state stability is that it is preserved in cascade connections.The property allows propagation of ISS properties from one agent to another,all the way up to the formation leader. This procedure will yield the global input gains of the leader and give a measure of the sensitivity of the formation shape with respect to the input applied at the leader.In the previous section it was shown that under the assumption of pure state feedback,a formation graph edge is input-to-state stable.The gain functions for the cascade interconnection˙x1=f1(t,x1,x2,u)˙x2=f2(t,x2,u)are given as:β(r,t)=β1(2β1(r,t2)+γ1(2β2(r,t2λM(P i)t+4¯γi¯βi¯βje−1−θ4λM(P j)t+¯βj e−1−θ{ζz 1=x 1z 2=x 2−x 1−d z 3=x 3−x 2−dthe formation equations can be written as:˙z 1=u ˙z 2=−kz 2−u ˙z 3=−kz 3+kz 2For the 1−2interconnection,a Lyapunov functioncandidate could be:V 2(z 2)12|z 2|2≤V 2(z 2)≤1k θ,θ∈(0,1).Then it follows that,|z 2|≤|z 2(0)|e−√k θ=β2(|z 2(0)|,t )+γ2(sup τ≤t|u |)The ISS input-gain function for agent v 2isγ2=sup τ≤t |u |∈z 23and its time derivative would then be˙V 3(z 3)=−kz 23+kz 3z 2For |z |3>sup τ≤t |z 2(τ)|θThen the formation,as a cascade connection of the subsystems of agents v 2and v 3,is input-to-state stable withγ(sup τ≤t|u |)=6+6k θ+θ2(1−θ)t,γ2=12(1−θ)t,γ3=1k θsup τ≤t|u |It can be shown analytically that the second formation can outperform the first in terms of the magnitude of relative errors with respect to the leader’s velocity.Specifically,if we denote by γs the input-to-state gain of the first interconnection connection and by γp the input-to-state gain of the second interconnection,γsk θ2≥6θ+6k θ+θk θ2=6k +7k θ≥2sup τ≤t |u |6.CONCLUSIONSIn this paper,the notion of formation input-to-state stability has been introduced.This form of stability can be used to characterize the internal state of a formation that has a leader-follower achitecture,and establishes a link between the motion of the leader of the formation or its external input and the shape of the formation.Formation ISS is a weaker form of stability than string stability,in the sense that it does not require inter-agent communication and relies entirely on position feedback only (as opposed to both position and velocity feedback)from each leader to its follower.Moreover,it establishes a link between the formation internal state and the outside world.In the proposed framework,different formation struc-tures can be analyzed and compared in terms of their stability properties.Future work is directed towards investigating the ef-fect of (limited)inter-agent communication on forma-tion stability and consistent ways of group abstrac-tions that are based on the formation ISS properties.Acknowledgment:This research is partially sup-ported by the University of Pennsylvania Research Foundation.7.REFERENCESBalch,T.and R.Arkin(1998).Behavior-based forma-tion control for multirobot systems.IEEE Trans-actions on Robotics and Automation.Beard,R.W.,wton and F.Y.Hadaegh(2000).A coordination architecture for spacecraft forma-tion control.IEEE Transactions on Control Sys-tems Technology.To appear.Desai,J.and V.Kumar(1997).Motion planning of nonholonomic cooperating mobile manipulators.In:IEEE International Conference on Robotics and Automation.Albuquerque,New Mexico. Egerstedt,Magnus(2000).Behavior based robotics using hybrid automata.In:Hybrid Systems:Com-putation and Control.Lecture Notes in Computer Science.Springer-Verlag.Egerstedt,Magnus and Xiaoming Hu(2001).Forma-tion constrained multi-agent control.In:Proceed-ings of the IEEE Conference on Robotics and Au-tomation.Seoul,Korea.pp.3961–3966.Fax,J.Alexander and Richard M.Murray(2001).Graph laplacians and vehicle formation stabiliza-tion.Technical Report01-007.CDS,California Institute of Technology.Fierro,R.,A.Das,V.Kumar,and J.P.Ostrowski (2001).Hybrid control of formations of robots.In:Proceedings of the IEEE International Con-ference on Robotics and Automation.Seoul,Ko-rea.pp.157–162.Khalil,Hassan,K.(1996).Nonlinear Systems.Pren-tice Hall.Krsti´c,Miroslav,Ioannis Kanellakopoulos and Petar Kokotovi´c(1995).Nonlinear and Adaptive Con-trol Design.John Willey and Sons.Lager,D.,J.Rosenblatt and M.Hebert(1994).A behavior-based systems for off-road navigation.IEEE Transaction on Robotics and Automation 10(6),776–783.McInnes, C.R.(1995).Autonomous ring forma-tion for a planar constellation of satellites.AIAA Journal of Guidance Control and Dynamics 18(5),1215–1217.Mesbahi,M.and F.Hadaegh(2001).Formationflying of multiple spacecraft via graphs,matrix inequal-ities,and switching.AIAA Journal of Guidance, Control and Dynamics24(2),369–377. Sontag,Eduardo D.and Yuan Wang(1995).On char-acterizations of the input-to-state stability prop-erty.Systems&Control Letters(24),351–359. Swaroop,D.and J.K.Hedrick(1996).Sting stability of interconnected systems.IEEE Transactions on Automatic Control41(3),349–357. Tabuada,Paulo,George J.Pappas and Pedro Lima (2001).Feasible formations of multi-agent sys-tems.In:Proceedings of the American Control Conference.Arlington,V A.pp.56–61.Tan,Kar-Han and M.Anthony Lewis(1997).Virtual structures for high-precision cooperative mobile robot control.Autonomous Robots4(4),387–403. Varaiya,P.(1993).Smart cars on smart roads:prob-lems of control.IEEE Transactions on Automatic Control38(2),195–207.Yanakiev,Diana and Ioannis Kanellakopoulos(1996).A simplified framework for string stability anal-ysis in AHS.In:Proceedings of the13th IFAC World Congress.San Francisco,CA.pp.177–182.Yun,Xiaoping,Gokhan Alptekin and Okay Albayrak (1997).Line and circle formations of distributed physical mobile robots.Journal of Robotic Sys-tems14(2),63–76.。

小学上册英语第一单元全练全测英语试题一、综合题(本题有100小题，每小题1分，共100分.每小题不选、错误，均不给分)1.This toy is _______ (破了).2.The beauty of floral displays can enhance any ______ or event. (花卉展示的美丽可以提升任何场合或活动的氛围。

)3.The ________ is a small bird that sings sweetly.4.Which animal is known for its ability to swim fast?A. DogB. DolphinC. CatD. ElephantB5.The _____ (train) goes very fast.6.How do you say "good afternoon" in Spanish?A. Buenos díasB. Buenas tardesC. Buenas nochesD. Adiós7.My pet rabbit has soft _______ (毛) that I like to pet.8.How many players are in a basketball team?A. 5B. 6C. 7D. 8A9.What do we call a large body of fresh water surrounded by land?A. OceanB. SeaC. LakeD. RiverC10. A _______ (小狼) learns to hunt from its parents.11.I think it’s fun to go ________ (赶集) on weekends.12.I like to _______ in the evening.13.The _______ (蝙蝠) flies at night.14.The __________ River flows through London.15.My friend is very ________.16.The __________ (历史的复杂性) demands thorough examination.17.This girl, ______ (这个女孩), loves to play the flute.18.What is 15 + 10?A. 20B. 25C. 30D. 3519.The kitten is ______ with a ball of yarn. (playing)20.What do we call an animal that eats only plants?A. CarnivoreB. HerbivoreC. OmnivoreD. Insectivore21.The weather is _____ (sunny/cloudy) today.22.In a reaction, the total mass of the reactants equals the total mass of the _____.23. A ______ (仙人掌) can store water for long periods.24.The dog is ______ with its ball. (playing)25.What is the capital of Egypt?A. CairoB. AlexandriaC. LuxorD. GizaA26.The __________ is a famous river in South America. (亚马逊河)27. A __________ is a mixture of liquids that do not mix.28.climate adaptation) prepares for climate impacts. The ____29.Which animal is known for its long neck?A. ElephantB. GiraffeC. LionD. Tiger30.What do you call the person who flies an airplane?A. PilotB. DriverC. CaptainD. Engineer31.n rainforest is rich in ______ (生物多样性). The Amer32.I have a _____ (compass) for navigation.33.The chemical formula for barium sulfate is __________.34.The ancient Egyptians made ______ (木乃伊) as part of their burial practices.35.What is the opposite of busy?A. FreeB. OccupiedC. EngagedD. Both A and BA36.I have a funny ________ (笑话) to tell you.37.The ______ (根系) plays a key role in stability.38. A ____(public forum) invites community input.39.Which animal is known for its ability to change color?A. ChameleonB. TigerC. ElephantD. Panda40.I like to draw pictures of ________.41.The flowers in the garden are _______ and cheerful, spreading happiness.42.The ________ is a bird that can talk.43.I always try to be ______ (诚实) and tell the truth. Honesty is very ______ (重要的) in friendships.44.The _____ (狮子) is a symbol of strength and bravery.45.Water is made up of hydrogen and ______.46.What is the capital of Egypt?A. CairoB. AlexandriaC. LuxorD. GizaA47. A _______ helps us understand how energy is transferred from one form to another.48.My mom loves __________ (知识分享).49.In a solution, the solvent is the substance that dissolves the _____.50.My favorite animal is the ______ (大象). It is very ______ (聪明) and gentle.51.The ________ was a famous document that outlined human rights.52.She has long ___. (hair)53.I can make my own ________ (玩具) from recycled materials.54.The direction of a force is called its ______.55.I want to grow a ________ for my mom's birthday.56.Which instrument has strings and is played by plucking?A. HarpB. FluteC. TrumpetD. TromboneA57.What is the term for a young horse?A. CalfB. PuppyC. FoalD. KittenC58.I like to ride my _______ (我喜欢骑我的_______).59.The __________ (植物的分类) system is complex and varied.60.The main gas released by burning fossil fuels is ______ dioxide.61. A _____ (植物适应性训练) can prepare plants for environmental changes.62.She speaks ________ languages.63.The ______ (植物的生态功能) is vital for balance.64.The smell of certain flowers can evoke strong ______. (某些花的香气可以引发强烈的情感。

Multivariable Feedback Control Analysis and Design1. IntroductionIn modern control systems, it is often necessary to control multiple variables simultaneously in order to meet specific performance requirements. This is known as multivariable control. Multivariable feedback control analysis and design involves the study of techniques for designing control systems that can handle multiple variables simultaneously. In this article, we will explore various aspects of multivariable feedback control analysis and design.2. Multivariable Control SystemsMultivariable control systems are systems that have multiple inputs and multiple outputs. These systems are typically more complex than single-input single-output (SISO) systems because the interactions between different variables can complicate the control design process. Understanding the characteristics and behavior of multivariable control systems is crucial for their effective analysis and design.2.1 System IdentificationBefore designing a multivariable control system, it is important to identify the dynamic behavior of the system. System identification techniques can be used to determine the mathematical models that describe the relationships between inputs and outputs of the system. This involves collecting input-output data and using various modeling techniques such as empirical modeling, transfer function modeling, or state-space modeling.2.2 Control ObjectivesIn multivariable control, there are often multiple conflicting control objectives that need to be satisfied simultaneously. These control objectives can include stability, desired transient response, disturbance rejection, and tracking of setpoints. Balancing theseobjectives and designing controllers that achieve them is a central aspect of multivariable control analysis and design.2.3 Interactions and CouplingOne of the key challenges in multivariable control is the presence of interactions and coupling between the different variables. Interactions occur when changes in one variable affect the behavior of another variable. These interactions can make it difficult to design controllers that do not interfere with each other. Understanding and mitigating interactions is essential for effective multivariable control.2.4 Controller StructureThe selection of an appropriate controller structure is critical to the success of multivariable control design. There are various types of controller structures that can be used, such as decentralized control, centralized control, and decentralized control with optimization. Each structure has its advantages and disadvantages, and the choice depends on the specific requirements of the control problem.3. Multivariable Control AnalysisMultivariable control analysis involves studying the behavior and performance of multivariable control systems. It aims to provideinsights into the system’s dynamics, stability, and robustness.3.1 Stability AnalysisStability is a fundamental requirement for any control system. In multivariable control, stability analysis becomes more complex due to the interactions and couplings between variables. Stability analysis techniques such as eigenvalue analysis, Nyquist stability criterion, and pole placement methods can be used to investigate and ensure thestability of a multivariable control system.3.2 Performance AnalysisPerformance analysis involves evaluating the performance of a multivariable control system in terms of its response to inputs and disturbance rejection. Performance measures such as rise time, settling time, overshoot, and steady-state error can be used to assess the system’s performa nce. Analysis techniques like frequency response analysis, time response analysis, and sensitivity analysis can provide valuable insights into the system’s performance characteristics.3.3 Robustness AnalysisRobustness analysis is concerned with the ability of a multivariable control system to withstand uncertainties and variations in the system’s parameters. Robust control techniques aim to designcontrollers that can provide satisfactory performance over a range of operating conditions and system uncertainties. Sensitivity analysis, robust stability analysis, and the use of optimal control techniques are commonly employed for robustness analysis in multivariable control.3.4 Interaction AnalysisInteraction analysis is crucial for understanding and managing the interactions between variables in a multivariable control system. Interaction measures such as relative gain array (RGA) and condition number matrix (CN) can be used to quantify the strength and direction of interactions. Analysis of interaction patterns can help in choosing appropriate control strategies and gain scheduling techniques to minimize interactions.4. Multivariable Control DesignMultivariable control design involves designing controllers that achieve the desired control objectives while taking into account the system’s dynamics, interactions, and constraints.4.1 Decentralized Control DesignDecentralized control design involves designing individual controllers for each variable in a multivariable control system. Each controller operates based on its local measurements and controls its respective variable. Decentralized control can be advantageous when theinteractions between variables are weak, and the system can be effectively decoupled.4.2 Centralized Control DesignCentralized control design aims to design a single controller that regulates all variables simultaneously. This approach considers the interactions between variables explicitly and can achieve better overall control performance. However, centralized control can be computationally complex and may require accurate modeling of the entire system.4.3 Decentralized Control with OptimizationDecentralized control with optimization is an intermediate approach that combines the advantages of both decentralized and centralized control. It involves designing decentralized controllers for individual variables while optimizing their performance collectively. This approach can provide a good balance between performance and complexity.4.4 Controller Tuning MethodsOnce the controller structure is selected, tuning methods are used to determine the controller parameters. Various tuning methods are available, such as PID tuning, gain scheduling, pole placement, and optimization-based methods. Each method has its advantages and limitations, and the choice depends on the specific control problem and requirements.5. ConclusionMultivariable feedback control analysis and design are essential for the effective control of systems with multiple variables. This article discussed key aspects of multivariable control systems, including systemidentification, control objectives, interactions and coupling, and controller structure selection. It also explored multivariable control analysis techniques, such as stability analysis, performance analysis, robustness analysis, and interaction analysis. Furthermore, the article covered various multivariable control design approaches like decentralized control, centralized control, and decentralized control with optimization. By understanding and applying these concepts and techniques, engineers can design robust and efficient multivariable control systems to meet desired control objectives.。

New Characterizations of Input to State StabilityEduardo D.SontagYuan WangAbstract—We present new characterizations of the Input to State Stability property.As a consequence of these re-sults,we show the equivalence between the ISS property and several(apparent)variations proposed in the literature.I.IntroductionThis paper studies stability questions for systems of the general formΣ:˙x=f(x,u),(1) with states x(t)evolving in Euclidean space R n and con-trols u(·)taking values u(t)∈U⊆R m,for some positive integers n and m(in all the main results,U=R m).The questions to be addressed all concern the study of the size of each solution x(t)—its asymptotic behavior as well as maximum value—as a function of the initial condition x(0)and the magnitude of the control u(·).One of the most important issues in the study of control systems is that of understanding the dependence of state trajectories on the magnitude of inputs.This is especially relevant when the inputs in question represent disturbances acting on a system.For linear systems,this leads to the consideration of gains and the operator-theoretic approach, including the formulation of H∞control.For not necessar-ily linear systems,there is no complete agreement as yet regarding what are the most useful formulations of system stability with respect to input perturbations.One candi-date for such a formulation is the property called“input to state stability”(ISS),introduced in[12].Various authors, (see e.g.[4],[5],[6],[10],[17]have subsequently employed this property in studies ranging from robust control and highly nonlinear small-gain theorems to the design of ob-servers and the study of parameterization issues;for expo-sitions see[14]and most especially the textbooks[7],[8]. The ISS property is defined in terms of a decay estimate of solutions,and is known(cf.[15])to be equivalent to the validity of a dissipation inequalitydV(x(t))dt≤σ(|u(t)|)−α(|x(t)|)holding along all possible trajectories(this is reviewed be-low),for an appropriate“energy storage”function V and comparison functionsσ,α.(A dual notion of“output-to-state stability”(OSS)can also be introduced,and leads to the study of nonlinear detectability;see[16].)E.Sontag is with SYCON-Rutgers Center for Systems and Control,Department of Mathematics,Rutgers University,New Brunswick,NJ08903.e-mail:sontag@.This re-search was supported in part by US Air Force Grant F49620-95-1-0101.Y.Wang is with Department of Mathematics,Florida Atlantic Uni-versity,Boca Raton,FL33431.e-mail:ywang@. This research was supported in part by NSF Grants DMS-9457826 and DMS-9403924In some cases,notably in[2],[6],[18],authors have suggested apparent variations of the ISS property,which are more natural when solving particular control problems. The main objective of this paper is to point out that such variations are in fact theoretically equivalent to the orig-inal ISS definition.(This does not in any way diminish the interest of these other authors’contributions;on the contrary,the alternative characterizations are of great in-terest,especially since the actual estimates obtained may be more useful in one form than another.For instance, the“small-gain theorems”given in[6],[2]depend,in their applicability,on having the ISS property expressed in a particular form.This paper merely states that from a the-oretical point of view,the properties are equivalent.For an analogy,the notion of“convergence”in R n is independent of the particular norm used—e.g.all L p norms are equiv-alent—but many problems are more naturally expressed in one norm than another.)One of the main conclusions of this paper is that the ISS property is equivalent to the conjunction of the following two properties:(i)asymptotic stability of the equilibrium x=0of the unforced system(that is,of the system defined by Equation(1)with u≡0)and(ii)every trajectory of(1) asymptotically approaches a ball around the origin whose radius is a function of the supremum norm of the control being applied.We prove this characterization along with many others.Since it is not harder to do so,the results are proved in slightly more generality,for notions relative to an arbitrary compact attractor rather than the equilibrium x=0.A.Basic Definitions and NotationsEuclidean norm in R n or R m is denoted simply as|·|. More generally,we will study notions relative to nonempty subsets A of R n;for such a set A,|ξ|A=d(ξ,A)= inf{d(η,ξ),η∈A}denotes the point-to-set distance from ξ∈R n to A.(So for the special case A={0},|ξ|{0}=|ξ|.) We also let,for eachε>0and each set A:B(A,ε):={ξ||ξ|A<ε},B(A,ε):={ξ||ξ|A≤ε}. Most of the results to be given are new even for A={0}, so the reader may wish to assume this,and interpret|ξ|A simply as the norm ofξ.(We prefer to deal with arbi-trary A because of potential applications to systems with parameters as well as the“practical stability”results given in Section VI.)The map f:R n×R m→R n in(1)is assumed to be locally Lipschitz continuous.By a control or input we mean a measurable and locally essentially bounded function u: I→R m,where I is a subinterval of R which contains theorigin,so that u (t )∈U for almost all t .Given a system with control-value set U ,we often consider the same system but with controls restricted to take values in some subset O ⊆U ;we use M O for the set of all such controls.Given any control u defined on an interval I and any ξ∈R n ,there is a unique maximal solution of the initial value problem ˙x =f (x,u ),x (0)=ξ.This solution is defined on some maximal open subinterval of I ,and it is denoted by x (·,ξ,u ).(For convenience,we allow negative times t in the expression x (t,ξ,u ),even though the interest is in behavior for t ≥0.)A forward complete system is one such that,for each u defined on I =R ≥0,and each ξ,the solution x (t,ξ,u )is defined on the entire interval R ≥0.The L m ∞-norm (possibly infinite)of a control u is denoted by u ∞.That is, u ∞is the smallest number c such that |u (t )|≤c for almost all t ∈I .Whenever the domain I of a control u is not specified,it will be understood that I =R ≥0.A function F :S →R defined on a subset S of R n containing 0is positive definite if F (x )>0for all x ∈S ,x =0,and F (0)=0.It is proper if the preimage F −1(−D,D )is bounded,for each D >0.A function γ:R ≥0→R ≥0is of class N (or an “N function”)if it is continuous and nondecreasing;it is of class N 0(or an “N 0function”)if in addition it satisfies γ(0)=0.A function γ:R ≥0→R ≥0is of class K (or a “K function”)if it is continuous,positive definite,and strictly increasing,and is of class K ∞if it is also unbounded (equivalently,it is proper,or γ(s )→+∞as s →+∞).Finally,recall that β:R ≥0×R ≥0→R ≥0is said to be a function of class KL if for each fixed t ≥0,β(·,t )is of class K and for each fixed s ≥0,β(s,t )decreases to zero as t →∞.(The notations K ,K ∞,and KL are fairly standard;the notations N and N 0are introduced here for convenience.)B.A Catalog of PropertiesWe catalog several properties of control systems which will be compared in this paper.Much of the terminology —except for “ISS”and the names for properties of unforced systems —is not standard,and should be considered ten-tative.A zero-invariant set A for a system Σas in Equation (1)is a subset A ⊆R n with the property that x (t,ξ,0)∈A for all t ≥0and all ξ∈A ,where 0denotes the control which is identically equal to zero on R ≥0.From now on,all definitions are with respect to a given forward-complete system Σas in Equation (1),and a given compact zero-invariant set A for this system.The main definitions follow.We first recall the definition of the (ISS)property:∃γ∈K ,β∈KL st :∀ξ∈R n ∀u (·)∀t ≥0|x (t,ξ,u )|A ≤β(|ξ|A ,t )+γ( u ∞).(ISS)This was the form of the original definition of (ISS)given in [12].It is known that a system is (ISS)if and only if it satisfies a dissipation inequality,that is to say,there exists a smooth V :R n →R ≥0and there are functions αi ∈K ∞,i =1,2,3and σ∈K so thatα1(|ξ|A )≤V (ξ)≤α2(|ξ|A )(2)and∇V (ξ)f (ξ,v )≤σ(|v |)−α3(|ξ|A )(3)for each ξ∈R n and v ∈R m .See [15],[14]for proofs and an exposition,respectively.A very useful modification of this characterization due to [11]is the fact that the (ISS)property is also equivalent to the existence of a smooth V satisfying (2)and Equation (3)replaced by an estimate of the type ∇V (ξ)f (ξ,v )≤−V (ξ)−α3(|ξ|A ).(This can be understood as:“for some positive definite and proper functions y =V (x )and v =W (u )of states and outputs respectively,along all trajectories of the system we have ˙y =−y +v ”.)The main purpose of this paper is to establish further equivalences for the (ISS)property.It will be technically convenient to first introduce a local version of the property (ISS),by requiring only that the estimate hold if the initial state and the controls are small,as follows:∃ρ>0,γ∈K ,β∈KL st :∀|ξ|A ≤ρ,∀ u ∞≤ρ|x (t,ξ,u )|A ≤β(|ξ|A ,t )+γ( u ∞)∀t ≥0.(LISS)Several standard properties of the “unforced”system ob-tained when u ≡0will appear as technical conditions.We review these now.The 0-global attraction property with re-spect to A (0-GATT)holds if every trajectory x (·)of the zero-input system(Σ0):˙x =f (x,0)(4)satisfies lim t →∞|x (t,ξ,0)|A →0;if this is merely requiredof trajectories with initial conditions satisfying |x (0)|A <ρ,for some ρ>0,we have the 0-local attraction property with respect to A (0-LATT).The 0-local stability property with respect to A (0-LS)means that for each ε>0there is a δ>0so that |ξ|A <δimplies that |x (t,ξ,0)|A <εfor all t ≥0.Finally,the 0-asymptotic stability property with respect to A (0-AS)is the conjunction of (0-LATT)and (0-LS),and the 0-global asymptotic stability property with respect to A (0-GAS)is the conjunction of (0-GATT)and (0-LS).Note that (0-GAS)is equivalent to the conjunction of (0-AS)and (0-GATT).It is useful (see e.g.[3],[12],[7])to express these properties in terms of comparison functions:∃β∈KL st :∀ξ∈R n ∀t ≥0|x (t,ξ,0)|A ≤β(|ξ|A ,t ).(0-GAS)and∃ρ>0,β∈KL st :∀|ξ|A <ρ∀t ≥0|x (t,ξ,0)|A ≤β(|ξ|A ,t )(0-AS)respectively.Next we introduce several new concepts.The limit prop-erty with respect to A holds if every trajectory must at some time get to within a distance of A which is a function of the magnitude of the input:∃γ∈N0st:∀ξ∈R n∀u(·)inft≥0|x(t,ξ,u)|A≤γ( u ∞).(LIM) Observe that,if this property holds,then it also holds with someγ∈K∞.However,the caseγ≡0will be of interest, since it corresponds to a notion of attraction for systems in which controls u are viewed as disturbances.The asymptotic gain property with respect to A holds if every trajectory must ultimately stay not far from A, depending on the magnitude of the input:∃γ∈N0st:∀ξ∈R n∀u(·)limt→+∞|x(t,ξ,u)|A≤γ( u ∞).(AG) Again,if the property holds,then it also holds with some γ∈K∞,but the caseγ≡0will be of interest later.The uniform asymptotic gain property with respect to A holds if the limsup in(AG)is attained uniformly with respect to initial states in compacts and all u:∃γ∈N0∀ε>0∀κ>0∃T=T(ε,κ)≥0st:∀|ξ|A≤κsupt≥T|x(t,ξ,u)|A≤γ( u ∞)+ε∀u(·).(UAG)The boundedness property with respect to A holds if bounded initial states and controls produce uniformly bounded trajectories:∃σ1,σ2∈N st:∀ξ∈R n∀u(·)supt≥0|x(t,ξ,u)|A≤max{σ1(|ξ|A),σ2( u ∞)}.(BND)(This is sometimes called the“UBIBS”or“uniform bounded-input bounded-state”property.)The global sta-bility property with respect to A holds if in addition small initial states and controls produce uniformly small trajec-tories:∃σ1,σ2∈N0st:∀ξ∈R n∀u(·)supt≥0|x(t,ξ,u)|A≤max{σ1(|ξ|A),σ2( u ∞)}.(GS)Observe that,if this property holds,then it also holds with bothσi∈K∞.The local stability property with respect to A holds if we merely require a local estimate of this type:∃δ>0,α1,α2∈N0st:∀|ξ|A≤δ∀ u ∞≤δsupt≥0|x(t,ξ,u)|A≤max{α1(|ξ|A),α2( u ∞)}.(LS)If this property holds,then it also holds with bothαi∈K∞,i=1,2Theorem1:Assume given any forward-complete system Σas in Equation(1),with U=R m,and a compact zero-invariant set A for this system.The following properties are equivalent:A.(ISS)B.(LIM)&(0-AS)C.(UAG)D.(LIM)&(0-GAS)E.(AG)&(0-GAS)F.(AG)&(LISS)G.(AG)&(LS)H.(LIM)&(LS)I.(LIM)&(GS)J.(AG)&(GS)This theorem will follow from a several technical facts which are stated in the next section and proved later in the paper.These technical results are of interest in themselves.C.List of Main Technical StepsWe assume given a forward-complete systemΣas in Equation(1),with U=R m,and a compact zero-invariant set A for this system.For ease of reference,wefirst list several obvious implications:(UAG)=⇒(AG).(5)(AG)=⇒(LIM).(6)(ISS)=⇒(0-GAS).(7)(LISS)=⇒(0-AS).(8)(LISS)=⇒(LS).(9) Because(LIM)implies(0-GATT)and(0-GAS)is the same as(0-AS)plus(0-GATT),we have:(LIM)&(0-GAS)⇐⇒(LIM)&(0-AS).(10)It was shown in[15]that(ISS)⇐⇒(UAG)&(LS).(11)It turns out that(LS)is redundant,so(UAG)is in fact equivalent to(ISS):Proposition I.1:(UAG)⇒(LS).This observation generalizes a result which is well-known for systems with no controls(for which see e.g.[1,Theo-rem1.5.28]or[3,Theorem38.1]).It should be noted that the standing hypothesis that A is compact is essential for this implication;in the general case of noncompact sets A, the local stability property with respect to A is not redun-dant.From Proposition I.1and Equation(7),we know then that:(UAG)=⇒(0-GAS).(12) We also prove these results:Lemma I.2:(0-GAS)=⇒(LISS).Lemma I.3:(BND)&(LS)⇐⇒(GS).Lemma I.4:(LIM)&(GS)⇐⇒(AG)&(GS). Lemma I.5:(LIM)⇒(BND).The converse of Lemma I.5is of course false,as illustrated by the autonomous system˙x=0(with n=m=1),which even satisfies(GS)but does not satisfy(LIM).From Lem-mas I.3and I.5,we have that:(LIM)&(LS)⇐⇒(LIM)&(GS).(13)The most interesting technical result will be this: Proposition I.6:(LIM)&(LS)⇒(UAG).We now indicate how the proof of Theorem1follows from all these technical facts.•(A⇐⇒C):by Proposition I.1and Equation(11).•(C⇒E):by(5)and(12).•(E⇒F):by Lemma I.2.•(F⇒G):by Equation(9).•(G⇒H):by Equation(6).•(H⇒I):by Equation(13).•(I⇒J):by Lemma I.4.•(J⇒G):obvious.•(H⇒C):this is Proposition I.6.•(E⇒D):by Equation(6).•(B⇐⇒D):by Equation(10).•(D⇒H):by Lemma I.2and Equation(9).A very particular consequence of the main Theorem is worth focusing upon:A⇐⇒J,i.e.(ISS)is equivalent to having both the global stability property with respect to A and the asymptotic gain property with respect to A. Consider this property:∃γ∈N0st:∀ξ∈R n∀u(·)lim t→+∞|x(t,ξ,u)|A≤γlimt→+∞|u(t)|(14)(the limsup being understood in the“essential”sense,of holding up to a set of measure zero;note also that sinceγis continuous and nondecreasing,the right-hand term equalslim t→+∞γ(|u(t)|)).It is easy to show(see Lemma(II.1))thatthis is equivalent to(AG).The conjunction of(14)and(GS)is the“asymptotic L∞stability property”proposedby Teel and discussed in the survey paper[2](in that paper, A={0});it thus follows that asymptotic L∞stability is precisely the same as(ISS).In[18],Tsinias considered the following property(in thatpaper,A={0}):∃γ∈K st:∀ξ∈R n∀u(·)[|x(t,ξ,u)|A≥γ(|u(t)|)∀t≥0]⇒limt→∞|x(t,ξ,u)|A=0(15)which obviously implies(LIM).The author considered the conjunction of(15)and(LS)(more precisely,the author also assumed a local stability property that implies(LS), namely f(x,u)=Ax+Bu+o(x,u),with A Hurwitz); because of the equivalence A⇐⇒H,this conjunction is also equivalent to(ISS).The outline of the rest of the paper is as follows.In Section II wefirst prove Proposition I.1,Lemmas I.2, I.3,and I.4,and the equivalence between Property(14) and(AG),all of which are elementary.Section III con-tains the proof of the basic technical step needed to prove the main result,as well as a proof of Lemma I.5.After this,Section IV establishes a result showing that uniform global asymptotic stability of systems with disturbances(or equivalently,of an associated differential inclusion)follows from the non-uniform variant of the concept;this would appear to be a rather interesting result in itself,and in any case it is used in Section V to provide the proof of Propo-sition I.6.Finally,in Section VI we make some remarks characterizing so-called“practical”ISS stability in terms of ISS stability with respect to compact sets.II.Some Simple ImplicationsWe start with the proof of Proposition I.1.Proof:We will show the following property,which is equivalent to(LS):∀ε>0∃δ>0st:∀|ξ|A≤δ∀ u ∞≤δsupt≥0|x(t,ξ,u)|A≤ε.(16)Indeed,assume givenε>0.Let T=T(ε/2,1).Pick anyδ1>0so thatγ(δ1)<ε/2.Then:for all|ξ|A≤1and u ∞≤δ1supt≥T|x(t,ξ,u)|A≤ε/2+γ( u ∞)<ε.(17)By continuity(at u≡0and states in A)of solutions with respect to controls and initial conditions,and compactness and zero-invariance of A,there is also someδ2=δ2(ε,T)> 0so that|η|A≤δ2and u ∞≤δ2⇒supt∈[0,T]|x(t,η,u)|A≤ε.Together with(17),this gives the desired property with δ:=min{1,δ1,δ2}.We now prove Lemma I.2.Proof:Wefirst note that the0-global asymptotic sta-bility property with respect to A implies the existence of a smooth function V such thatα1(|ξ|A)≤V(ξ)≤α2(|ξ|A)∀ξ∈R n,for someα1,α2∈K∞,and∇V(ξ)f(ξ,0)≤−α3(|ξ|A)∀ξ∈R n,for someα3∈K∞(this is well-known;see for instance,[9] for one such a converse Lyapunov theorem).Following ex-actly the same steps as in the proof of Lemma3.2in[13], one can show that there exists some functionχ∈K∞such that for allχ(|v|)≤|ξ|A≤1,∇V(ξ)f(ξ,v)≤−α3(|ξ|A)/2.(18) (Here we note that in the proof of Lemma3.2in[13],the function g(s)=1for s∈[0,1].)Using exactly the same arguments used on page441 of[12],one can show that there exist a KL-functionβand a K∞-functionγso that if|x(t,ξ,u)|A≤1for all t∈[0,T) for some T>0,then it holds that|x(t,ξ,u)|A≤max{β(|ξ|A,t),γ( u ∞)}(19) for all t∈[0,T).Letρ=min{κ−1(1/2),γ−1(1/2)},where κ(r)=β(r,0)for r≥0.Note here thatρ≤κ−1(1/2)≤1/2.We now show that the(LISS)property holds with theseβ,γ,andρ.Fix anyξand u with|ξ|A≤ρandu ∞≤ρ.First note that|x(t,ξ,u)|<1for t small enough.Claim:|x(t,ξ,u)|A≤1for all t≥0.Assume the claim is false.Then witht1=inf{t:|x(t,ξ,u)|A≥1},it holds that0<t1<∞.Note then that|x(t,ξ,u)|A<1 for all t∈[0,t1).This then implies that|x(t,ξ,u)|A≤max{β(ρ,0),γ(ρ)}≤1/2∀t∈[0,t1). By continuity,|x(t1,ξ,u)|A<1,contradicting to the defi-nition of t1.This shows that t1=∞,i.e.,|x(t,ξ,u)|A≤1 for all t≥0.Thus the estimate in(19)holds for all t,as desired.Next we prove Lemma I.3:boundedness property with respect to A and local stability property with respect to A implies global stability property with respect to A(the converse is obvious).Proof:Assume that Equations(BND)and(LS)hold, for a given choice ofδ,σ1,σ2,α1,α2.Pick a constant c≥0and two class-K functionsβ1andβ2so that,for each i=1,2,σi(s)≤βi(s)+c for all s≥0.Pick two class-K functionsγ1andγ2so that,for each i=1,2,it holds that:γi(s)≥αi(s)∀0≤s≤δ,γi(s)≥2βi(s)∀s≥0,γi(s)≥2[βi(s)+2c]∀s≥δ.Consider anyξand u.Then Equation(GS)holds.In-deed,if both|ξ|A≤δand u ∞≤δthen this follows from Equation(LS).Assume now that|ξ|A>δ.Thus Equation(BND)implies that,for all t≥0,|x(t,ξ,u)|A≤σ1(|ξ|A)+σ2( u ∞)≤β1(|ξ|A)+c+β2( u ∞)+c≤β1(|ξ|A)+2c+(1/2)γ2( u ∞)≤(1/2)[γ1(|ξ|A)+γ2( u ∞)]≤max{γ1(|ξ|A),γ2( u ∞)}.The case u ∞>δis similar.Lemma I.4says that the limit property with respect to A plus the global stability property with respect to A imply the asymptotic gain property with respect to A;it is shown as follows.Proof:Letσ1,σ2,γ∈Nbe as in(LIM)and(GS). We claim that(AG)holds with:γ(s):=max{(σ1◦γ)(s),σ2(s)}.Pick anyξ,u,and anyε>0.By(LIM),there is some T≥0so that|x(T,ξ,u)|A≤γ( u ∞)+ε.Applying(GS) to the initial state x(T,ξ,u)and the control v(t):=u(t+T) we conclude thatlim t→+∞|x(t,ξ,u)|A≤supt≥T|x(t,ξ,u)|A≤max{σ1(γ( u ∞)+ε),σ2( u ∞)}and takingε→0provides the conclusion.Finally,we show:Lemma II.1:Property(14)is equivalent to(AG).Proof:Sinceγ(limt→+∞|u(t)|)≤γ( u ∞),Property(14)implies(AG),with the sameγ.Conversely,assume that(AG)holds;we next show that Property(14)holds withthe sameγ.Pick anyξ∈R n,control u,andε>0.Letr:=limt→+∞|u(t)|.Let h>0be such thatγ(r+h)−γ(r)<ε.Pick T>0so that|u(t)|≤r+h for almost all t≥T,andconsider the functions z(t):=x(t+T)and v(t):=u(t+T)defined on R≥0.Note that v is a control with v ∞≤r+hand that z(t)=x(t,ζ,v),whereζ=x(T,ξ,u).By thedefinition of the asymptotic gain property with respect toA,applied with initial stateζand control v,limt→+∞|x(t,ξ,u)|A=limt→+∞|z(t,ζ,v)|A≤γ( v ∞)≤γ(r+h)<γ(r)+ε.Lettingε→0gives Property(14).III.Uniform Reachability TimeLet(1)be a forward-complete system.For each subsetO of the input-value space U,each T≥0,and each subsetC⊆R n,we denoteR T O(C):={x(t,ξ,u)|0≤t≤T,u∈M O,ξ∈C}andR O(C):={x(t,ξ,u)|t≥0,u∈M O,ξ∈C}=T≥0R T O(C).In[9,Proposition5.1],it is shown that:Fact III.1:Let(1)be a forward-complete system.Foreach bounded subset O of the input-value space U,eachT≥0,and each bounded subset C⊆R n,R T O(C)isbounded.PGiven afixed system(1)which is forward-complete,apointξ∈R n,a subset S⊆R n,and a control u,one mayconsider the“first crossing time”τ(ξ,S,u):=inf{t≥0|x(t,ξ,u)∈S}with the convention thatτ(ξ,S,u)=+∞if x(t,ξ,u)∈Sfor all t≥0.The following result and its corollary are central.Theystate in essence that,for bounded controls,ifτ(ξ,S,u)isfinite for all u then this quantity is uniformly bounded overu,up to small perturbations ofξand S,and(the Corollary)uniformly on compact sets of initial states as well.(Observethat we are not making the assumption that f is convex oncontrol values and that the set of such values is compactand convex,which would make the result far simpler,bymeans of a routine weak- compactness argument.)Theresult will be mainly applied in the following special case:O is a closed ball in R m,W=R n,and for a given compactset A,C(in the Corollary)is a closed ball of the typeB (A ,2s ),p ∈C ,Ω=B (A ,2s ),and K =B (A ,(3/2)s ).But the general case is not harder to prove,and it is of independent interest.Lemma III.2:Let (1)be a forward-complete system.As-sume given:•an open subset Ωof the state-space R n,•a compact subset K ⊂Ω,•a bounded subset O of the input-value space U ,•a point p ∈R n,and •a neighborhood W of p ,so thatsup u ∈M Oτ(p,Ω,u )=+∞.(20)Then there is some point q ∈W and some v ∈M O suchthatτ(q,K,v )=+∞.(21)Proof:Let p 0=p be as in the hypotheses.Thus for each integer k ≥1we may pick some d k ∈M O so that x (t,p 0,d k )∈Ωfor all 0≤t ≤k .For each j ≥1,we let θj (t )=x (t,p 0,d j ),t ≥0.Consider first {θj (t )}j ≥1as a sequence of functions de-fined on [0,1].From Fact III.1we know that there exists some compact subset S 1of R n such that x (t,p 0,d j )∈S 1for all 0≤t ≤1,for all j ≥1.Let M =sup {|f (ξ,λ)|:ξ∈S 1,λ∈O}.Then d θj (t ) ≤M for all j and almostall 0≤t ≤1.Thus the sequence {θj (t )}j ≥1is uniformly bounded and equicontinuous on [0,1],so by the Arzela-Ascoli Theorem,we may pick a subsequence {σ1(j )}j ≥1of {j }j ≥1with the property that {θσ1(j )(t )}j ≥1converges to a continuous function κ1(t ),uniformly on [0,1].Now we consider {θσ1(j )(t )}j ≥1as a sequence of functions defined on [1,2].Using the same argument as above,one proves that there exists a subsequence {σ2(j )}j ≥1of {σ1(j )}j ≥1such that {θσ2(j )(t )}j ≥1converges uniformly to a func-tion κ2(t )for t ∈[1,2].Since {σ2(j )}is a subsequence of {σ1(j )},it follows that κ2(1)=κ1(1).Repeating the above procedure,one obtains inductively on k ≥1a subse-quence {σk +1(j )}j ≥1of {σk (j )}j ≥1such that the sequence {θσk +1(j )(t )}j ≥1converges uniformly to a continuous func-tion κk +1on [k,k +1].Clearly,κk (k )=κk +1(k )for all k ≥1.Let κbe the continuous function defined by κ(t )=κk (t )for t ∈[k −1,k )for each k ≥1.Then on each interval [k −1,k ],κ(t )is the uniform limit of {θσk (j )(t )}.Since the complement of Ωis closed and the θj ’s have images there,it is clear that κremains outside Ω,and hence outside K .If κwould be a trajectory of the system corresponding to some control v ,the result would be proved (with q =p 0).The difficulty lies,of course,in the fact that there is no reason for κto be a trajectory.However,κcan be well approximated by trajectories,and the rest of the proof consists of carrying out such an approximation.Some more notations are needed.For each control d with values in O ,we will denote by ∆d the control given by ∆d (t )=d (t +1)for each t in the domain of d (so,for instance,the domain of ∆d is [−1,+∞)if the domain of d was R ≥0).We will also consider iterates of the ∆operator,∆k d ,corresponding to a shift by k .Since K is compact and Ωis open,we may pick an r >0such thatB (K,r )⊆Ω.We pick an r 0smaller than r/2and so that the closed ball of radius r 0around p 0is included in the neighborhood W in which q must be found.Finally,let p k =κ(k )for each k ≥1.Observe that both p 0and p 1are in S 1by construction.Next,for each j ≥1,we wish to study the trajectory x (−t,p 1,∆d σ1(j ))for t ∈[0,1].This may be a priori un-defined for all such t .However,since S 1is compact,wemay pick another compact set S1containing B (S 1,r )in its interior,and we may also pick a function f:R n ×R m →R n which is equal to f for all (x,u )∈ S1×O and has compact support;now the system ˙x = f(x,u )is complete,meaning that solutions exist for all t ∈(−∞,∞).We use x (t,ξ,u )to denote solutions of this new system.Observe that foreach trajectory x (t,ξ,u )which remains in S1,x (t,ξ,u )is also defined and coincides with x (t,ξ,u ).In particu-lar, x (−t,θσ1(j )(1),∆d σ1(j ))=x (−t,θσ1(j )(1),∆d σ1(j )),for each j ,since these both equal x (1−t,p 0,d σ1(j )),for each t ∈[0,1].The set of states reached from S 1,using the modified system,in negative times t ∈[−1,0],is in-cluded in some compact set (because the modified system is complete,and again appealing to Fact III.1).Thus,by Gronwall’s estimate,there is some L ≥0so that,for all j ≥1and all t ∈[0,1],x (−t,p 1,∆d σ1(j ))−x (−t,θσ1(j )(1),∆d σ1(j ))≤L p 1−θσ1(j )(1) ,(no “∼”needed in the second solution,since it is also a solution of the original system).Since θσ1(j )(1)→p 1,it follows that there exists some j 1such that for all j ≥j 1,x (−t,p 1,∆d σ1(j ))−x (−t,θσ1(j )(1),∆d σ1(j )) <r 02(22)for all t ∈[0,1].Note that this means in particular thatx (−t,p 1,∆d σ1(j ))∈B (S 1,r/4)⊆ S1for all such t ,for all j ≥j 1,so “∼”can be dropped in Equation (22)for all j ≥j 1.Now let 0<r 1<r 0be such thatx (−t,p,∆d σ1(j 1))−x (−t,p 1,∆d σ1(j 1)) <r 02(23)for all t ∈[0,1],for all p ∈B (p 1,r 1).As this impliesin particular that x (−t,p,∆d σ1(j 1))∈B (S 1,r/2)⊆ S1,again tildes can be bining (22)and (23),it follows that for each p ∈B (p 1,r 1),x (−t,p,∆d σ1(j 1))is defined for all t ∈[0,1]andx (−t,p,∆d σ1(j 1))−x (−t,θσ1(j 1)(1),∆d σ1(j 1)) <r 0(24)for all t ∈[0,1].Let w 1(t )=d σ1(j 1)(t ).Then (24)implies that for each p ∈B (p 1,r 1)it holds that x (−1,p 1,∆w 1)∈B (p 0,r 0),and,since x (−t,θσ1(j 1),∆d σ1(j 1))∈Ωfor all t ∈[0,1],x (−t,p,∆w 1)∈B (K,r/2)∀t ∈[0,1].In what follows we will prove,by induction,that for each i ≥1,there exist 0<r i <r i −1and w i of the form。

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROLInt.J.Robust Nonlinear Control 2003;13:1035–1056(DOI:10.1002/rnc.753)Input-to-state stability with respect to inputsand their derivativesDavid Angeli 1,n ,y ,Eduardo D.Sontag 2,z and Yuan Wang 3,}1Dip.Sistemi e Informatica,Universit "adi Firenze,Via di Santa Marta 3,50139Firenze,Italy 2Department of Mathematics,Rutgers University,New Brunswick,NJ 08903,USA3Department of Mathematics,Florida Atlantic University,Boca Raton,FL 33431,USASUMMARYA new notion of input-to-state stability involving infinity norms of input derivatives up to a finite order k is introduced and characterized.An example shows that this notion of stability is indeed weaker than the usual iss .Applications to the study of global asymptotic stability of cascaded non-linear systems are discussed.Copyright #2003John Wiley &Sons,Ltd.KEY WORDS :input-to-state stability;Lyapunov methods;dissipative systems;cascaded systems1.INTRODUCTIONA central question in control theory is how to formulate,for general non-linear systems,notions of robustness and stability with respect to exogenous input disturbances.The linear case is by now very well understood,and,at least in a finite-dimensional set-up,most ‘reasonable’definitions of ‘input-to-state’or ‘input–output’stability (provided in this last case that additional reachability and observability assumptions are met)boil down to local asymptotic stability,viz.to the classical condition on the systems poles lying in the complex open left half-plane.However,for non-linear systems the range of possibilities is much broader,and the goal of coming up with an effective classification for many different behaviours that might be labeled as ‘stable’together with methods which would allow to establish relationships between such stability notions has attracted a substantial research effort within the past years.In this respect,input to state stability (iss )and integral iss ,as well as H 1theory,have proven to be powerful tools,used successfully in order to tackle problems both of robustness analysis and control synthesis,[1–5].Received 29October 2001Revised 6August 2002Published online 31March 2003Accepted 4September 2002Copyright #2003John Wiley &Sons,Ltd.y E-mail:angeli@dsi.unifi.it n Correspondence to:David Angeli,Dipartimento di Sistemi e Informatica,Universit "adi Firenze,Via di Santa Marta 3,50139Firenze,Italy.z E-mail:sontag@ }E-mail:ywang@Contract/grant sponsor:U.S.Air Force;contract/grant number:F-49620-01-0063.Contract/grant sponsor:NSF;contract/grant number:DMS-0072620In the iss -related literature,a ‘disturbance’is a locally essentially bounded measurable function.Such an extremely rich set of possible input perturbations is well suited to model Gaussian and random noises,as well as constant or periodic signals,slow parameters drifts,and so on.If,on one side,this makes the notion of iss extremely powerful,on the other it is known that iss might sometimes be too strong a requirement [6].In the output regulation literature [7],instead,the focus is on ‘deterministic’disturbances,i.e.signals that can be generated by a finite dimensional non-linear systems,(usually smooth),when the state is evolving in a neighbour-hood of a neutrally stable equilibrium position.This is an extremely interesting class of persistent disturbances for which,roughly speaking,the following is true:jj d jj 1small )jj ’djj 1and derivatives of arbitrary order are also small Under similar circumstances,for instance when cascading asymptotically stable systems,regarding the ‘forcing’system’s state as a disturbance typically yieldslim sup t !þ1j d ðt Þj ¼0)lim sup t !þ1j ’d ðt Þj ¼0Nevertheless,the classical definition of input-to-state stability completely disregards such additional information.Tracking of output references,see Reference [8],is another area where ‘derivative’knowledge is usually disregarded (the analysis is often performed only taking into account constant set-points),whereas such information could be exploited to get tighter estimates for the steady-state tracking error due to time-varying,smooth reference signals.Analogous situations arise when parameters variations are taken into account (i.e.in adaptive control)and we expect the system to have nice and stable behaviour for slow parameters drifts.The study of systems with slowly varying parameters has long been an interesting topic in the literature,see e.g.References [9,10].The analysis of such a system is usually carried out by first considering the systems corresponding to ‘frozen’parameters.If for all the frozen parameters,the corresponding frozen systems possess certain stability property uniformly,then it is reasonable to expect that the system with slowly varying parameters will possess the same property.See,for instance,Reference [10]for a result of this type.A more general question is how the magnitudes of the time derivatives of the time varying parameters affect the behaviour of the systems.The main contribution of this note is to show how,in the context of iss ,stability notions can be adjusted in order to take into account robustness with respect to disturbances and their time derivatives.The new notion of D k iss is defined through an iss -like estimate which involves the magnitudes of the inputs and their derivatives up to the k th order.We also propose several properties related to the D k iss notion.All these properties serve to formalize the idea of ‘stable’dependence upon the inputs and their time derivatives.They differ in the formulation of the decay estimates which make precise how the magnitudes of derivatives affect the system.We illustrate by means of several interesting examples how these properties differ from each other and from the well known iss property.One of our main objectives is to provide equivalent Lyapunov characterizations for these properties.Interestingly enough,one of our Lyapunov formulations already appeared in Reference [9],(see formula (5)in that reference).In this work,we provide a stability property that is equivalent to the existence of this type of Lyapunov functions.As a key step in establishing the Lyapunov formulations,we show how the D k iss property can be treated as a Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1036special case of the input–output-to-state stability property(for detailed discussions on this property,see Reference[11]).A second objective is to discuss some applications of the newly introduced notions to the analysis of cascaded non-linear systems.The well known result that cascading preserves the iss property is generalized to the D k iss property.The paper is organized as follows:Section2provides the basic definitions.Sections3–5 contain the Lyapunov characterizations of the D k iss property and some other related properties. Sections6and7are devoted to the study of cascaded systems.Sections8and9provide discussions on the relation between the newly introduced stability notions and the well established iss property.2.BASIC DEFINITIONSConsider non-linear systems of the following form:’xðtÞ¼fðxðtÞ;uðtÞÞð1Þwhere xðtÞ2R n and uðtÞ2R m for each t50:The function f:R nÂR m!R n is locally Lipschitz continuous.Thus,for any measurable,locally essentially bounded function uðtÞ:R!R m;and any initial condition x2R n;there exists a unique solution xðt;x;uÞof(1)satisfying the initialcondition xð0;x;uÞ¼x;defined on some maximal intervalðTÀx;u ;Tþx;uÞ:Recall that system(1)is input-to-state stable(iss for short)if there exist g2K1n and b2KL so that the following holds:j xðt;x;uÞj4bðj x j;tÞþgðjj u½0;tÞjj1Þð2Þfor all t50;all x2R n;and all input signals u;where for any interval I;u I denotes the restrictionof u to I;and where jj v jj denotes the usual L m1-norm(possibly infinite)of v:Usually one can thinkof u as an exogenous disturbance entering the system.Note that if(2)holds for any trajectory on any interval where the trajectory is defined,then the system is automatically forward complete. We denote by W k;1ðJÞ;for any integer k51and any interval J;the space of all functions u:J!R m for which theðkÀ1Þst derivative uðkÀ1Þexists and is locally Lipschitz.For k¼0;we define W0;1ðJÞas the set of locally essentially bounded u:J!R m:When J¼½0;þ1Þ;we omit J and write simply W k;1:(Since absolutely continuous functions have essentially bounded derivatives if and only if they are Lipschitz,the definition of W k;1ðJÞ;for positive k;amounts to asking that theðkÀ1Þst derivative uðkÀ1Þexists and is absolutely continuous,and hence,its derivative,that is,uðkÞ;is locally essentially bounded.Thus W k;1ðJÞis a standard Sobolev space, justifying our notation.)Definition2.1System(1)is said to be k th derivative input-to-state stableðD k iss)if there exist some KL-function b;and some K-functions g0;g1;...;g k such that,for every input u2W k;1;the*A function F:S!R is positive definite if FðxÞ>0;8x2S;x=0and Fð0Þ¼0:A function g:R50!R50is ofclass K if it is continuous,positive definite,and strictly increasing.It is of class K1if it is also unbounded.Finally, b:R50ÂR50!R50is of class KL if for eachfixed t50;bðÁ;tÞis of class K and for eachfixed s>0;bðs;tÞdecreases to0as t!1:An important fact concerning K1functions which will often be used in the following sections is the so-called‘weak triangular inequality’gðaþbÞ4gð2aÞþgð2bÞfor all a;b50:Copyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056INPUT-TO-STATE STABILITY1037following holds:j x ðt ;x ;u Þj 4b ðj x j ;t Þþg 0ðjj u jjÞþg 1ðjj ’ujjÞþÁÁÁþg k ðjj u ðk ÞjjÞð3Þfor all t 50:As with iss ,we remark that if estimate (3)was instead only required to hold on the maximal interval of definition of the solution x ðt ;x ;u Þ;then j x ðt ;x ;u Þj is uniformly bounded on any subinterval of the maximal interval.Hence,the solution must be globally defined if u 2W k ;1;and the same definition results.We say simply that the system is D iss when it is D 1iss and,of course,iss is the same as D k iss for k ¼0:It is also clear that a system is D k iss if and only if there exist some b 2KL and some g 2K such thatj x ðt ;x ;u Þj 4b ðj x j ;t Þþg ðjj u jj ½k Þð4Þfor all t 50;where jj u jj ½k ¼max 04i 4k fjj u ði Þjjg :Lemma 2.2System (1)is D k iss if and only if property (4)holds for all smooth input functions (with the same b ;g ).ProofOne implication is trivial.To prove the non-trivial implication,assume for some b 2KL and g 2K ;estimate (4)holds for all smooth input functions.By causality,one may replace jj u jj ½k by jj u ½0;t Þjj ½k in (4).Let u 2W k ;1:Fix T >0such that x ðt ;x ;u Þis defined on ½0;T :Note that u ðk Þis essentially bounded on ½0;T :It is a routine approximation fact (reviewed in Corollary A.2in the appendix)that there exists an equibounded sequence of C 1functions f u j g such that*u j !u pointwise on ½0;T ;and *lim sup j !1jjðu j Þ½0;T Þjj ½k 4jj u ½0;T Þjj ½k :Applying (4)to the trajectories with the input function u j ;and then taking the limits,we get j x ðt ;x ;u Þj 4b ðj x j ;0Þþg ðjj u jj ½k ½0;T þx ;u ÞÞð5ÞHence,T þx ;u ¼1;that is,x ðt ;x ;u Þis defined on ½0;1Þ:Thus T can be picked arbitrarily,and (5)holds for all t 50where jj u jj ½k ½0;T þx ;uÞbecomes by jj u jj ½k :&3.A LYAPUNOV CHARACTERIZATION OF D k issFix k 51:For system (1),consider the auxiliary system’x¼f ðx ;z 0Þ;’z 0¼z 1;...;’z k À1¼v ð6ÞLet#x ðt ;x ;Z ;v Þ:¼x ðt ;x ;Z ;v Þz ðt ;Z ;v Þ!Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1038INPUT-TO-STATE STABILITY1039 denote the trajectory of(6)with the initial state xð0Þ¼x;zð0Þ¼Z;(note that the z-component of the solution is independent of the choice of x:)Observe that,if property(4)is known to hold for all inputs in W k;1;then,for the trajectories ðxðt;x;z0;vÞ;zðt;x;z0;vÞÞof the auxiliary system,the following property holds:j xðt;x;Z;vÞj4bðj x j;tÞþ*g0ðjj z jj½0;t Þþ*g1ðjj v jjÞð7Þfor all measurable,locally essentially bounded inputs v:Given the fact that j zðtÞj4jj z jj½0;tÞis always true,we getj#xðt;x;Z;vÞj4bðjðx;ZÞj;tÞþ#g0ðjj z jj½0;t Þþ#g1ðjj v jjÞfor some#g0;#g12K:This shows that if(1)is D k iss;then(6)is input–output-to-state stable,i.e., ioss,with v as input and z¼ðz0;z1;...;z kÀ1Þas outputs(cf.Reference[11]).On the other hand,if the auxiliary system(6)is ioss,then there exist some b2KL and g0;g2K such thatj#xðt;x;Z;vÞj4bðj x jþj Z j;tÞþg0ðjj z jj½0;tÞÞþgðjj v jjÞfor all locally essentially bounded inputs v:Observe thatbðj x jþj Z j;tÞ4bð2j x j;tÞþbð2j Z j;0Þ4bð2j x j;tÞþbð2jj z jj½0;tÞ;0ÞIt follows thatj#xðt;x;Z;vÞj4%bðj x j;tÞþ%g0ðjj z jj½0;tÞÞþgðjj v jjÞholds for all locally essentially bounded v;where%bðs;tÞ¼bð2s;tÞ;and%g0ðsÞ¼bð2s;0Þþg0ðsÞ:In particular,j xðt;x;Z;vÞj4%bðj x j;tÞþ%g0ðjj z jj½0;tÞÞþgðjj v jjÞThis implies that for any u2W k;1;the trajectory of system(1)with initial state x satisfies the estimate:j xðt;x;uÞj4bðj x j;tÞþg1ðjj u jj½k Þwhere g1ðsÞ¼%g0ðsÞþgðsÞ:We have therefore proved the following result that underlies the proofs of Theorems1and2to be given later.Lemma3.1Let k51:System(1)is D k iss if and only the associated auxiliary system(6)is ioss with v as input and z¼ðz0;z1;...;z kÀ1Þas output.By the main result in Reference[11],System(6)is ioss if and only if it admits an ioss-Lyapunov function,that is,a smooth function V:R nÂR km!R50such thata;%a2K1;it holds that*for some%aðjðx;zÞjÞ4Vðx;zÞ4%aðjðx;zÞjÞ8ðx;zÞ%Copyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056*for some a ;r 2K 1;@V @x ðx ;z Þf ðx ;z Þþ@V @z 0ðx ;z Þz 1þÁÁÁþ@V @z k À1ðx ;z Þv 4Àa ðV ðx ;z ÞÞþr ðjðz ;v ÞjÞfor all x ;z and v :Interpreting z as the input and its derivatives for system (1),we get the following:Theorem 1Let k 51:System (1)is D k iss if and only if there exists a smooth function V :R n ÂR km !R 50such that*there exist some %a ;%a 2K 1such that for all ðx ;m ½k À1 Þ2R n ÂR km ;it holds that %a ðjðx ;m ½k À1 ÞjÞ4V ðx ;m ½k À1 Þ4%a ðjðx ;m ½k À1 ÞjÞð8Þ*there exist some a 2K 1;r 2K 1such that for all x 2R n and all m ½k 2R m ðk þ1Þwith m ½k ¼ðm 0;m 1;...;m k Þ;it holds that@V @x ðx ;m ½k À1 Þf ðx ;m 0Þþ@V @m 0ðx ;m ½k À1 Þm 1þ@V @m 1ðx ;m ½k À1 Þm 2þÁÁÁþ@V @m k À1ðx ;m ½k À1 Þm k 4Àa ðV ðx ;m ½k À1 ÞÞþr ðj m ½k jÞð9ÞRemark 3.2Note that inequality (8)implies that%a ðj x jÞ4V ðx ;m ½k À1 Þ4%a ðjðx ;m ½k À1 ÞjÞð10ÞSuppose a system (1)admits a Lyapunov function V satisfying (9)and (10).Then it can be seen that,along any trajectory x ðt Þwith u 2W k ;1as the input,(9)yieldsd d tV ðx ðt Þ;u ðt Þ;...;u ðk À1Þðt ÞÞ4Àa ðV ðx ðt Þ;u ðt Þ;...;u ðk À1Þðt ÞÞÞþr ðjj u jj ½k Þfor almost all t 50:From this it follows that for some b 2KL and g 2K ;it holds thatV ðx ðt Þ;u ðt Þ;...;u ðk À1Þðt ÞÞ4b ðj V 0j ;t Þþg ðjj u jj ½k Þ8t 50where V 0¼V ðx ð0Þ;u ð0Þ;...;u ðk À1Þð0ÞÞ:Combining this with (10),one sees that system (1)is D k iss :Hence,an equivalent Lyapunov characterization of D k iss is the existence of a smooth function Vsatisfying (9)and (10)for some %a ;%a ;a 2K 1and some r 2K :4.ASYMPTOTIC GAINSClearly,if a system is D k iss ;then it is forward complete (for u 2W k ;1)and for some g 0;g 1;...;g k 2K it holds thatlim sup t !1j x ðt ;x ;u Þj 4g 0ðjj u jjÞþg 1ðjj ’ujjÞþÁÁÁþg k ðjj u ðk ÞjjÞ:ð11ÞWe say that a forward complete system satisfies the asymptotic gain (AG)property in u ;...;u ðk Þif,for some g 0;...;g k 2K ;(11)holds for all x 2R n and all u 2W k ;1:Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG 1040INPUT-TO-STATE STABILITY1041 By Lemma3.1,the system(1)is D k iss if and only if the associated auxiliary system(6)is ioss with v as input and z¼ðz0;z1;...;z kÀ1Þas output.Applying the main result in Reference[12] about asymptotic gains for the ioss property to the auxiliary system(6),one can prove the following:Theorem2For a forward complete system as in(1),the following are equivalent:1.it is D k iss;2.it satisfies the AG property in u;...;uðkÞ;and the corresponding zero-input system’x¼fðx;0Þis(neutrally)stable.5.RELATED NOTIONSIn this section,we consider two properties related to D iss:We focus specifically on D iss(rather than D k iss)as it seems to be the most relevant in applications.As a matter of fact,the authors were not able tofind any example of a D2iss system not being D iss and it is therefore an open question whether or not D k iss(k51)is equivalent to D iss:We say that system(1)is iss in’u if,for some b2KL and some g2K;the following estimate holds for all trajectories with inputs in W1;1:j xðt;x;uÞj4bðj x j;tÞþgðjj’u jjÞ8t50ð12ÞWe say that system(1)is iss in constant inputs if,for some b2KL and g2K;the following estimate holds for all trajectories corresponding to constant inputs u:j xðt;x;uÞj4bðj x j;tÞþgðjj u jjÞ8t50:ð13ÞIt is obvious that if a system is iss in’u;then it is gas uniformly in all constant inputs,that is, for some b2KL;the following holds for all trajectories with constant inputs:j xðt;x;uÞj4bðj x j;tÞ8t50Also note thatðiss in’uÞ)ðD issÞ:The converse is in general false.This can be seen through the following argument.Suppose D iss implies iss in’u:Then we would haveðissÞ)ðD issÞ)ðiss in’uÞand hence,ðissÞ)ðiss in’uÞ:But this is false,as one can see that the linear system’x¼Àxþu is iss but not iss in’u:It is also clear that,for any k50;ðD k issÞ)ðiss in constant uÞAgain,the converse implication is in general false as shown by examples in ing similar arguments as in the proof of Lemma3.1,we get the following:*System(1)is iss in’u if and only if the auxiliary system’x¼fðx;zÞ;’z¼vð14ÞCopyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056is state-independent-input-to-output stable,i.e.siios (see Reference [13])with v as inputs and x as outputs;and *System (1)is iss in constant inputs if and only if the auxiliary system’x ¼f ðx ;z Þ;’z ¼0ð15Þis output-to-state-stable,i.e.oss (see Reference [11])with z as outputs.Applying Theorem 1.2of Reference [14]in conjunction with Remark 4.1in Reference [14]to the siios property for system (14),we get the following:Proposition 5.1System (1)is iss in ’uif and only if there exists a smooth Lyapunov function V :R n ÂR m !R 50satisfying the following:*for some %a ;%a 2K 1;%a ðj x jÞ4V ðx ;m 0Þ4%a ðj x jÞ8x 2R n ;8m 02R m ð16Þ*for some w 2K 1and some continuous,positive definite function a ;V ðx ;m 0Þ5w ðj m 1jÞ)@V @x ðx ;m 0Þf ðx ;m 0Þþ@V @m 0ðx ;m 0Þm 14Àa ðV ðx ;m 0ÞÞð17Þfor all x 2R n and all m 0;m 12R m :Observe that if one restricts the set where the input functions take values to be a bounded set U (as in the case of Reference [9]),then the Lyapunov characterization in Proposition 5.1isequivalent to the existence of a smooth Lyapunov function V satisfying (16)for some %a ;%a 2K 1such that for some a 2K 1and s 2K ;@V @x ðx ;m 0Þf ðx ;m 0Þþ@V @m 0ðx ;m 0Þm 14Àa ðV ðx ;m 0ÞÞþs ðj m 1jÞfor all x 2R n ;all m 02U ;and all m 12R m :Such a Lyapunov estimate was used in [9]to analyze the asymptotic behaviour of systems with slowly varying parameters.Applying Theorem 2of Reference [11]to the oss property for system (15),we have the following:Proposition 5.2System (1)is iss with respect to constant inputs if and only if it admits a smooth Lyapunov function V :R n ÂR m !R 50such that*for some %a ;%a 2K 1;%a ðjðx ;m ÞjÞ4V ðx ;m Þ4%a ðj V ðx ;m ÞjÞ8x 2R n ;8m 2R m ð18Þ*for some a 2K 1;s 2K ;@V @xðx ;m Þf ðx ;m Þ4Àa ðV ðx ;m ÞÞþs ðj m jÞð19Þfor all x 2R n ;m 2R m :Copyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1042Remark5.3It may also be interesting to consider the D iss property with different indexes on different components of the inputs.For instance,for a system’x¼fðx;u;vÞð20Þwithðu;vÞas inputs,one may consider the property that for some b2KL;g u2K and g v2K; it holds thatj xðt;x;u;vÞj4bðj x j;tÞþg uðjj u jjÞþg vðjj v jjÞþg vðjj’v jjÞð21ÞOne can also get a Lyapunov characterization for such a property by using the same argument as in the proof of Theorem1with the ioss results.For instance,a system as in(20)satisfies property(21)if and only if there exists a smooth Lyapunov function V such that*for some%a;%a2K1;%aðjðx;n0ÞjÞ4Vðx;n0Þ4%aðjðx;n0ÞjÞ*for some a2K1;some r u;r v2K;it holds that@V @x ðx;n0Þfðx;m0;n0Þþ@V@n0ðx;n0Þn14ÀaðVðx;n0ÞÞþr uðj m0jÞþr vðj n0jÞþr vðj n1jÞfor all x;m0;n0and n1:6.APPLICATION OF D iss TO THE ANALYSIS OF CASCADE SYSTEMSAn interesting feature of iss,which makes it particularly useful in feedback design,is that the property is preserved under cascades,(see Reference[15]).Unfortunately,this is not the case for the weaker notion of integral iss,as remarked in Reference[1](but,see Reference[16]for related work).Interestingly,however,although D iss is also a weaker property than iss,it is preserved under cascades,as shown in this section.For a system’x¼fðx;v;uÞwithðv;uÞas inputs,we say that the system is D k iss in v and D l iss in u if there exist b2KL and g2K such that the following holds along any trajectory xðt;x;v;uÞwith initial state x;any input ðv;uÞfor which v2W k;1and u2W l;1:j xðt;x;v;uÞj4bðj x j;tÞþgðjj v jj½k Þþgðjj u jj½l Þ8t50Lemma6.1Consider a cascade system’x¼fðx;z;uÞ’z¼gðz;uÞð22Þwhere xðÁÞand zðÁÞevolve on R n1and R n2;respectively,the input u takes values in R m;and where f is locally Lipschitz and g is smooth.Let k50:Suppose that the z-subsystem is D k iss with u as input,and that the x-subsystem D kþ1iss in z and D k iss in u:Then the cascade system(22)is D k iss:Copyright#2003John Wiley&Sons,Ltd.Int.J.Robust Nonlinear Control2003;13:1035–1056INPUT-TO-STATE STABILITY1043ProofBy assumption,there exist b z 2KL and g z 2K such that,along any trajectory z ðt Þof the z -subsystem with input u ;it holds thatj z ðt Þj 4b z ðj z ð0Þj ;t Þþg z ðjj u jj ½k Þ8t 50ð23Þand there exist some b x 2KL and g x 2K such that,for any trajectory x ðt ;v ;u Þof the system ’x¼f ðx ;v ;u Þ;j x ðt ;x ;v ;u Þj 4b x ðj x ð0Þj ;t Þþg x ðjj v jj ½k þ1 Þþg x ðjj u jj ½k Þ8t 50ð24ÞTo prove Lemma 6.1,we need to find a suitable estimate for the x -component of solutions of(22).For this purpose,we define by induction for 14i 4k þ1:g i ða ;b 0;b 1;...;b i À1Þ¼@g i À1@a g ða ;b 0ÞþX i À2j ¼0@g i À1@b j b j þ1where g 1ða ;b 0Þ¼g ða ;b 0Þ:It can be seen that g i ð0;0;...;0Þ¼0for all 04i 4k ;hence,there exists some s i 2K such thatg i ða ;b 0;...;b i À1Þ4s i ðj a jÞþs i ðj b ½i À1 jÞAgain,by induction,one can show that,along any trajectory z ðt Þof the z -subsystem of (22)with an input u 2W k ;1;it holds thatd i d tz ðt Þ¼g i ðZ ðt Þ;d ðt Þ;’d ðt Þ;...;d ði À1Þðt ÞÞfor all 14i 4k þ1:It then follows thatjj z ½k þ1 jj 4s ðjj z jjÞþs ðjj u ½k jjÞð25Þfor some s 2K :It then follows from (24)and (25)that,for some r 2K ;it holds that along any trajectory ðx ðt Þ;z ðt ÞÞof (22),j x ðt Þj 4b x ðj x ð0Þj ;t Þþr ðjj z jjÞþr ðjj u jj ½k Þ8t 50ð26ÞApplying a standard argument to (26)and (23)as in the proof of the result that a cascade of iss systems is again iss ,one can show that system (22)is D k iss :To be more precise,(26)implies that j x ðt Þj 4b x j x ðt =2Þj ;t 2þr ðjj z jj ½t =2;t ÞÞþr ðjj u jj ½k ½t =2;t ÞÞ8t 50ð27Þalong any trajectory of (22).Fix an input u and pick any trajectory ðx ðt Þ;z ðt ÞÞof (22)with the input u :Let x 1¼x ðt =2Þ:We also have j x 1j 4b x j x j ;t 2þr ðjj z jj ½0;t =2ÞÞþr ðjj u jj ½k ½0;t =2ÞÞ8t 50Hence,there exist some *b x ;*b z and some *r 2K (which depend only on b x ;r )such that b x j x 1j ;t 24*b x ðj x j ;t Þþ*b z ðjj z jj ½0;t =2Þ;t Þþ*r ðjj u jj ½k ½0;t =2ÞÞð28Þfor all t 50:By (23),j z ðt Þj 4b z ðj z ð0Þj ;0Þþg z ðjj u jj ½k Þfor all t50;hence,*b z ðjj z jj ½0;t =2Þ;t Þ4%b z ðj z ð0Þj ;t Þþ%g z ðjj u jj ½k Þ8t 50ð29ÞCopyright #2003John Wiley &Sons,Ltd.Int.J.Robust Nonlinear Control 2003;13:1035–1056D.ANGELI,E.D.SONTAG AND Y.WANG1044。

DATA SHEETSKY85747-11: 5 GHz High-Power WLAN Front-End ModuleApplications∙ 802.11ax networking systems∙ WLAN-enabled wireless video streaming systemsFeatures∙ Integrated high-performance 5 GHz PA, LNA with bypass, and T/R switch∙ Fully matched input and output∙ Integrated logarithmic power detector and directional coupler ∙ Transmit gain: 34.5 dB ∙ Receive gain: 16 dB∙ Supports 802.11ax output power: +18.5 dBm, -43 dB DEVM, MCS11∙ Integrated, temperature- compensated log detector ∙ Highly sensitive, jammer-tolerant LNA ∙ Small LGA (24-pin, 3 x 5 mm) package (MSL4, 260 ︒C per JEDEC J-STD-020)S kywork s G reen TM product s are compliant with all applicable legi s lation and are halogen-free.For additional information, refer to Skyworks Definition of Green TM , document number SQ04–0074.Figure 2. SKY85747-11 Pinout(Top View)Figure 1. SKY85747-11 Block DiagramDescriptionThe SKY85747-11 is a highly integrated, 5 GHz front-end module (FEM) incorporating a transmit/receive (T/R) switch, a high-gain low-noise amplifier (LNA) with bypass, and a power amplifier (PA) intended for high-power 802.11ax applications and systems. The LNA and PA disable functions ensure low leakage current in the off mode. An integrated logarithmic power detector is included to provide closed-loop power control over 20 dB of dynamic range.The device is provided in a compact, 24-pin 3 x 5 mm Land Grid Array (LGA) package, which may reduce the front-end board space by more than 50 percent. A functional block diagram is shown in Figure 1. The pin configuration and package are shown in Figure 2. Signal pin assignments and functional pin descriptions are provided in Table 1.Table 1. SKY85747-11 Signal DescriptionsPin Name Description Pin Name Description1 GND Ground 13 LNA_EN RX control pin2 TX_IN Transmit Input 14 RX_OUT Switch RX output3GND Ground 15GND Ground4 N/U Not used (recommend GND) 16 LNA_IN LNA input5 N/U Not used (recommend GND) 17 LNA_OUT LNA output6 DET Detectoroutput 18 GND Ground7 GND Ground 19 VDD LNA supply voltage8 CPLR DPD coupler output 20 GND Ground9GND Ground 21GND Ground10 PA_EN TX control pin 22 VCC3 PA third stage supply voltage11 GND Ground 23 VCC2 PA second stage supply voltage12 ANT Antenna 24 VCC1 PA first stage supply voltageTechnical DescriptionThe SKY85747-11 comprises a high-power 5 GHz PA, a 5 GHz LNA, and a low-loss broadband switch to provide the T/R switching function. The device is fully matched, and requires few external components for optimal performance, which makes it ideal for small portable or high stream-count applications. The FEM provides over +32 dB of transmit gain over the frequency band. The LNA supports active and bypass modes, which can operate in the presence of jammers by offering +10 dBm input third order intercept (IIP3). The power amplifier, low-noise amplifier, and T/R switch can be controlled as shown in Table 5. Electrical and Mechanical SpecificationsThe absolute maximum ratings of the SKY85747-11 are provided in Table 2. The recommended operating conditions are specified in Table 3 and electrical specifications are provided in Table 4. The state of the SKY85747-11 is determined by the logic provided in Table 5.Table 2. SKY85747-11 Absolute Maximum Ratings1Parameter Symbol Minimum Maximum Units Supply voltage VCC1, VCC2, VCC3, and VDD -0.3 +6.0 VDC input on control pins (LNA_EN and PA_EN) V IN -0.3+3.6V Tx input power (50 Ω load) TX IN+10dBm Tx supply current TX_I CC800mA Storage temperature T ST -40+150︒C Junction temperature T J160︒CElectrostatic discharge:Human Body Model (HBM), Class 1C ESD1000 V1Exposure to maximum rating conditions for extended periods may reduce device reliability. There is no damage to device with only one parameter set at the limit and all other parameters set at or below their nominal value. Exceeding any of the limits listed here may result in permanent damage to the device.ESD HANDLING: Although this device is designed to be as robust as possible, electrostatic discharge (ESD) can damage this device.This device must be protected at all times from ESD when handling or transporting. Static charges may easily producepotentials of several kilovolts on the human body or equipment, which can discharge without detection.Industry-standard ESD handling precautions should be used at all times.Table 3. SKY85747-11 Recommended Operating ConditionsParameter Symbol Min Typ Max Units Supply voltage VCC1, VCC2, VCC3, and VDD 4.2 5.0 5.5 V Control logic:H igh Low V IHV IL1.63.60.4VVPA enable current I ENABLE1020μALNA bias current I DD2535mA LNA_EN enable current 10 μAOperating temperature T OP -40 +85︒C(V CC1 = V CC2 = V CC3 = V DD = 5.0 V, T OP = 25 °C, Unless Otherwise Noted)Parameter Symbol Test Condition Min Typ Max Units Frequency range f Main frequency band 5.15 5.925 GHz Transmit ModeGain GG3.9At 3.9 GHz32 34.5-1836.5-13dBdBGain flatness Over any 80 MHz bandwidth 1 dBOutput power P OUT 11ax, MCS10/11, HE20-HE80, -47 dB DEVM11ax, MCS10/11, HE20-HE80, -43 dB DEVM11ac, MCS8/9, VHT20-VHT80, -35 dB DEVM11n, MCS7, HT20-HT40, -30 dB DEVMMCS0, HT20, 2dB mask margin+12+16+22+23+26+17+18.5+23+24.5+27dBmdBmdBmdBmdBmBand edge power OOB 5180 MHz HT20, -47 dBm/MHz +21 dBmCurrent consumption Modulatedsignal:@ quiescent@ +21 dBm@ +25 dBm@ +27 dBmLeakage, EN off20028037045013104105001.3mAmAmAmAmA2nd harmonics 2fo +27 dBm MCS0 -50 -45 dBm/MHz 3rd harmonics 3fo +27 dBm MCS0 -50 -45 dBm/MHzIsolation From ANT to RX in TX modeFrom TX to RX in TX mode-45-15dBdBInput return loss |S11| 10 dB Output return loss |S22| 6 dBPower detector output V DET No RF@ +5 dBm@ +21 dBm@ +28 dBm0.250.630.790.20.330.720.90.380.810.99VVVVPower detector slope Slope +5 dBm to +28 dBm 20 24 mV/dBPower detector error ERR DET +10 dBm < P OUT < +28 dBmΔP OUT vs ideal VDET, 5.15 GHz to 5.85 GHz:50 Ω2:1 VSWR1.52dBpk-pkdBpk-pkPower detector output impedance Z OUT_DET RF output = -30 dBm 200 Ω Coupling factor CPLG -19 -17 -15.5 dB Coupler directivity Dir 17.5 dBStability S TAB+27 dBm MCS0, 0.1 GHz to 20 GHz, loadVSWR = 6:1All non-harmonic related outputs < -45 dBm/MHzRuggedness Ru TX_IN = +10 dBm, 10:1 mismatch, all phases No permanent damage(V CC1 = V CC2 = V CC3 = V DD = 5.0 V, T OP = 25 °C, Unless Otherwise Noted)ParameterSymbol Test Condition Min Typ Max UnitsReceive ModeGain GLNA activeLNA bypass 14.5 16 -7 dBdB 1 dB input compression point IP1dB LNA active LNA bypass -5 0 +19 dBm dBm Gain step19 21 24 dB Gain flatness Over any 80 MHz bandwidth -0.25 +0.25 dB Noise figure NF End to end 2.02.4dBInput return loss |S 11|LNA active LNA bypass99 dBdB Output return loss|S 22| 6 dB Third order input intercept pointIIP3 LNA active +7+11dBmSwitching timet SWState 2 ↔ State 3 State 2 ↔ State 1200 500 220 nsns1 Performance is guaranteed only under the conditions listed in this table.Table 5. SKY85747-11 LogicModeState PA_EN (Pin 10)LNA_EN (Pin 13)TX to ANT1 1 0 ANT to RX port (LNA mode)2 0 1 Not supported- 1 1 ANT to RX port (Bypass mode)3Evaluation Board DescriptionThe SKY85747-11 Evaluation Board is used to test the performance of the SKY85747-11 FEM. A suggested application schematic diagram is shown in Figure 3. A photograph of the Evaluation Board is shown in Figure 4. Table 6 provides the Bill of Materials (BOM) list for the Evaluation Board components. Evaluation Board Setup Procedure1.Connect power supply ground to the J5 header, pin 1.2.Apply 5 V to the J5 header, pin 2 and to the J6 header, pins 15, 17, 19, and 21 using jumpers (pin 15 to pin 16; pin 17 to pin 18; pin 19 to pin 20; and pin 21 to pin 22, respectively).3.Select a path according to the information in Table 5(L = 0 V, H = 3 V), either by placing jumpers at theJ6 header, pins 9 and 13, or applying signals from a controller.4.Detector output can be measured on the J6 header, pin 4. Circuit Design ConsiderationsThe following design considerations are general in nature and must be followed regardless of final use or configuration:∙Paths to ground should be made as short as possible.∙The ground pad of the SKY85747-11 has special electrical and thermal grounding requirements. This pad is the main thermal conduit for heat dissipation. Because the circuit board acts as the heat sink, it must shunt as much heat as possible from the device.Therefore, design the connection to the ground pad to dissipate the maximum wattage produced by the circuit board. Multiple vias to the grounding layer are required.∙TX_IN is DC shorted to GND. There is no DC leaking from the chip, but if there is DC on the line interfacing with the TX_IN pin, a 10 pF blocking capacitor is recommended.∙ANT, RX_OUT, and LNA_IN are DC blocked and do not require blocking capacitors.∙LNA_OUT is DC blocked but if there is > 1.5 V DC on the line connected to the LNA_OUT pin, a 10 pF blocking capacitor is recommended.NOTE: A poor connection between the ground pad and ground increases junction temperature (T J), which reduces thelife of the device.MAMA9J1S203929-003 Note: Some DNI and 0Ω components are not shown in this schematic.Figure 3. SKY85747-11 Application SchematicFigure 4. SKY85747-11 Evaluation BoardTable 6. SKY85747-11 Evaluation Board Bill of MaterialsComponent Value Size Vendor Part Number Description C1 10uF0805MurataGRM21BR71A106KE51LCeramic C2 1 nF 0402 Murata GRM1555C1H102JA01 Multilayer ceramicC3, C6, C9 100 nF 0402 Murata GRM155R71C104KA88D Monolithic ceramicC5, C7, C8 10 pF 0402 Murata GRM1555C1H100JZ01 Multilayer ceramicC10, C11 1 pF 0402 Murata GRM1555C1H1R0CZ01 Multilayer ceramicC16 10 pF 0402 Murata GJM1555C1H100GB01 RF, high Q, low lossR1, R2 0 Ω 0402Panasonic ERJ2GE0R00 Thick film chip resistorU1 SKY85747-11 MCM3x5-24 Skyworks Solutions Inc. SKY85747-11 5 GHz 11ax 5V FEM in 3x5 package with DPD couplerPackage DimensionsThe PCB layout footprint for the SKY85747-11 is shown in Figure 5. Typical part markings are shown in Figure 6. Package dimensions for the are shown in Figure 7, and tape and reel dimensions are provided in Figure 8. Package and Handling InformationSince the device package is sensitive to moisture absorption, it is baked and vacuum packed before shipping. Instructions on the shipping container label regarding exposure to moisture after the container seal is broken must be followed. Otherwise, problems related to moisture absorption may occur when the part is subjected to high temperature during solder assembly.The SKY85747-11 is rated to Moisture Sensitivity Level 4 (MSL4) at 260 C. It can be used for lead or lead-free soldering. For additional information, refer to the Skyworks Application Note, Solder Reflow Information, document number 200164.Care must be taken when attaching this product, whether it is done manually or in a production solder reflow environment. Production quantities of this product are shipped in a standard tape and reel format.Figure 5. SKY85747-11 PCB Layout Footprint(Top View)SkyworksSolutions,Inc.•Phone[781]376-3000•Fax[781]376-3100•*********************•203929N • Skyworks Proprietary Information • Products and Product Information are Subject to Change Without Notice • December 19, 201911Figure 6. Typical Part Markings(Top View)Figure 7. SKY85747-11 Package DimensionsSkyworksSolutions,Inc.•Phone[781]376-3000•Fax[781]376-3100•*********************•12December 19, 2019 • Skyworks Proprietary Information • Products and Product Information are Subject to Change Without Notice • 203929NFigure 8. SKY85747-11 Tape and Reel DimensionsOrdering InformationPart Number Product Description Evaluation Board Part Number SKY85747-11 5 GHz High-Power WLAN Front-End Module SKY85747-11EK1Copyright © 2016-2019 Skyworks Solutions, Inc. All Rights Reserved.Information in this document is provided in connection with Skyworks Solutions, Inc. (“Skyworks”) products or services. These materials, including the information contained herein, are provided by Skyworks as a service to its customers and may be used for informational purposes only by the customer. Skyworks assumes no responsibility for errors or omissions in these materials or the information contained herein. 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专业英语复习Lesson3Microprocessors (1)Lesson4Operational Amplifiers (2)Lesson8Clock Sources (3)Lesson12Personal Computer Systems (4)Lesson13Overview of Modern Digital Design (5)Lesson16Basic Concepts of DSP (6)Lesson19High Fidelity Audio (8)Lesson22Digital Image Fundamentals (9)Lesson25Choosing the right core (10)Lesson26Design Languages for Embedded Systems (11)Lesson27Choosing a Real-Time Operating System (12)Lesson28Signal Sources (13)Lesson3Microprocessors1.micron是“微米(百万分之一米)”2.data width是指算术逻辑单元ALU的字长3.MIPS Million Instructions Per Second每秒百万条指令4.Reset复位5.tri-state buffer三态缓冲器A tri-state buffer is a device that allows you to control when an output signal makes it to the bus.When the tri-state buffer's control bit is active,the input of the device makes it to the output.When it's not active,the output of the device is Z,which is high-impedance or,equivalently,nothing.There is no electrical signal is allowed to pass to the output.6.PipeliningA technique where the microprocessor fetches the next instruction before completing execution of the previous instruction,in order to increase processing speed.）流水线是一种在前一条指令全部执行完之前就开始取下一条指令，以提高处理速度的技术。

2004-21-0073 Concept and Functionality of the Active Front Steering SystemWilly Klier, Gerd Reimann and Wolfgang ReineltZF Lenksysteme GmbH, Schwäbisch Gmünd, Germany Copyright © 2003 SAE InternationalABSTRACTActive Front Steering (AFS) provides an electronically controlled superposition of an angle to the steering wheel angle. This additional degree of freedom enables a continuous and driving-situation dependent adaptation of the steering characteristics. Features like steering comfort, effort and steering dynamics are optimized and stabilizing steering interventions can be performed. After the successful introduction of AFS (or active steering) together with the new BMW 5-series into the international market, ZF Lenksysteme focuses on aspects like system modularization and integration. For that reason the system bounds, its functionality, and the required system interface are defined to provide a compatibility to several overall chassis control concepts. This paper focuses on a modular system concept and its respective advantages and requirements.1. INTRODUCTIONThis steering system developed by ZF Lenksysteme and BMW AG enables driver dependent as well as automatic steering interventions without loss of the mechanical connection between steering wheel and road wheels [1,2,3] (see Figure 1).This fact together with current definitions for steering systems imply that AFS is not a steer by wire system. The AFS system provides (compare [3,4,5,6]):•an improved steering comfort (reduced steering effort),•an enhanced dynamic behavior of the steering system (quick response to driver’s input) and •vehicle stabilization (active safety).After a short description of the steering system and respective components in Section 2, the modular concept, its functionality and the required system interface will be illustrated in Section 3. Some conclusions and an outlook will be presented in Section 4.2. COMPONENTS AND FUNCTIONALITYThe electrical and mechanical components as well as the functionality of the AFS system will be briefly described in this section. Figure 2 shows the following AFS components and subsystems:•Rack and pinion power steering system including (see Figure 2) the main gear (1), a Servotronic valve(2), a steering pump (9), an oil reservoir with filter(10) and the respective hoses (11),•AFS actuator including the synchronous motor (3) with its respective electrical connections, the superposition gear system (4) and theelectromagnetic locking unit (7),Figure 1: Principle of the angle superposition Figure 2 : Schematic representation of the AFS-system components•AFS electronic control system with the AFS ECU (5),the pinion angle sensor (8), the motor angle sensor (6), the respective electrical connections of the ECU and the required software modules.COMPONENTSThe electric motor (see Figure 3) generates the required electrical torque for the desired motion of the AFS actuator. This synchronous motor has a wound stator, a permanent magnet rotor assembly and a sensor to determine the rotor position. The motor torque is controlled by a field oriented control. This control strategy transform the stator currents into the torque-and rotor-flux-producing components. These current components can be controlled separately and do not depend on the rotor angle. The motor angle sensor is based on a magneto-resistive principle and includes a signal amplification and a temperature compensation.This sensor signal is used for control and monitoring purposes.In analogy to the motor angle sensor, the pinion angle sensor is also based on a magneto-resistive principle and includes a signal amplification and a temperature compensation. This sensor also includes a CAN-interface which enables other control systems like ESP to directly use the raw signal. The pinion angle is used as an input to the steering assistance functions and for monitoring purposes.The metal stud of the electromagnetic locking unit (ELU)is pressed towards the worm-locking gear by a spring.This mechanism is unlocked by a specific current supplied by the ECU. The ELU locks the worm (Figure 3) if the system is shut down and in case of a safety relevant malfunction (compare [7,8,9]). In this case the driver is able to further steer with a constant steering ratio (i.e. the mechanical ratio).The electronic control unit developed for the AFS system establishes the connection between the electrical system of the vehicle, the vehicle CAN – bus, the AFSsensors and the electric motor.Figure 3 : Electric Motor and Electromagnetic Locking UnitThe core components of the ECU are two microprocessors. They perform the computations required for control, monitoring and safety purposes. Via the integrated power output stages, the electric motor,the ELU, the ECO–pump and the Servotronic subsystem are controlled. The microprocessors also perform redundant computations and monitoring.The basis of the AFS system is the well-tried and reliable rack and pinion power steering system of ZF Lenksysteme.The core subsystem of AFS is the mechatronic actuator which is placed between the steering valve and the steering gear (see Figure 4). The actuator includes the planetary gear set with two mechanical inputs and a single mechanical output. The servo-valve connects the input shaft of the planetary gear with the steering column and the steering wheel. The second input shaft is driven by the electric motor and is connected to the planetary gear by the worm and worm wheel. The pinion angle sensor is mounted on the output shaft, which is the mechanical input for the steering gear. The relation between the input of the steering gear (pinion) and the road wheel angle is a nonlinear kinematic relation.FUNCTIONALITYThe functionality of AFS is defined by the so-called hardware oriented (low level) and the user oriented (high level) functions. These functions can also be classifiedinto application and safety functions (see Figure 5).Figure 4 : AFS actuatorApplication functions are those functions, that are required for the normal operation of the system. All other functions are part of the safety system. High level application functions can be classified into kinematic and kinetic functions (see Figure 6).Figure 7 shows the signal flow of the AFS system in the vehicle-driver overall closed loop. With the vehicle signals as input, the stabilization (e.g. yaw rate control)and the assistance functions (e.g. variable steering ratio)compute a desired superposition angle. This angle serves as command input signal to the controlled actuator. A safety system monitors the function and the components of the steering system (compare [7] and [8]). Every failure or error, that may lead to a safety relevant situation, is identified and suitable actions are initiated in order to keep the system in a well definedstate.Figure 5 : Structure of the AFS FunctionalityFigure 6 : Structure of high level functionsFigure 7 : Block diagram including the overall signal flow in the AFS systemThese actions reach from partial deactivations of single functions to shutting off the AFS system (fail silent behavior).In the next subsections, some high level functions of the AFS system will be described.KINEMATIC STEERING ASSISTANCE FUNCTIONS Kinematic steering assistance functions are feedforward controllers which adapt the static and dynamic steering characteristics to the current driving/vehicle situation as functions of the steering activity. This functionality is restricted by the actuator dynamics and the steering feel.These functions are part of the steering system (see Section 3). They are developed and implemented by ZF Lenksysteme.Currently, the variable steering ratio (VSR) provides the most noticeable benefit for the driver. This kinematic function adapts the steering ratio i V (1), between the steering wheel angle and an average road wheel angle,to the driving situation as a function of e.g. the vehicle velocity (see Figure 8). Under normal road conditions at low and medium speeds, the steering becomes more direct, requiring less steering effort (see Figure 9) of the driver which increases the agility of the vehicle in city traffic or when parking. At high speeds the steering becomes less direct, offering improved directional stability. Additional to the velocity dependency, the variable steering ratio developed by ZF Lenksysteme includes a dependency of the pinion angle i.e. rack displacement. This feature provides a reduced steering effort for large steering angles and a more precise steering for small steering angles.The principle of this function is based on the definition of the steering ratioFmSV :i δδ=.(1)Figure 8 : Example of the variable steering ratio as function of vehicle velocityInserting the nonlinear kinematic relation ()()G sk Fm f δ=δ between pinion angle δG , average roadwheel angle δFm and the linear kinematic relation ()S S M M G k k δ⋅+δ⋅=δ between pinion angle,steering wheel angle δS and motor angle δM into (1)yields the relation()S S M M sk SV k k f i δ⋅+δ⋅δ=.(2)The core algorithm of the VSR function computes a motor angle VSRd M δthat fulfils (2) for a predefined desired steering ratio i V and a measured steering wheel angle δS .Another steering assistance function that is evident for the driver in usual driving conditions is the so-called steering lead function (SLD). This kinematic function adapts the steer response to the driving/vehicle situation as a function of suitable vehicle and steering measured signals. The ZF Lenksysteme approach includes a differentiating prefilter for the steering wheel angle (see Figure 10). The weighted steering wheel angular velocitySSLD T δ⋅& defines then the desired motor angle (outputof the SLD function) for the controlled AFS actuator.Figure 9 : Slalom ride (cones distance: 16m andvehicle velocity approx. 50 kph) with AFS/VSR andwith a conventional mechanical ratioFigure 10 : Overall block diagram of the steering lead functionThis algorithm represents an insertion of a zero 1 in the transfer function between steering wheel angle and average front wheel angle. This additional zero is placed so that the delay due to the dynamic of the steering system is reduced, partially compensated or if desired increased. Figure 11 shows the results of a double lane change manouver on asphalt at a vehicle speed of approx. 85 [km/h]. The increased steering dynamic reduces the required steering interventions in order to perform the driving task.KINETIC STEERING ASSISTANCE FUNCTIONS Kinetic steering assistance functions also include feedforward controllers. Besides the primary task of providing the usual steering torque assistance like in conventional steering systems, these functions the additional task is providing a reduction/compensation of the reaction torque caused by the AFS actuator motion.These functions are restricted by the steering feel and the dynamics of the steering system. They are part of the steering system (see Section 3) and are developed and implemented by ZF Lenksysteme.The first kinetic function is the servotronic control function (SVT). The function algorithms include the computation of the desired current for the electro-hydraulic converter of the Sercotronic 2 component. The torque assistance is adapted to the driving/vehicle situation as a function of the vehicle velocity and the pinion angle velocity (actuator activity) (see Figure 12).The first dependency is the well-known vehicle-velocity dependent assistance torque, that provides the highest assistance torques for low velocities (i.e. steer comfort)and low assistance torques at high velocities in order toimprove the lateral stability of the vehicle.Figure 11 : Double lane change with and without the SLD function1in terms of control engineeringThe second dependency is AFS specific and sets a reduction/compensation of the reaction torque.Due to the possible high rack-displacement velocities, a higher 2 flow rate is required in order to take fully advantage of the AFS functionality. On the other hand thermal strains and a high fuel consumption have to be avoided. For that reason an electronic controlled orifice pump that modifies the flow rate in the hydraulic system has been included into the steering system. Another important kinetic function includes the control of the electronic controlled orifice pump (ECO). The main task of this function is to compute a desired current for the ECO-pump as a function of the vehicle velocity and the pinion angle velocity (actuator activity). These dependencies have been chosen in analogy to dependencies for the Servotronic control.KINEMATIC STABILIZATION FUNCTIONSThe stabilization functions represent another kind of consumer value increment. These functions include closed loop control algorithms that generate automatic 3steering interventions to stabilize the vehicle (see Figure13).Figure 12 : Example of the dependencies of thedesired current for the servotronic controlFigure 13 : Lane change / ABS-braking with different steering functions (µ≈0.2)2higher than the required flow rate for similar vehicles with conventional steering systems 3Automatic in a sense of an explicit independency from the steering wheel angle defined by the driverThey are not part of the steering system (see Section 3),they are developed and implemented by the car manufacturer. Some examples of this kind of functions are (see [4,6]):• yaw rate control,• yaw torque control and• disturbance rejection function.SAFETY AND MONITORING FUNCTIONSThe above described functions imply high requirements for the safety integrity of the system [8,9]. For this reason ZF Lenksysteme has developed a suitable safety concept for the steering system that includes several safety and monitoring functions on high and low level (see [7]).3. MODULAR CONCEPTIn the first phase of the market introduction of AFS, ZF Lenksysteme developed the rack and pinion steering components, the mechatronic actuator as well as the electronic control unit which includes the low level software (see Figure 14). BMW developed the safety concept, the application and associated safety high level functions and also took the system responsibility [4,9](see Figure 14). In the second phase of the AFS development ZF Lenksysteme focuses on a modular concept that simplify the combination and integration of the AFS system with other chassis control systems and in different vehicle platforms [10]. The modular concept implies a clear distribution of responsibilities and the associated functionality and safety distribution (see Figure 15). Hereby, the steering system has to be autonomous and keep the complete steer functionality even in case of failure or absence of several vehicle dynamic control systems (including the kinematic stability functions). The simplest approach to achieve this autonomy is a separation of vehicle and steeringfunctionality and safety in a hardware level.Figure 14 : Overall block diagram of the first system conceptThis implies running the kinematic stabilization function on a separated ECU, e.g. the ESP control unit taking into account that several required vehicle motion signals are available and even parts of the required algorithms are already implemented.An essential requirement for the modular concept is a new system interface that allows an external intervention for stabilization purposes. Such a system interface has been developed together with involved car manufacturers and component suppliers based on well-known principles like the Cartronic approach. This provides a compatibility with current and future system concepts and development organization structures (e.g.integration of the system by a third party). Moreover, the modular concept with the mentioned interface allows a parallel development of the stabilization and steering assistance functions and reduces the required testing activities for the integrated steering system.SYSTEM INTERFACEIn order to simplify the description of the interface for the modular AFS system, it will be defined in three phases (see Figure 16)• assistance,• assistance and stabilization and• assistance, stabilization, manual configuration anddiagnosis.The pure assistance interface exclusively includes input signals (I 1):• signed road wheel speeds: input signals of safetyand steering assistance functions,• status of the road wheel speeds: requirement forutilization of the road wheel speeds,• steering wheel angle: input signal of a single safetyfunction and several kinematic assistance functions,Figure 15 : Overall block diagram of the modular system concept• ESP and ABS intervention flags: binary signal foreach road wheel including a brake intervention flag used in safety functions,• engine revolutions: input signal of the systemdynamic monitoring function,• current gear: this signal is required only if the sign ofthe road wheel velocities is not available,The interface required for assistance and stabilization interventions includes besides I1 additional input signals (I2):• desired superposition angle for vehicle dynamicstabilization: input signal which includes a relative superposition angle, represented as an average road wheel angle or pinion angle. This angle is relative to the current absolute assistance superposition angle,• execute flag of the stabilization intervention:condition for performing the stabilization intervention. This signal also includes the associated safety information about the intervention command.This interface also includes an output (O2) required by the overall vehicle dynamics controller and defined by the following signals:• current average front wheel angle: this signal iscomputed from the measured pinion angle and the known nonlinear steering kinematics,• requested steering angle: this angle is computedfrom the measured steering wheel angle and the current desired steering ratio, represented as an average road wheel angle or pinion angle,• desired superposition assistance angle: output fromthe kinematic steering assistance functions,represented as an average road wheel angle or pinion angle,• dynamic capacity: estimated maximal additionalangular speed that can be demanded by an external vehicle controller,• system status: this signal includes information aboutthe current system mode (e.g. initialization, on, etc.),Figure 16 : Interface for the modular AFS concept•raw pinion angle: raw signal of the pinion angle sensor. The receiver of this signal has to perform own plausibility checks.Finally the complete system interface includes the inputs I1 and I2 as well as I3 with the signals:•VSR flag: signal for switching the mode of the VSR(e.g. sport, comfort),•SVT/ECO flag: signal for switching the mode of the kinetic steering assistance functions (e.g. sport, comfort).The complete interface also includes besides the outputs O1 and O2, the output O3 with the signals:•current superposition angle: this signal provides a redundant information that can be used by the overall vehicle dynamics controller for diagnosis/monitoring purposes,•failure code: this signal includes information about all failures/errors that are relevant for diagnostics. 4. CONCLUSIONThe market introduction of the Active Front Steering system represents an important step towards an entire chassis control in a series vehicle. The high equipment rate of AFS in the new BMW 5-series shows the enormous interest of the customers in the system due to the evident and continuous benefit experienced. Consequently, ZF Lenksysteme had to focus on a modular system concept that allows an independent development of assistance and stabilization (vehicle control) functions.Moreover, the enclosure and autonomy of the steering system improves the availability and allows reuse of functions and components for several vehicle platforms. The defined system interface minimizes the application and testing time and costs. The protection of the OEM and supplier know-how is also supported by the modular concept, allowing an overall system integration by a third party.ACKNOWLEDGMENTSWe would like to thank our colleagues Reinhard Grossheim, Wolfgang Schuster, Ralf Redemann and Christian Lundquist for their excellent work developing the assistance and safety functions as well as the failure strategy for the modular concept of ZF Lenksysteme. We also would like to thank Peter Brenner and Gerd Mueller for making possible the software development of the high level functions. REFERENCES1. Klier, W., Reinelt, W., Active Front Steering (Part 1)– Mathematical Modeling and Parameter Estimation, SAE technical paper 2004-01-1102, SAE World Congress, Steering & Suspension Technology Symposium. Detroit, USA, March 2004.2. Klier, W., Reimann, G., Reinelt, W., Active FrontSteering – Systemvernetzung und Funktionsumfang, Steuerung und Regelung von Fahrzeugen und Motoren – AUTOREG 2004, March 2004, pp. 569 –583, 2004.3. Reinelt, W., Klier, W., Lundquist, Ch., Reimann, G.,Schuster, W., Großheim, R., Active Front Steering for Passenger Cars – System Modelling and Functions, IFAC Symposium – Advances in Automotive Control. Italy, April 2004, pp. 697 – 702, 2004.4. Knoop, M., Leimbach, K.-D. und Verhagen, A.,Fahrwerksysteme im Reglerverbund, Tagung Fahrwerktechnik, Haus der Technik, Essen, 1999. 5. Köhn, P., Baumgarten, G., Richter, T., Schuster, M.und Fleck, R., Die Aktivlenkung - Das neue Fahrdynamische Lenksystem von BMW, Tagungsband Aachener Kolloquium Fahrzeug- und Motorentechnik 2002, pp. 1093 – 1109, 2002.6. Fleck, R., Aktiv-Lenkung – Ein wichtiger ersterSchritt zum Steer-by-Wire, Tagung PKW-Lenksysteme – Vorbereitung auf die Technik von morgen, Haus der Technik e.V., Essen, 2003.7. Reinelt, W., Klier, W., Reimann, G., Active FrontSteering (Part 2) – Safety and Functionality, SAE technical paper 2004-01-1101, SAE World Congress, Steering & Suspension Technology Symposium. Detroit, USA, March 2004.8. Reinelt, W., Klier, W., Reimann, G.,Systemsicherheit des Active Front Steering, Steuerung und Regelung von Fahrzeugen und Motoren – AUTOREG 2004, March 2004, pp. 49 –58, 2004.9. Eckrich, M., Pischinger, M., Krenn, M., Bartz, R. undMunnix, P., Aktivlenkung – Anforderungen an Sicherheitstechnik und Entwicklungsprozess, Tagungsband Aachener Kolloquium Fahrzeug- und Motorentechnik 2002, pp. 1169 – 1183, 200210. Kirchner, A., Schwitters, F., Vernetzte und modulareAuslegung von Fahrerassistenzfunktionen, VDI –Tagung Elektronik im Kraftfahrzeug 25. Und 26.September, Baden-Baden, 2004.CONTACTDr. Willy Klier, ZF Lenksysteme GmbH, Dept. ZEMF Active Front Steering – (Team Leader) Safety and Algorithms, Richard-Bullingerstr. 77, 73527 Schwäbisch Gmünd, Germany, Tel.: +49/7171312589, Email: Willy.Klier@。

脉冲系统与脉冲控制及其应用1导论在现实世界中，存在许多实际的工程和自然系统，在某些时间区间连续渐变，而又由于某种原因，在某些时刻内会系统状态会遭到突然的改变。

由于变化时间往往非常短，其突变或跳跃过程可以视为在某时刻瞬间发生的。

我们把这种现象称为脉冲现象。

这些系统不能单靠传统的连续系统或单靠离散系统能解决的，可以找到许多具有这种现象的例子，如，生态学中的种群增长［1-3］，传染病防治［4-6］，数字通信系统［7-9］，金融［10］，经济学中优化控制问题［11］等等都具有这种脉冲现象。

这种例子在很多领域中也能找到，如，自动控制，计算机网络、供应链系统以及通信系统等等。

这种状态在某些瞬间发生突然变化的系统是不能用单用连续动力系统或者离散动力系统来描述的，这就很很自然的人们就提出了脉冲系统来描述这类具有脉冲现象的动力系统。

一般来说，一个脉冲系统包括三个元素［12］:(1) 一个连续的常微分系统，控制系统在脉冲或重置事件间的动态行为。

(2) 一个离散的差分系统，在脉冲或重置事件发生的时候，状态瞬间改变的情况。

(3) 一个判据，决定什么时候发生重置事件。

通常连续时间非线性脉冲系统可以描述为x&(t)fx(t),u(t),ttk, k{1,2 ,L}(1)x Ck(t,x), t tk, k {1,2,L}.其中脉冲时间{tl,t2,ts丄}是一个严格递增的时间序列，X Rn为系统状态变量，U为系统控制输入，X x(tk) x(tk)。

类似的离散时间脉冲系统可以描述为X(t 1)f x(t),u(t) , ttk, k{1,2,L},(2)x(tk1) Ck(t,x), ttk,k{1,2,L},其中t Z，Z代表非负整数。

2国内外研究现状脉冲系统的研究最早可以追溯到上世纪60时年代Miliman, VD , Myshkis, A D[13]。

近些年来，脉冲系统作为一个非常活跃的研究方向，吸引了一大批来自不同领域的学者进行研究。

耕种土地的英语Title: The Art and Science of Farming the LandFarming, an age-old practice deeply rooted in human civilization, stands as a testament to our symbiotic relationship with nature. It is not merely a means of sustenance but an intricate dance between man, soil, water, and sunlight, where each step is guided by wisdom passed down through generations and innovations born out of necessity. As we delve into the art and science of farming the land, we uncover a world where tradition harmoniously blends with modernity, cultivating not just crops but also a sustainable future for generations to come.At the heart of successful farming lies an intimate understanding of the soil—its composition, fertility, and microbial life. Soil health is paramount; it acts as both a cradle for seeds and a provider of nutrients necessary for plant growth. Organic matter enriches the soil, improving its structure, water retention capacity, and nutrient availability. Composting, crop rotation, and cover cropping are practices that enhance soil fertility naturally, reducing reliance on synthetic fertilizers which can degrade soil quality over time. By nurturing the soil, farmers ensure a thriving ecosystembeneath their feet, fostering resilience against pests, diseases, and environmental stressors.Water management is another crucial aspect of farming, especially in regions facing scarcity or erratic rainfall patterns. Efficient irrigation systems like drip or sprinkler methods minimize water wastage while delivering precise amounts directly to plant roots. Rainwater harvesting techniques and the construction of check dams help conserve and utilize natural water resources effectively. Moreover, adopting drought-tolerant crop varieties and implementing conservation tillage practices further mitigate water usage, promoting sustainability in agriculture.The selection of crops plays a pivotal role in farming strategies. Crop diversity not only ensures food security but also enhances ecosystem stability by breaking pest and disease cycles. Polyculture—growing multiple crops simultaneously—mimics natural ecosystems, reducing the need for chemical inputs and promoting biodiversity. Agroforestry, which integrates trees and shrubs into agricultural landscapes, provides shade, windbreaks, and habitat for beneficial organisms, enhancing overall farm productivity and ecological balance.Innovation has revolutionized traditional farming practices, ushering in an era of precision agriculture. Advanced technologies such as GPS-guided machinery, drones for monitoring crop health, and sensors embedded in soil for real-time data collection enable farmers to optimize resource use, predict yields, and manage fields remotely. These tools empower farmers with actionable insights, allowing them to make informed decisions that maximize efficiency and minimize environmental impact.Sustainable farming goes beyond production; it encompasses social equity and economic viability. Fair trade initiatives ensure that farmers receive fair compensation for their labor, fostering community development and poverty alleviation. Local food networks strengthen connections between consumers and producers, promoting transparency and supporting small-scale farmers who often employ environmentally friendly practices. Education and training programs equip farmers with knowledge on sustainable practices, fostering a culture of continuous learning and adaptation.In conclusion, farming the land is a delicate balance between harnessing nature's bounty and preserving itsintegrity for future generations. It requires a deep respect for the environment, a commitment to innovation, and a recognition of the interconnectedness of all life forms. As we stand at the crossroads of technological advancement and ecological stewardship, embracing sustainable farming practices becomes imperative. Through this harmonious coexistence with nature, we can cultivate a greener, more resilient planet, nourishing both humanity and the earth that sustains us.。

自适应控制Adaptive control1.关于控制2.关于自适应控制3.模型参考自适应控制4.自校正控制5.自适应替代方案6.预测控制参考文献主要章节内容说明：第一部分：第一章自适应律的设计§1.参数最优化方法§2.基于Lyapunov稳定性理论的方法§3.超稳定性理论在自适应控制中的应用第二章误差模型§1.Narendra误差模型§2.增广矩阵§3.线性误差模型第三章MRAC的设计和实现第四章小结第二部分：第一章模型辨识及控制器设计§1.系统模型：CARMA模型§2.参数估计：LS法§3.控制器的设计方法：利用传递函数模型§4.自校正第二章最小方差自校正控制§1.最小方差自校正调节器§2.广义最小方差自校正控制第三章极点配置自校正控制§1.间接自校正§2.直接自校正1.About control engineering education1)control curriculum basic concept(1)dynamic system●The processes and plants that are controlled have responses that evolvein time with memory of past responses●The most common mathematical tool used to describe dynamic system isthe ordinary differential equation (ODE).●First approximate the equation as linear and time-invariant. Thenextensions can be made from this foundation that are nonlinear 、time-varying、sampled-data、distributed parameter and so on.●Method of building model (or equation )a)Idea of writing equations of motion based on the physics andchemistry of the situation.b)That of system identification based on experimental data.●Part of understanding the dynamical system requires understanding theperformance limitations and expectation of the system.2.stabilityWith stability, the system can at least be used●Classical control design method, are based on a stability test.Root locus 根轨迹Bode‟s frequency response 波特图Nyquist stability criterion 奈奎斯特判据●Optimal control, especially linear-quadratic Gaussian (LQG) control (线性二次型高斯问题) was always haunted by the fact that method did notinclude a guarantee of margin of stability.The theory and techniques of robust (鲁棒)design have been developedas alternative to LQG●In the realm of nonlinear control, including adaptive control, it iscommon practice to base the design on Lyapunov function in order to beable to guarantee stability of final result.3.feedbackMany open-loop devices such as programmable logic controllers (PLC) are in use, their design and use are not part of control engineering.●The introduction of feedback brings costs as well as benefits. Among thecosts are need for both actuators and sensors, especially sensors.●Actuator defines the control authority and set the limits of speed indynamic response.●Sensor via their inevitable noise, limit the ultimate(最终) accuracy ofcontrol within these limits, feedback affords the benefit of improveddynamic response and stability margins, improved disturbancerejection(拒绝) ，and improved robustness to parameter variability.●The trade off between costs and benefits of feedback is at the center ofcontrol design.4.Dynamic compensation●In beginning there was PID compensation, today remaining a widely usedelement of control, especially in the process control.●Other compensation approaches : lead-and-log networks (超前-滞后)observer-based compensators include : pole placement, LQG designs.●Of increasing interest are designs capable of including trade-off amongstability, dynamic response and parameter robustness.Include: Q parameterization, adaptive schemes.Such as self-tuning regulators, neural-network-based-controllers.二、historical perspectives (透视)●Most of early control manifestations appear as simple on-off (bang-bang)controllers with empirical (实验；经验性的) setting much dependent uponexperience.●The following advances such as Routhis and Hurwitz stability analysis(1877).Lyapunov‟s state model and nonlinear stability criteria(判据) (1890) .Sperry‟s early work on gyroscope and autopilots (1910), and Sikorsky‟swork on ship steering (1923)Take differential equation, Heaviside operators and Laplace transform astheir tools.●电机工程(electrical engineering)The largely changed in the late 1920s and 1930s with Black‟s developmentof the feedback electronic amplifier, Bush‟s differential analyzer, Nyquist‟sstability criterion and Bode‟s frequency response methods.The electrical engineering problems faced usually had vary complex albeitmostly linear model and had arbitrary (独立的；随机的) and wide-ringingdynamics.●过程控制(process control in chemical engineering)Most of the progress controlled were complex and highly nonlinear, butusually had relatively docile (易于处理的) dynamics.One major outcome of this type of work was Ziegler-Nichols‟PIDthres-term controller. This control approach is still in use today, worldwidewith relatively minor modifications and upgrades (including sampled dataPID controllers with feed forward control, anti-integrator-windupcontrollers :抗积分饱和，and fuzzy logic implementations).●机械工程(mechanical engineering)The application of controls in mechanical engineering dealt mostly in thebeginning with mechanism controls, such as servomechanisms, governorsand robots.Some typical control application areas now include manufacturing processcontrols, vehicle dynamic and safety control, biomedical devices and geneticprocess research.Some early methodological outcomes were the olden burger-Kahenbugerdescribing function method of equivalent linearization, and minimum-time,bang-bang control.●航空工程（aeronautical engineering ）The problems were generally a hybrid (混合) of well-modeled mechanicsplus marginally understood fluid dynamics. The models were often weaklynonlinear, and the dynamics were sometimes unstable.Major contributions to framework of controls as discipline were Evan‟s rootlocus (1948) and gain-scheduling.●Additional major contributions to growth of the discipline of control over thelast 30-40 years have tended to be independent of traditional disciplines.Examples include:Pontryagin‟s maximum principle (1956) 庞特里金Bellman‟s dynamic programming (1957)贝尔曼Kalman‟s optimal estimation (1960)And the recent advances in robust control.三、Abstract thoughts on curriculum●The possibilities for topic to teach are sufficiently great. If one tries topresent proofs of all theoretical results. One is in danger of giving thestudents many mathematical details with little physical intuition orappreciation for the purposes for which the system is designed.●Control is based on two distinct streams of thought. One stream is physicaland discipline-based. Because one must always be controlling some thing.The other stream is mathematics-based, because the basis concepts ofstability and feedback are fundamentally abstract concepts best expressedmathematically. This duality(两重性) has raised, over the years, regularcomplaints about the …gap‟ between theory and practice.●The control curriculum typically begins with one or two courses designed topresent an overview of control based on linear, constant, ODE models,s-plane and Nyquist‟s stability ideas, SISO feedback and PID, lead-lay andpole-placement compensation.These introductory courses can then be followed by courses in linear systemtheory, digital of control, optimal control, advanced theory of feedback, andsystem identification.四、Main control courses●Introduction to controlLumped system theoryNonlinear controlOptimal controlAdaptive controlRobot controlDigital controlModeling and simulationAdvanced theoryStochastic processesLarge scale multivariable systemManufacturing systemFuzzy logic Neural Networks外文期刊：《Automatic》IFAC 国际自动控制联合会Computer and control abstractsIEEE translations on Automatic controlAutomation●Specialized \ experimental courses✓Intelligent controlApplication of Artificial IntelligenceSimulation and optimization of lager scale systems robust control ✓System identification✓Microcomputer-based control systemDiscrete-event systemsParallel and Distributed computationNumerical optimization methodsNumerical system theory●Top key works from 1963-1995 in IIACAdaptive control 305Optimal control 277Identification 255Parameter estimation 244Stability 217Linear system 184Non-linear systems 168Robust control 158Discrete-time systems 143Multivariable systems 140Robustness 140Multivariable systems control systems 110Optimization 110Computer control 104Large-scale systems 103Kalman filter 102Modeling 107为什么自适应 《Astrom 》chapter 1✓ 反馈可以消除扰动。