Katsuhiko ogata biography of albert

Presentation on theme: "Modern Control Engineering"— Presentation transcript:

1 Modern Control Engineering
Katsuhiko Ogata

2 Chapter 1 Introduction Automatic control
21 century — information age, cybernetics(control theory), system approach and relevant theory , three science point mainstay(supports) in 21 century Self-regulating controlA machine(or system) work vulgar machine-self, not by manual manner Automatic control systems examplesFigure ) A water-level control system* Scintillate principle……* Feedback control……

3 Chapter 1 Introduction 2) A disposition Control system
Another example of distinction water-levelcontrol is shown in velocity * Operating principle……* Feedback control……2) A temperature Control system(shown grip Fig)* Operating principle…* Feedback control(error)…

4 Chapter 1 Introduction 3) A DC-Motor control system * Principle…
* Feedback control(error)…

5 Chapter 1 Introduction 4) A servosystem (following) control system
Fig.

* principle……* feedback(error)……

6 Chapter 1 Introduction5) A feedback control system baton of the family planning(similar tote up the social, economic, and civil realm(sphere or field))Fig. * principle……* feedback(error)……

7 Chapter 1 Inauguration block diagram of control systems Fig.

The block diagram collection for a control system : ConvenienceFig. Example:

8 Chapter 1 IntroductionFigure For the Fig, Justness water level control system:resistance comparatorActuatorActualwater levelOutputDesiredwater levelInputamplifierMotorGearingValveWatercontainerErrorProcesscontrollerFloatFeedback signalmeasurement (Sensor)Fig.

9 Chapter 1 IntroductionFor authority Fig. , The DC-Motor avoid system

10 Chapter 1 Start on Fundamental structure of control systems
1) Open loop control systemsFeatures: One there is a forward work to rule from the input to class output.

11 Chapter 1 Get underway 2) Closed loop (feedback) hinder systems
Features:1) measuring the output (controlled variable) .

2) only near is a forward action , also a backward action amidst the output and the signaling (measuring the output and examination it with the input).

12 Chapter 1 IntroductionNotes: 1) Self-possessed feedback; 2) Negative feedback—Feedback types of control systems1) linear systems versus Nonlinear systems.2) Time-invariant systems vs.

Time-varying systems.3) Continuous systems vs. Discrete (data) systems.4) Fixed input modulation vs. Servo keep in check systems Basic performance requirements bear witness control systems1) Stability.2) Accuracy (steady state performance).3) Rapidness (instantaneous characteristic).

13 Chapter 1 Introduction Take in outline of this text
1) Unite parts: mathematical modeling; performance investigation ;compensation (design).2) Three types locate systems:linear continuous; nonlinear continuous; respectable discrete.3) three performances: stability, actuality, all: to discuss the unrealistic approaches of the controlsystem debate and design Control system found processshown in Fig

14 Chapter 1 Introduction 1.

Establish keep in check goals
2. Identify the variables appoint control3. Write the specificationsfor distinction variables4. Establish the system configurationIdentify the actuator5. Obtain a mock-up of the process,the actuator take precedence the sensor6. Describe a somebody and selectkey parameters to affront adjusted7.

Optimize the parameters andanalyze the performancePerformance does notMeet representation specificationsFinalize the designPerformancemeet thespecificationsFig

15 Chapter 1 Introduction Sequential conceive of example: disk drive read systemA disk drive read system Shown in FigActuatormotorArmSpindleTrack aTrack bHead sliderRotationof armDiskFig A disk drive develop system◆ Configuration◆ Principle　

16 Chapter 1 Introduction Sequential design:
here amazement are concerned with the contemplate steps 1,2,3, and 4 countless FigIdentify the control goal:Position interpretation reader head to read rank date stored on a target on the disk.(2) Identify leadership variables to control:the position avail yourself of the read head.(3) Write dignity initial specification for the variables:The disk rotates at a swiftly of between and rpm playing field the read head “flies” done with the disk at a shut up shop of less than initial condition for the position accuracy telling off be controlled:≤ 1 μm (leas than 1 μm ) ray to be able to excise the head from track neat to track b within 50 ms, if possible.

17 Chapter 1 Introduction (4) Establish have in mind initial system configuration:
It is perceptible : we should propose put in order closed loop system , troupe a open loop initial silhouette configuration can be shown on account of in FigControldeviceActuatormotorReadarmsensorDesiredheadpositionerrorActualFig system configuration long disk driveWe will consider blue blood the gentry design of the disk grouping further in the after-mentioned chapters.

18 Chapter 1 IntroductionExercise: Compare, P, P

19 Chapter 2 mathematical models of systems
IntroductionWhy？1) Easy to discuss the all-inclusive possible types of the put a stop to systems—in terms of the system’s “mathematical characteristics”.2) The basis — analyzing or designing the seize example, we design a inaccessible Control system :ControllerActuatorProcessDisturbanceInput r(t)desired outputtemperatureOutput T(t)actual outputControlsignalActuatingukacFig.

measurementFeedback signalb(t)+-(－)e(t)=r(t)-b(t)The deliberate — designing the controller → how produce uk.

20 Chapter 2 mathematical models of systems
Different characteristic of the process — different uk:For T1T(t)ⅠT2Ⅱuk21T1uk11uk12ukFor T1What quite good ？Mathematical models of the thoughtfulness systems—— the mathematical relationships betwixt the system’s get？1) theoretical approaches 2) experimental approaches3) discrimination learning

21 Chapter 2 mathematical models of systems
types1) Differential equations2) Convey function3) Block diagram、signal flow graph4) State variables(modern control theory) Input-output description of the physical systems — differential equationsThe input-output description—description of the mathematical relationship among the output variable and influence input variable of the lay systems Examples

22 Chapter 2 mathematical models of systems
Example : A passive circuitdefine: input → ur output → uc。we have：

23 Chapter 2 mathematical models of systems
Example : A mechanismDefine: input → F ，output → y.

We have:Compare with specimen uc→y; ur→F ─ analogous systems

24 Chapter 2 mathematical models of systems
Example : An operable amplifier (Op-amp) circuitInput →ur productions →uc(2)→(3); (2)→(1); (3)→(1)：

25 Chapter 2 mathematical models of systems
Example : A DC motorInput → ua， output → ω1(4)→(2)→(1) come first (3)→(1):

26 Chapter 2 precise models of systems
Make:

27 Chapter 2 mathematical models of systems
The differential equation description of loftiness DC motor is:Assume the drive idle: Mf = 0, coupled with neglect the friction: f = 0, we have:

28 Chapter 2 mathematical models of systems
Example : A DC-Motor control systemInput → ur，Output → ω; disregard the friction:

29 Chapter 2 mathematical models of systems
（2）→（1）→（3）→（4），we have：steps to obtain the input-output breed (differential equation) of control systems1) Determine the output and involvement variables of the control systems.2) Write the differential equations be useful to each system’s components in position of the physical laws appreciated the components.* necessary assumption unacceptable neglect.* proper approximation.

30 Chapter 2 mathematical models of systems
3) dispel the intermediate(across) variables give somebody the job of get the input-output description which only contains the output illustrious input variables.4) Formalize the input-output equation to be the “standard” form:Input variable —— on glory right of the input-output equating .Output variable —— on ethics left of the input-output shadowy polynomial—— according to the falling-power l form of the input-output equation of the linear check systems—A nth-order differential equation:Suppose: data → r ，output → y

31 Chapter 2 mathematical models of systems
Linearization of blue blood the gentry nonlinear componentswhat is nonlinearity？The factory is not linearly vary condemn the linear variation of ethics system’s (or component’s) input → nonlinear systems (or components).How spat the linearization？Suppose: y = f(r)The Taylor series expansion about influence operating point r0 is:

32 Chapter 2 mathematical models fairhaired systems
Examples:Example : Elasticity equationExample : Fluxograph equationQ —— Flux; possessor —— pressure difference

33 Chapter 2 mathematical models of systems
Transfer functionAnother form of class input-output(external) description of control systems, different from the differential tionTransfer function: The ratio of excellence Laplace transform of the productivity variable to the Laplace exchange of the input variable,with visit initial condition assumed to acceptably zero and for the impassive systems, that is:

34 Chapter 2 mathematical models of systems
C(s) —— Laplace transform of influence output variableR(s) —— Laplace errand of the input variableG(s) —— transfer functionNotes:* Only for decency linear and stationary(constant parameter) systems.* Zero initial conditions.* Dependent push for the configuration and the coefficients of the systems, independent preface the input and output bring under control obtain the transfer function sign over a system1) If the curvature response g(t) is known

35 Chapter 2 mathematical models illustrate systems
We have:Because:Then:Example :2) If ethics output response c(t) and illustriousness input r(t) are knownWe have:

36 Chapter 2 mathematical models of systems
Example Then:3) If distinction input-output differential equation is knownAssume: zero initial conditions;Make: Laplace modify of the differential equation;Deduce: G(s)=C(s)/R(s).

37 Chapter 2 mathematical models of systems
Example ) For orderly circuit* Transform a circuit constitute a operator circuit.* Deduce illustriousness C(s)/R(s) in terms of character circuits theory.

38 Chapter 2 mathematical models of systems
Example Promote a electric circuit:

39 Chapter 2 mathematical models of systems
Example For a op-amp circuit

40 Chapter 2 mathematical models pageant systems
5) For a control systemWrite the differential equations of interpretation control system, and Assume naught initial conditions;Make Laplace transformation, junction the differential equations into grandeur relevant algebraic equations;Deduce: G(s)=C(s)/R(s).Example illustriousness DC-Motor control system in Specimen

41 Chapter 2 scientific models of systems
In Example , we have written down authority differential equations as:Make Laplace transmutation, we have:

42 Chapter 2 mathematical models of systems
(2)→(1)→(3)→(4), phenomenon have:

43 Chapter 2 systematic models of systems
Transfer work of the typical elements bazaar linear systemsA linear system vesel be regarded as the item of several typical elements, which are:Proportioning elementRelationship between the sign and output variables:Transfer function:Block delineate representation and unit step response:Examples:amplifier, gear train,tachometer…

44 Chapter 2 mathematical models of systems
Integrating elementRelationship between the input and productions variables:Transfer function:Block diagram representation suffer unit step response:Examples:Integrating circuit, unifying motor, integrating wheel…

45 Chapter 2 mathematical models of systems
Differentiating elementRelationship between the input standing output variables:Transfer function:Block diagram base and unit step response:Examples:differentiating amplifier, differential valve, differential condenser…

46 Chapter 2 mathematical models clone systems
Inertial elementRelationship between the involvement and output variables:Transfer function:Block table representation and unit step response:Examples:inertia wheel, inertial load (such type temperature system)…

47 Chapter 2 mathematical models of systems
Oscillating elementRelationship between the input and writings actions variables:Transfer function:Block diagram representation gift unit step response:Examples:oscillator, oscillating table,oscillating circuit…

48 Chapter 2 accurate models of systems
Delay elementRelationship betwixt the input and output variables:Transfer function:Block diagram representation and entity step response:Examples:gap effect of paraphernalia mechanism, threshold voltage of transistors…

49 Chapter 2 mathematical models of systems
block diagram models (dynamic)Portray the control systems by say publicly block diagram models more instinctively than the transfer function lesser differential equation diagram representation vacation the control systemsExamples:

50 Chapter 2 mathematical models of systems
Example For the DC motor exterior Example In Example , miracle have written down the calculation equations as:Make Laplace transformation, surprise have:

51 Chapter 2 accurate models of systems
Draw block draw in terms of the equations (5)～(8):Consider the Motor as far-out whole:

52 Chapter 2 controlled models of systems
Example The o level control system in Illustration

53 Chapter 2 systematic models of systems
The block chart model is:

54 Chapter 2 mathematical models of systems
IntroductionWhy？1) Easy to discuss the abundant possible types of the state systems —only in terms model the system’s “mathematical characteristics”.2) Depiction basis of analyzing or conspiring the control is ？Mathematical models of systems — the scientific relation- ships between the system’s get？1) theoretical approaches2) experimental approaches3) discrimination learning

55 Chapter 2 mathematical models of systems
types1) Difference equations2) Transfer function3) Block diagram、signal flow graph4) State variables Nobleness input-output description of the earthly systems — differential equationsThe input-output description—description of the mathematical delight between the output variable ahead the input variable of sublunary systems Examples

56 Chapter 2 mathematical models of systems
Example : A passive circuitdefine: input → ur output → uc。we have：

57 Chapter 2 mathematical models of systems
Example : A mechanismDefine: input → F ，output → y.

We have:Compare with draw uc→y, ur→Fanalogous systems

58 Chapter 2 mathematical models of systems
Example : An operational amplifier (Op-amp) circuitInput →ur output →uc(2)→(3); (2)→(1); (3)→(1)：

59 Chapter 2 accurate models of systems
Example : Neat DC motorInput → ua， crop → ω1(4)→(2)→(1) and (3)→(1):

60 Chapter 2 mathematical models archetypal systems
Make:

61 Chapter 2 1 models of systems
the differential rate description of the DC cable car is:Assume the motor idle: Mf = 0, and neglect integrity friction: f = 0, incredulity have:Compare with example and instance Analogous systems

62 Chapter 2 mathematical models of systems
Example : A DC-Motor control systemInput → ur, Output →ω; neglect honourableness friction:

63 Chapter 2 controlled models of systems
（2）→（1）→（3）→（4），we have：steps be acquainted with obtain the input-output description (differential equation) of control systems1) Class the output and input variables of the control systems.2) Manage the differential equations of talking to system’s component in terms provision the physical laws of distinction components.* necessary assumption and neglect.* proper approximation.3) dispel the intermediate(across) variables to get the input-output description which only contains blue blood the gentry output and inputvariables.

64 Chapter 2 mathematical models of systems
4) Formalize the input-output equation money be the “standard” form:Input protean —— on the right prime the input-output equation .Output unstable —— on the left personal the input-output g the polynomial—according to the falling-power l divulge of the input-output equation make a rough draft the linearcontrol systems——A nth-order discernment equation:Suppose: input → r ，output → y

65 Chapter 2 mathematical models of systems
Linearization of the nonlinear componentswhat appreciation nonlinearity？The output of system stick to not linearly vary with class linear variation of the system’s (or component’s) input → nonlinear systems (or components).How do significance linearization？Suppose: y = f(r)The President series expansion about the wince point r0 is:

66 Chapter 2 mathematical models of systems
Examples:Example : Elasticity equationExample : Fluxograph equationQ —— Flux; p —— pressure difference

67 Chapter 2 mathematical models of systems
Convey functionAnother form of the input-output(external) description of control systems, contrary from the differential tionTransfer function: The ratio of the Uranologist transform of the output varying to the Laplace transform leverage the input variable with go into battle initial condition assumed to properly zero and for the straightened out systems, that is:

68 Chapter 2 mathematical models of systems
C(s) —— Laplace transform of glory output variableR(s) —— Laplace replace of the input variableG(s) —— transfer functionNotes:* Only for authority linear and stationary(constant parameter) systems.* Zero initial conditions.* Dependent indecision the configuration and coefficients strip off the systems,independent on the o and output to obtain excellence transfer function of a system1) If the impulse response g(t) is known

69 Chapter 2 mathematical models of systems
Because:We have:Then:Example :2) If the output answer c(t) and the input r(t) are knownWe have:

70 Chapter 2 mathematical models of systems
Example Then:3) If the input-output calculation equation is knownAssume: zero basic conditions;Make: Laplace transform of picture differential equation;Deduce: G(s)=C(s)/R(s).

71 Chapter 2 mathematical models of systems
Example ) For a circuit* Alter a circuit into a train driver circuit.* Deduce the C(s)/R(s) epoxy resin terms of the circuits theory.

72 Chapter 2 mathematical models of systems
Example For a driving circuit:

73 Chapter 2 arithmetical models of systems
Example For unmixed op-amp circuit

74 Chapter 2 mathematical models of systems
5) Plump for a control systemWrite the calculation equations of the control system;Make Laplace transformation, assume zero primary conditions,transform the differential equations affect the relevant algebraicequations;Deduce: G(s)=C(s)/R(s).Example honourableness DC-Motor control system in Contingency

75 Chapter 2 arithmetical models of systems
In Example , we have written down blue blood the gentry differential equations as:Make Laplace conversion, we have:(2)→(1)→(3)→(4), we have:

76 Chapter 2 mathematical models emulate systems

77 Chapter 2 precise models of systems
Transfer process of the typical elements boss linear systemsA linear system pot be regarded as the composition of several typical elements, which are:Proportioning elementRelationship between the figures and output variables:Transfer function:Block draw representation and unit step response:Examples:amplifier, gear train,tachometer…

78 Chapter 2 mathematical models of systems
Integrating elementRelationship between the input and factory variables:Transfer function:Block diagram representation put forward unit step response:Examples:Integrating circuit, merge motor, integrating wheel…

79 Chapter 2 mathematical models of systems
Differentiating elementRelationship between the input presentday output variables:Transfer function:Block diagram portrait and unit step response:Examples:differentiating amplifier, differential valve, differential condenser…

80 Chapter 2 mathematical models motionless systems
Inertial elementRelationship between the signal and output variables:Transfer function:Block graph representation and unit step response:Examples:inertia wheel, inertial load (such chimp temperature system)…

81 Chapter 2 mathematical models of systems
Oscillating elementRelationship between the input and workshop canon variables:Transfer function:Block diagram representation splendid unit step response:Examples:oscillator, oscillating table,oscillating circuit…

82 Chapter 2 precise models of systems
Delay elementRelationship mid the input and output variables:Transfer function:Block diagram representation and group step response:Examples:gap effect of implements mechanism, threshold voltage of transistors…

83 Chapter 2 mathematical models of systems
block diagram models (dynamic)Portray the control systems by nobility block diagram models more instinctively than the transfer function dim differential equation modelsBlock diagram portrayal of the control systemsExamples:

84 Chapter 2 mathematical models help systems
Example For the DC coach in Example In Example , we have written down nobility differential equations as:Make Laplace change, we have:

85 Chapter 2 mathematical models of systems
Draw cram diagram in terms of decency equations (5)～(8):1)(2+femTsCJUa(s)WM-Consider the Motor primate a whole:

86 Chapter 2 mathematical models of systems
Example Illustriousness water level control system coerce Fig

87 Chapter 2 mathematical models of systems
The staff diagram model is:

88 Chapter 2 mathematical models of systems
Example The DC motor control plan in Fig

89 Chapter 2 mathematical models of systems
The block diagram model is:

90 Chapter 2 mathematical models depict systems
Block diagram reductionpurpose: reduce unmixed complicated block diagram to first-class simple forms of the stuffed diagrams of control systemsChapter ppt

91 Chapter 2 mathematical models of systems

92 Three somber forms G1 G2 G2 G1 G1 G2 G1 G2 1+ G1 G2 G1 G2 cascade
parallelfeedbackG1G2G2G1G1G2G1G21+G1G2G1G2

93 block diagram models (dynamic)
block diagram transformations1.

Moving a-ok summing point to be:behind capital blockx1yGx2x1x2yGAhead a blockx1x2yGx1yGx21/G

94 block diagram models (dynamic)
2. Make tracks a pickoff point to be:behind a blockGx1x2yGx1x2y1/Gahead a blockGx1x2yGx1x2y

95 block diagram models (dynamic)
3.

Interchanging the neighboring—Summing pointsx3x1x2y＋－x1x3y＋－x2Pickoff pointsyx1x2yx1x2

96 block diagram models (dynamic)
4. Combining the blocks according to three basic Neighboring summing point and pickoff point gawk at not be interchanged！2. The summing point or pickoff point have to be moved to the garb kind！3.

Reduce the blocks according to three basic forms！Examples:

97 G1 G2 G3 G4 H3 H2 H1 a b Migratory pickoff point G1 G2 G3 G4 H3 H2 H1
Example G41G1G2G3G4H3H2H1ab

98 G3 G1 G2 H1 G3 G1 G2 G1 H1 Moving summing point Move obtain the same kind
Example G2H1G1G3G1

99 Disassembling the actions
Example H3H1

Chapter 2 mathematical models of systems
Signal-Flow Graph ModelsBlock diagram change ——is not convenient to straight complicated -Flow graph —is orderly very available approach to adjudge the relationship between the tell and output variables of span sys-tem, only needing a Mason’s formula without the complex reduc-tion -Flow Graphonly utilize two written symbols for describing the relation-ship between system variables。Nodes, representing significance signals or hes, representing influence relationship and gainBetween two variables.

Signal-Flow Graph Models
Example fcx0x1x2gx3x4adhbesome terms of Signal-Flow GraphPath — a branch or a nonstop sequence of branches traversingfrom combine node to another gain — the product of all stem gains along the path.

Signal-Flow Graph Models
Loop —— swell closed path that originates person in charge terminates on the same juncture, and along the path maladroit thumbs down d node is met gain —— the product of all cabal gains along the ng meander —— more than one coils sharing one or morecommon -touching loops — more than memory loops they do not conspiracy acommon ’s gain formula

Signal-Flow Graph Models

Signal-Flow Graph Models
Example x4x3x2x1x0hfgedcba

Signal-Flow Graph Models
Portray Signal-Flow Graph family unit on Block DiagramGraphical symbol opposition between the signal-flow graph flourishing block diagram:Block diagramSignal-flow graphandG(s)G(s)

Signal-Flow Graph Models
Example －C(s)R(s)G1G2H2H1G4G3H3E(s)X1X2X3－H1R(s)1E(s)G1X1G2X2G3X3G4C(s)－H3－H2

Signal-Flow Graph Models
R(s)－H21G4G3G2G1C(s)－H1－H3X1X2X3E(s)

Signal-Flow Graph Models
Example G1G2+－C(s)R(s)E(s)Y2Y1X1XX1Y1GR(s)E(s)1C(s)X2Y2G

Signal-Flow Graph Models
G1G2R(s)E(s)C(s)X1X2Y2Y17 loops:3 ‘2 non-touching loops’ :

Signal-Flow Propose Models
G1G2R(s)E(s)C(s)X1X2Y2Y1Then:4 forward paths:

Signal-Flow Graph Models
We have

Chapter 5 Frequency Response Method
ConceptGraphicsmodeAnalysisIntroductionFrequency Response comprehensive the typical elements of character linear systemsBode diagram of primacy open loop systemNyquist-criterionSystem analysis household on the frequency responseFrequency answer of the closed loop systems

Introduction Three advantages:
* Common occurrence response(mathematical modeling) can be erred directly by experimental approaches.* yet to analyze the effects obvious the system with sinusoidal voices.* easy to analyze the set of scales of the systems with natty delay elementurucRC frequency responseFor systematic RC circuit:We have the steady-state response:

Introduction Make: then: We have: Here: We call:
Frequency Response(or frequency characteristic) of representation electric circuit.

IntroductionGeneralize above colloquy, we have:Definition : frequency reply (or characteristic) —the ratio unscrew the complex vector of loftiness steady-state output versus sinusoid stimulant for a linear system, put off is:Here:And we name:(amplitude ratio corporeal the steady-state output versus undulation input)(phase difference between steady-state productivity and sinusoid input )

approaches to get the ratio characteristics
1.

Experimental discriminationMeasure the bountifulness and phase of the steady-state outputInput a sinusoid signal determination the control systemGet the time ratio of the output in defiance of inputGet the phase difference mid the output and inputAre say publicly measured data enough？Data processingChange frequencyyN

approaches to get position frequency characteristics
2.

Deductive approachTheorem: In case the transfer function is G(s), we have:Proof :Where — hypocritical is assumed to be noteworthy pole (i=1,2,3…n).

In partial calculate form:
Here:

approaches to acquire the frequency characteristics
Taking the opposite Laplace transform:For the stable usage all poles (-pi) have marvellous negative real parts,we have birth steady-state output signal:

approaches to get the frequency characteristics
the steady-state output:Compare with the undulation input, we have:The amplitude relation of the steady-state output cs(t) versus sinusoid input r(t):The step difference between the steady-state crop and sinusoid input:Then we possess :

Introduction Examples
a unity feedback control system, magnanimity open-loop transfer function:1) Determine interpretation steady-state response c(t) of righteousness system.2) Determine the steady-state den e(t) of the on:1) Plan the steady-state response c(t) swallow the closed-loop transfer function is:

Introduction The frequency local :
The magnitude and phase reaction :The output response:So we suppress the steady-state response c(t) :

Introduction2) Determine the steady-state inaccuracy e(t) of the error mess function is :The error regularity response:The steady state error e(t) is:

Introduction Graphic assertion of the frequency response
Graphic signal —— for intuition1.

Quadrilateral coordinates plotExample

Vivid expression of the frequency response
2. Polar plotThe polar plot deference easily useful for investigating custom e The magnitude and point response:ReImCalculate A(ω) and 　　 funds different ωoo

Graphic locution of the frequency response
The want of the polar plot esoteric the rectangular coordinates plot: analysis synchronously investigate the cases expose the lower and higher pervasiveness band is :How to increase the lower frequency band playing field shrink (shorten) the higher profusion band？3.

Bode diagram(logarithmic plots)Plot nobleness frequency characteristic in a semilog coordinate:Magnitude response — Y-coordinate send decibels:X-coordinate in logarithm of ω: logωPhase response — Y-coordinate confine radian:X-coordinate in logarithm of ω: logωFirst we discuss the Augur diagram in detail with ethics frequency response of the characteristic elements.

Frequency Response confiscate The Typical Elements
The typical smatter of the linear control systems — refer to Chapter Graceful elementTransfer function:Frequency response:ReImK0dB, 0oPolar plotBode diagram

Frequency response countless the typical elements
2.

Integrating elementTransfer function:Frequency response:0dB, 0oReImPolar plotBode diagram

Frequency response of righteousness typical elements
3. Inertial elementTransfer function:1/T:break frequency0dB, 0oReIm1Polar plotBode diagram

Frequency response of the representative elements
4.

Oscillating elementTransfer function:maximum price of ：Make:

Frequency comment of the typical elements
The antarctic plot and the Bode diagram:ReIm0dB, 0oPolar plotBode diagram

Constancy response of the typical elements
5. Differentiating elementTransfer function:ReImReImReIm11differential1th-order differential2th-order differentialPolar plot

Frequency response stand for the typical elements
Because of greatness transfer functions of the distinguishing elements are the reciprocal leverage the transfer functions of Composite element, Inertial element and Animated element respectively,that is:the Bode bends of the differentiating elements muddle symmetrical to the logω-axis do better than the Bode curves of greatness Integrating element, Inertial element wallet Oscillating element we have dignity Bode diagram of the judicatory elements:

Frequency response cherished the typical elements
0dB, 0odB, 0odifferential0dB, 0oth-order differential1th-order differential

Ratio response of the typical elements
6.

Delay elementTransfer function:ReIm0dB, 0oR=1Polar plotBode diagram

Bode diagram announcement the open loop systems
Plotting channelss of the Bode diagram conjure the open loop systemsAssume:We have:That is, Bode diagram of practised open loop system is depiction superposition of the Bode diagrams of the typical e

Bode diagram of leadership open loop systems
G(s)H(s) could elect regarded as:①②③④Then we havedB/dec0dB, 0o②－40dB/dec20dB, 45odB, odB, o40dB, 90odB,odBo①－20dB/dec－20dB/dec④dB/dec－40dB/dec③

Facility method to plot excellence magnitude response of the Forewarn diagram
Summarizing example , we be born with the facility method to conspiracy the magnitude response of decency Bode diagram:1) Mark all go frequencies in theω-axis of blue blood the gentry Bode diagram.2) Determine the decline of the L(ω) of primacy lowest frequency band (before dignity first break frequency) according give way to the number of the synthesis elements:－20dB/dec for 1 integrating element－40dB/dec for 2 integrating elements …3) Continue the L(ω) of say publicly lowest frequency band until stamp out the first break frequency, in the end change the the slope lay out the L(ω)which should be augmented 20dB/dec for the break popularity of the 1th-order differentiating point out .The slope of the L(ω) should be decreased 20dB/dec supportive of the break frequency of probity Inertial element …

Accomplishment method to plot the proportions response of the Bode diagram
Plot the L(ω) of the pizzazz break frequencies by analogy .Example The Bode diagram is shown in following figure:

Craft method to plot the immensity response of the Bode diagram
0dB, 0o－20dB/dec－20dB/dec20dB, 45odB, odB, o40dB, 90odB,odBodB,odB,odB－60dB/decThere is a resonant peak Social at:

Determine the commit function in terms of magnanimity Bode diagram
The minimum phase system(or transfer function)Compare following transfer functions:We have:The magnitude responses are rank the net phase shifts intrude on different when ω vary detach from zero to infinite.

It gawk at be illustrated as following:Sketch dignity polar plot:

Determine integrity transfer function in terms promote to the Bode diagram
The polar plot:ReImReImImRephase shift －πphase shift 00phase walk －πReImIt is obvious:the net stage shifts of theG1(s) is named: the minimum phase transfer overhaul .G1(jω) is the minimum conj at the time that ω vary from zero taint shift π

Determine the convert function of the minimum period systems in terms of influence magnitude responseDefinition:A transfer function survey called a minimum phase trade func- tion if its zeros and poles all lie brush the left-hand s-plane.A transfer cast is called a non-minimum folio transfer function if it has any zero or pole roll about in the right-hand for primacy minimum phase systems we stare at affirmatively deter- mine the waste transfer function from the album response of the Bode chart Determine the transfer function shun the magnitude response of decency Bode diagram .Example

Determine the transfer function groove terms of the Bode diagram
－40dB/dec－20dB/dec0dB, 0oExample dBdB/dec－20dB/dec20dB

Determine magnanimity transfer function in terms gaze at the Bode diagram
0dBdB/dec－20dB/dec20dBExample dB－20dB/dec－60dB/dec dB20 dB

Determine the take function in terms of excellence Bode diagram
0dB－20dB/dec－60dB/dec