《线性系统的根轨迹分析法 (7).pdf》由会员分享，可在线阅读，更多相关《线性系统的根轨迹分析法 (7).pdf（19页珍藏版）》请在taowenge.com淘文阁网|工程机械CAD图纸|机械工程制图|CAD装配图下载|SolidWorks_CaTia_CAD_UG_PROE_设计图分享下载上搜索。
1、1 Chapter 3 Time Domain Analysis of Automatic Control Systems 3.1 Typical Input Function and Time Domain Performance Indexes 3.2 Transient Response of A First-order System 3.3 Transient Response of A Second-order System 3.4 Analysis of high-order system 3.5 System Stability and Algebraic Stability C
2、riteria 3.6 Analysis of Steady-state Error 2 1 1 2 2 Definition of error and steady-state error System error analysis under the given input 3 3 System error analysis under the disturbance input 3.6 Analysis of Steady-state Error 4 4 Error analysis of compound control system 3 1 Definition of error a
3、nd steady-state error 3.6 Analysis of Steady-state Error Steady-state error of a control system is a measure of control precision,which is often referred to as steady-state performance.Steady-state error is an important technical index in control system.Only when the system is stable,it is meaningfu
4、l to study the steady-state error;for unstable systems,steady-state errors do not exist at all.If the step function is utilized,the system without steady-state error is usually called error-free system,while systems that exist error in the steady-state is called system with steady-state error.The pr
5、inciple steady-state error：the steady-state error caused by system structure,input action form and type.The structural steady-state error：the steady-state error of the system caused by nonlinear factors.4 System error：the difference between the desired value and the actual value ,i.e.,System deviati
6、on：the difference between the input signal and the main feedback signal of the system,i.e.,Steady-state error of the system：when t,the system error is called steady-state error,expressed by ,i.e.,the steady-state deviation of the system：when t,the stable value of the system deviation,expressed by ,i
7、.e.,U s()N s()Y s()G s2()G s1()-+E s()H s()B s()y t()0y t()ty ty t()()()0u t()b t()e tu tb t()()()ssttsslim()essee tsE stssslim()lim()03.6 Analysis of Steady-state Error 1 Definition of error and steady-state error 5 NOTE：1）Later in this course,error is the deviation of the system，steady-state error
8、 is actually the steady-state deviation of the system.2）Only in stable systems can steady-state errors be calculated.U s()N s()Y s()G s2()G s1()-+E s()H s()B s()E sU sB sH s Y sH s Y sH ss()()()()()()()()()0The relationship between the deviation signal and the error：3.6 Analysis of Steady-state Erro
9、r 1 Definition of error and steady-state error 6 ssK sK23(1 0.5)032E.g.The system structure diagram is shown in the figure,when the input signal is a unit ramp function,try to find the steady-state error of the system under the action of input signal,and is it possible to adjust the value of k to ma
10、ke the steady-state error less than 0.1?Solution:Calculating the steady-state error is only meaningful in a stable system,So stability should be judged first.The characteristic equation like this:K06According to the Routh criterion,we have:The transfer function like this：U sG s G s H ss ssKssE ss ss
11、E()1()()()(1)(21)(0.51)()()1(1)(21)12When the unit ramp is input,the system error signal can be expressed as follows：s ssKssE ss U ss ssE(1)(21)(0.51)()()()(1)(21)12According to the final value theorem,the steady-state error like this.KesE sssslim()10The value of K should be 06 to make it stable,i.e
12、.,So we cant adjust the value of k to make the steady-state error less than 0.1.ess613.6 Analysis of Steady-state Error 1 Definition of error and steady-state error(0.51)(1)(21)Kss ss()U s)(sY()E s7 2 Analysis of the system error under the given input G s G s H sE sU s1()()()()()12Does not consider
13、the influence of disturbance,let ,the system error can be expressed as follows：N s()0According to the final value theorem,the steady-state error can be expressed as follows：GG HG see tsE ssU ssU sktsssssr11()lim()lim()limlim()()12000In the formula is called the open-loop transfer function.G sGG Hk()
14、12U s()N s()Y s()G s2()G s1()-+E s()H s()B s()Obviously，is related to the input and the open-loop transfer function.essr3.6 Analysis of Steady-state Error 8 T sTsTsssG sG skksssjljlllnnkikikkkmm(1)(21)()()(1)(21)11201121212Suppose that the open-loop transfer function of the system can be expressed a
15、s G sk()Here：k is the open-loop amplification factor.v refers to the number of integral elements,it is called the type of the system or non-differential step number,too.After the open-loop transfer function is written in the form of time constant,is the remaining part after removing integral and pro
16、portional elements.mmmnnn2,21212G s()03.6 Analysis of Steady-state Error 2 Analysis of the system error under the given input 9 sG sG sG sKkesU ssskkpsssr1 lim()1()1 lim()1lim()1110000 When the input is a unit step function,that is sU s()1 Here：,kp is the position error coefficient of the system.Whe
17、n ，kKkGskespssr1lim(),100v0KG sspklim()0When ，sKGsekspssrlim(),000v1The value of reflects the steady-state accuracy of the system under unit step input.The larger the is,the smaller the is,therefore,reflects the ability of the system to tract the step input.KpessrKpKp3.6 Analysis of Steady-state Err
18、or 2 Analysis of the system error under the given input 10 sG sG ss G sKkesU ssskkvsssrlim()1()lim()lim()11101000 When the input is ,that is,the input is a unit ramp function.sU s()12Here：,Kv is called the velocity error constant of the system.When ，KskG sesvssrlim()0,00v0Ks G ssvklim()0 When ，sKG s
19、eksvssrlim(),000v2The value of reflects the steady-state accuracy of the system under ramp input,the larger the is,the smaller the is.So the reflects the system ability to track the ramp input.Kvessr When ，kKkGskesvssrlim(),100v1KvKv3.6 Analysis of Steady-state Error 2 Analysis of the system error u
20、nder the given input 11 2 sG sG ssG sKkesU ssskkasssrlim()1()lim()lim()111020002 When the input is a function of unit acceleration,that is,：sU s()13Here：,Ka is called the acceleration error coefficient.When ，KskG sesassrlim()0,00(1,2)v0,1Ks G ssaklim()02When ，sKG seksassrlim(),000v3The value of refl
21、ects the steady-state accuracy of the system under parabolic input,the larger the is,the smaller the is,therefore,reflects the ability of the system to track the acceleration input.KaessrWhen ，kKkG skesassrlim(),100v2KaKa3.6 Analysis of Steady-state Error 2 Analysis of the system error under the giv
22、en input 12 The steady-state error of the system under typical input：is the open-loop amplification coefficient(open-loop gain when the open-loop transfer function is written as a time constant).K3.6 Analysis of Steady-state Error 2 Analysis of the system error under the given input 13 Steady-state
23、error of the system under given action is related to external effects.The steady-state error is different when different input is added to the same system.The steady-state error of the system is related to the open-loop gain k in the form of time constant,for a systems with errors，when k increases,t
24、he steady-state error of the system will decrease;however,the stability and dynamic performance of the system will become worse at the same time.The steady-state error is also related to the number of integral elements,when the number of integral elements increases,the steady-state error will decrea
25、se,but the increased number of integral elements will lead to poor stability and dynamic performance of the system,so systems belong to type III and above are rarely used.Summary 3.6 Analysis of Steady-state Error 2 Analysis of the system error under the given input 14 3 System error analysis under
26、the disturbance input G s G s H sE sN sG s H s1()()()()()()()122We can ignore the influence of the given input,i.e.,the error of the system can be expressed as follows.U s()0According to the final value theorem,the steady-state error of the system can be expressed as follows：GG HG see tsE ssN ssN sG
27、 HG Hktsssssn11()lim()lim()lim()lim()1200022Here：refers to the open-loop transfer function.G sGG Hk()12U s()N s()Y s()G s2()G s1()-+E s()H s()B s()Obviously,the is related to the form of the disturbance input and open-loop transfer function,and it is also related to the .essnG H23.6 Analysis of Stea
28、dy-state Error 15 s TsG sk k kk(1)()123E.g.：Consider the following two systems.The transfer function of the two system are exactly the same.So for a given input,the steady-state errors of the two systems are the same;but for disturbance,due to the different action points of the disturbance,the trans
29、fer function of the forward disturbance channel is different,so its disturbance error is also different.U s()N s()Y s()k12kskTs31a()+-U s()N s()Y s()k12kskTs31b()+-E s()E s()3.6 Analysis of Steady-state Error 3 System error analysis under the disturbance input 16 GsesGsksssn1()lim0()03E.g.：Consider
30、the following two systems.If an integral element is added between the disturbance action point and the deviation point,we can reduce or eliminate the system steady-state error under the disturbance.When the steady-state error of the system under disturbance is considered,we can analyze it below.G sk
31、esN sG s G sksssn1()lim()1()()1023NOTE：In a system where the given input and the disturbance co-exist,the total steady-state error of the system should be equal to the sum of the given error and the disturbance error,the error point is defined on the same point.3.6 Analysis of Steady-state Error Let
32、 the N(s)=1/s and U(s)=0,figure a shows the error as 3 System error analysis under the disturbance input 17 4 Error analysis of compound control system Compound control system：If an additional control term related to a given action or disturbance action is introduced into the control system,it is ca
33、lled complex control,it can further reduce the given error and disturbance error.The feed-forward control system.U s()E s()Y s()G s1()G s2()(a)U s()E s()Y s()G s1()G s2()G s3()B s()(b)Add a element on the basis of figure(a),we constitute a feed-forward control system.G s3()The error of figure(a)can
34、be expressed as G s GsE sU s1()()()()112The error of figure(b)can be expressed as GGGGE sU sU sGGG GG G11()(1)()()11212122323When ,this is the condition that the feed-forward system is completely invariant to the given action.GsG s()()123E s()03.6 Analysis of Steady-state Error 18 The feedforward co
35、ntrol system.Let the system input ,because the system is a unit feedback system,the error caused by the disturbance like this U s()0Before feedforward element is added,the error of the system can be expressed as follows.E sY s()()After adding the feedforward element,the system error can be expressed
36、 as follows：GGE sY sN sG1()()()122 G s G sE sY sN sG sG s G s G s1()()()()()()()()()122123U s()N s()Y s()E s()G s3()G s1()G s2()When ,this is the condition under which feedforward is completely invariant to disturbance.G sG s()()113E s()03.6 Analysis of Steady-state Error 4 Error analysis of compoun
37、d control system 19 Chapter 3 Summary 3.1 Typical Input Function and Time Domain Performance Indexes 3.2 Transient Response of A First-order System 3.3 Transient Response of A Second-order System 3.4 Analysis of high-order system 3.5 System Stability and Algebraic Stability Criteria 3.6 Analysis of Steady-state Error