外文版自动控制理论课件.ppt

1、外文版自动控制理论 多媒体课件制作2022-12-61Automatic Control System 1.Automatic Control System 1.1 Introduction 1.2 An example 1.3 Types of control system 2.Mathematical Foundation 2.1 The transfer function concept 2.2 The block diagram.2.3 Signal flow graphs 2.4 Construction of signal flow graphs 2.5 General input

2、-output gain transfer function 2022-12-62Automatic Control System 3.Time-Domain Analysis Of Control System 3.1 Introduction 3.2 Typical test signals for time response of control systems 3.3 First Order Systems 3.4 Performance of a Second-Order System 3.5 Concept of Stability 4.The Root Locus Techniq

3、ues 4.1 Introduction 4.2 Root Locus Concept 2022-12-63Automatic Control System 4.3 The Root Locus Construction Procedure for General System 4.4 The zero-angle(negative)root locus 5.Frequency-Domain Analysis of Control System 5.1 Frequency Response 5.2 Bode Diagrams 5.3 Bode Stability Criteria 5.4 Th

4、e Nyquist Stability Criterion 2022-12-64Automatic Control System 6.Control system design 6.1 Introduction 6.2 Cascade Lead Compensation 6.3 Properties of the Cascade Lead Compensator 6.4 Parameter Design by the Root Locus Method 2022-12-651.Introduction 2022-12-661.1 Introduction In recent years,aut

5、omatic control systems have assumed an increasingly important role in the development and advancement of modern civilization and technology.Domestically,automatic controls in heating and air conditioning systems.Industrially,automatic control systems are found in numerous applications,such as qualit

6、y control of manufactured products,machine tool control,modern space technology and weapon systems.Even such problems as inventory control,social and economic systems control,and environment and hydrological systems control may be approached from the theory of automatic control.2022-12-671.2 An exam

7、ple The human being is a feedback control system,let us consider that the objective is to reach for an object on a desk,As one is reaching for the object,the brain sends out a signal to the arm to perform the task.The eyes serve as a sencing device which feed back continuously the position of the ha

8、nd.The distance between the hand and the object is the error,which is eventually brought to zero as the hand reaches the object.This is a typical example of closed-loop control.2022-12-681.3 Types of control system Control systems are classified in terms that described either the system itself or it

9、s variables.These descriptive terms are mainly of the either-or form.1 An open-loop system is shown in Fig1-1-1a and is characterized by the input entering directly into the control elements unaffected by the output;the output is related to the input solely by the characteristics of the plant and co

10、ntrol elements.In the closed-loop system of Fig.1-1-2b,however the input is modified by the actual output before entering the control elements.2022-12-69control elementsplantoutputinputFig1-1-1aControl elementsFeedbackelementsplantinputoutputFig1-1-1b2022-12-6102 Lumped parameter system are those fo

11、r which the physical characteristics are assumed to be concentrated in one or more“lumps”and thus independent of any special distribution.For example springs are massless and electrical leads resistanceless.3 A stationary or time-invariant system is one whose parameters do not vary with time.4 A sin

12、gle-variable system is defined as one with only one output for one reference or command input(SISO)5 A multivariable(MIMO)system has any number of inputs and outputs.6 A continuous-variable system is one for which all the system variables are continuous functions of time.2022-12-6112.Mathematical Fo

13、undation 2022-12-6122.1 The transfer function concept From the mathematical standpoint,algebraic and differential or difference equations can be used to describe the dynamic behavior of a system.In systems theory,the block diagram is often used to portray system of all types.For linear systems,trans

14、fer functions and signal flow graphs are valuable tools for analysis as well as for design If the input-output relationship of the linear system of Fig.1-2-1 is known,the characteristics of the system itself are also known.The transfer function of a system is the ratio of the transformed output to t

15、he transformed input.2022-12-613systeminputoutputaTF(s)inputoutputb)()()()()(srscsinputsoutputsTFFinger 1-2-1 input-output relationships(a)general(b)transfer function (2-1)2022-12-614Summarizing over the properties of a function we state:1.A transfer function is defined only for a linear system,and

16、strictly,only for time-invariant system.2.A transfer function between an input variable and output variable of a system is defined as the ratio of the Lap lace transform of the output to the input.3.All initial conditions of the system are assumed to zero.4.A transfer function is independent of inpu

17、t excitation.2022-12-6152.2 The block diagram.Figure 2-3-1 shows the block diagram of a linear feedback control system.The following terminology often used in control systems is defined with preference to the block diagram.R(s),r(t)=reference input.C(s),c(t)=output signal(controlled variable).B(s),b

18、(t)=feedback signal.E(s),e(t)=R(s)-C(s)=error signal.G(s)=C(s)/c(s)=open-loop transfer function or forward-path transfer function.M(s)=C(s)/R(s)=closed-loop transfer function H(s)=feedback-path transfer function.G(s)H(s)=loop transfer function.G(s)H(s)Fig2-2-12022-12-616The closed loop transfer func

19、tion can be expressed as a function of G(s)and H(s).From Fig.2-2-1we write:C(s)=G(s)c(s)(2-2)B(s)=H(s)C(s)(2-3)The actuating signal is writtenC(s)=R(s)-B(s)(2-4)Substituting Eq(2-4)into Eq(2-2)yieldsC(s)=G(s)R(s)-G(s)B(s)(2-5)Substituting Eq(2-3)into Eq(2-5)givesC(s)=G(s)R(s)-G(s)H(s)C(s)(2-6)Solvin

20、g C(s)from the last equation,the closed-loop transfer function of the system is given by M(s)=C(s)/R(s)=G(s)/(1+G(s)H(s)(2-7)2022-12-6172.3 Signal flow graphsFundamental of signal flow graphs A simple signal flow graph can be used to represent an algebraic relationIt is the relationship between node

21、 i to node with the transmission function A,(it is also represented by a branch).jijiXAX(2-8)2022-12-6182.3.1 Definitions Let us see the signal flow graphs2022-12-619Definition 1:A path is a Continuous,Unidirectional Succession of branches along which no node is passed more than once.For example,to

22、to to ,and back to and to to are paths.1X2X3X4X32,XX2X1X2X4XDefinition 2:An Input Node Or Source is a node with only outgoing branches.For example,is an input node.1XDefinition 3:An Output Node Or Sink is a node with only incoming branches.For example,is an output node.4X2022-12-620Definition 4:A Fo

23、rward Path is a path from the input node to the output node.For example,to to to and to to are forward paths.Definition 5:A Feedback Path or feedback loop is a path which originates and terminates on the same node.For example,to ,and back to is a feedback path.Definition 6:A Self-Loop is a feedback

24、loop consisting of a single branch.For example,is a self-loop.1X2X3X4X1X2X4X2X3X2X33A2022-12-621Definition 7:The Gain of a branch is the transmission function of that branch when the transmission function is a multiplicative operator.For example,is the gain of the self-loop if is a constant or trans

25、fer function.Definition 8:The Path Gain is the product of the branch gains encountered in traversing a path.For example,the path gain of the forward path from,to to to is Definition 9:The Loop Gain is the product of the branch gains of the loop.For example,the loop gain of the feedback loop from to

26、and back is 33A33A1X2X3X4X.2332AA2X2X3X2022-12-6222.4 Construction of signal flow graphs A signal flow graph is a graphical representation of a set of algebraic relationship,and it is a directed graph.The arrow represents the relationship between variables.In general,a variable can be represented by

27、 a node.Example:A typical feedback system.(In this case,a dummy node and a branch are added because the output node C has all outgoing branch).2022-12-6232022-12-624Example:Consider the following resistor network.There are five variables,.,21321iiVVVWe can write 4 linear equations:243322222332211111

28、111iRvvRvRiiRiRvvRvRii2022-12-625Let as input node,the output node can be found as follows:1v3v2022-12-626 2.5 General input-output gain transferfunction Let denote the ration between the input and the output.For the signal flow diagram representation,it becomes Definition 10:Non-touching two loops,

29、paths,or loop and path are said to be non-touching if they have no nodes in common.RCT lnXXT 2022-12-627iPjkPjjkkkP1)1(1 jjjjjjPPP3211=the ith forward path gain=jth possible product of k non-touching loop gainsDefinition 11:Signal Flow Graph Determinant(or characteristic function);is defined as foll

30、ows:2022-12-6281-(sum of all loop gains)+(sum of all gain-products of 2 non-touching loops)-(sum of all gain-products of 3 non-touching loops)+iiiiPRCTThe general formula for any signal flow graph is=evaluated with all loops touching Pi eliminated.2022-12-629Example:0321 PPGP,1,1,011kjPGHPjkThere is

31、 only one forward path;henceThere is only one(feedback)loop.Hence2022-12-630 GHP11111011GHGPRCT111thenandfinal2022-12-631Example:The signal flow graph of the resistance network,determine the voltage gain There is one forward path 13vvT2022-12-632Hence the forward path gain is 21431RRRRPThere are thr

32、ee feedback loops:Hence the loop gains are ,1311RRP,2321RRP2431RRP2022-12-63312P21433111RRRRPP 12312111)(1PPPP21432423131RRRRRRRRRR214332413121RRRRRRRRRRRRThere are no three loops that do not touch.ThereforeThere are two non-touching loops,loops one and three.HenceGain-Product of the only two non-to

33、uchingloops=2022-12-634 Since all loops touch the forward path,Finally,114332413121431113RRRRRRRRRRRRPvvT2022-12-635Problems1.A control system has the block diagram of Fig1.(a)Find the system transfer function c/r(b)Redraw the block diagram to show the control force u as the output and find the tran

34、sfer function u/r(c)Redraw the diagram with the actuating signal as the output and find the transfer function/r.ccGpGH-+rFig 1u2022-12-6363.Time-Domain Analysis Of Control System 2022-12-6373.1 Introduction Since time is used as an independent variable in most control systems,it is usually of intere

35、st to evaluate the time response of the systems.In the analysis problem,a reference input signal is applied to a system,and the performance of the system is evaluated by studying the response in the time domain.The time response of a control system is usually divided into two parts:The transient res

36、ponse and the steady-state response,if c(t)denotes a time response,then,in general,it may be written C(t)=ct(t)+css(t)Where ct(t)=transient response2022-12-6383.2 Typical test signals for time response of control systems Unlike many electrical circuits and communication systems,The input excitations

37、 to many practical control systems are not known ahead of time.In many cases,the actual inputs of a control system may vary in random fashions with respect to time.For the purpose of analysis and design,it is necessary to assume some basic types of input functions so that the performance of a system

38、 can be evaluated with respect to this signals.By selecting these basic test signals properly,not only the mathematical treatment of the problem is systematized,but the responses due to this inputs allow the prediction of the systems performance to other more complex inputs.In a design problem,perfo

39、rmance criteria may be specified with respect to these test signals so that a system may be designed to meet the criteria.To facilitate the time-domain analysis,the following deterministic test signal are often used.2022-12-639Test signalsr(t)R(s)PurposeImpulsestability testSteptransientresponse tes

40、tRamptrackingcapability testParabolicfast trackingcapability test ,00,0r tAttt A ,00,0r tAtt As ,00,0r tAttt 2As 2,00,0r tAttt32 As2022-12-6403.3 First Order Systems Unit-impulse response of the first-order system may be found by assuming,i.e.,the intensity of the impulse is equal to one,1111Y sR ss

41、s 1ty te (3-3-1)(3-3-2)2022-12-6412022-12-642Unit-step response of the first-order system may be found by assuming as 1R ss 1111111Y sR ssssss Unit-ramp response of the first-order system may be found by assuming as 21R ss 2221111111Y sR sssssss 1ty tte 1ty te (3-3-3)(3-3-4)(3-3-5)(3-3-6)2022-12-643

42、The error signal is then 1te tr ty te As approaches the infinity,the error signal approaches ,i.e.,t e (3-3-7)2022-12-6443.4 Performance of a Second-Order System Lets consider a unity feedback system shown below.The output can be found as Y s2022-12-645 222212nnnGsYsRsGsKRsspsKRsss nK 2pK where(3-3-

43、8)2022-12-646Unit-step response:Let .Then 1R ss 2222112nnnKY sspsKssss 11sin11sinnntntdy tetet ,21 1cos 01 21dnn (3-3-9)(3-3-10)(3-3-11)where2022-12-6472022-12-648Unit-impulse response:Let .Then 1Rs 2222nnnKYsspsKss nK 2pK sinntndy tet wherehence(3-3-12)(3-3-13)2022-12-6492022-12-650Standard perform

44、ance measures for the second-order system are defined by using unit-step response as shown below.2022-12-651Rise time:The rise time is the time required for the response to rise from 10%to 90%,5%to 95%,or 0%to 100%of its final value.For under-damped second-order systems,the 0%to 100%rise time is nor

45、mally used.For over-damped systems,the 10%to 90%rise time is commonly used.The 0%to 100%rise time may be calculated by using Eqn(3.13).rT1rTrT 11sin1n rTrdry TeT Solving for yields rTrdT On the other hand,it is difficult to obtain the exact analytic expressions for the 10%to 90%rise time .The follow

46、ing graph shows for .1rT1rT0.050.95 (3-3-14)(3-3-15)2022-12-652The linear approximation formula for effective for may be found as 1rT0.30.8 12.160.6rnT The swiftness(or fastness)of a response to a step input is dependent on both and .For a given ,the response is faster for a larger as n n(3-3-16)202

47、2-12-653For a given ,the response is faster for lower as shown below.n 2022-12-654Peak time:The peak time is the time required for the response to reach the first peak of the overshoot.We may obtain the peak time by differentiating given in Eqn.(3-3-13)and letting this derivative equal to zero,orpT

48、y tsin0nppTndptTdyeTdt This yields the following equation:sin0dpT Since the peak time corresponds to the first peak overshoot,we havepdT (3-3-17)(3-3-18)(3-3-19)2022-12-655Maximum overshoot:The maximum overshoot is the maximum peak value of the response curve.Substituting obtained above into Eqn.(3.

49、10),the may be found as tpMpTtpM211tpMePercent overshoot:Percent overshoot is the measure of the relative overshoot defined by.100%tpvvMfP Of where is the final value of the response.The amount of the maximum(percent)overshoot indicates the relative stability of the system.Hence,the percent overshoo

50、t may be given by vf21.100P Oe(3-3-20)(3-3-21)(3-3-22)2022-12-656Settling time:The settling time is the time required for the response curve to reach and stay within a range about the final value of size of absolute percentage of the final value(usually 2%or 5%).sT 44snTT33snTT (2%criterion)(5%crite